funqcionaluri sivrceebi leqcia ta krsi · li sivrcis cnebamde – tanamedrove matematikis ert-ert...

Teimuraz axobaZe

funqcionaluri sivrceebi

(saleqcio kursi)

2

maTematikuri analizis, geometriisa da wrfivi algebris cnebebis gan-

zogadebam usasrulo SemTxvevebisaTvis dasabami misca sxvadasxva zogad

Teorias, romelTa erTobliobasac “funqcionalur analizs” uwodeben.

gasuli saukunis pirvel naxevarSi Seiqmna zogad sivrceTa Teoria,

rogorebicaa: metrikuli, wrfivi normirebuli, topologiuri sivrceebi.

daiwyo wrfivi funqciebis (wrfivi funqcionalebisa da operatorebis)

Seswavla, romelTac SemdgomSi mravali gamoyeneba hpoves maTematikis

sxvadasxva dargebSi, fizikasa da teqnikaSi. miRebul iqna mTeli rigi Se-

degebi arawrfiv funqcionalur analizSic. Seswavlil iqna arawrfivi

asaxvebi, ganviTarda diferencialuri da integraluri aRricxva wrfiv

sivrceebSi, gamokvleul iqna arawrfivi funqcionalebis minimumisa da mi-

nimizaciis sakiTxebi. gasuli saukunis SuaxanebSi daiwyo wrfiv sivrce-

ebSi diferencialuri gantolebebis kvleva SemosazRvruli da Semousaz-

Rvreli operatorebisaTvis. kvlevisTvis gansakuTrebiT Zneli aRmoCnda

diferencialuri gantolebebi SemousazRvreli (wrfivi da arawrfivi)

operatorebisaTvis. es dargi karga xania intensiurad viTardeba.

Tanamedrove maTematika warmoudgenelia funqcionaluri analizis ga-

reSe. dReisaTvis funqcionaluri analizis ideebi, meTodebi, terminolo-

gia, aRniSvnebi da stili maTematikis TiTqmis yvela dargSi iWreba da

erT mTlianobad aerTianebT maT. funqcionaluri analizis meTodebs di-

di mniSvneloba aqvs Teoriuli dargebisaTvis, magram misi roli gamoye-

nebiTi dargebisaTvis sruliad gansakuTrebulia. TviT funqcionaluri

analizis warmoSoba ukavSirdeba sxvadasxva gamoyenebiTi sakiTxebis Ses-

wavlas.

funqcionaluri analizis warmoSobasa da ganviTarebaSi gansakuTrebu-

li roli miekuTvneba lebegis zomis Teorias, romelmac stimuli misca

mTeli maTematikis ganviTarebas me-20 saukuneSi. es Teoria gaxda maTema-

tikaSi kvlevebis erT-erTi mTavari iaraRi. am Toriis umniSvnelovanes

nawils warmoadgens lebegis integralis klasikuri Teoria.

Cveni mizani araa am Teoriebis Seviswavla mTeli sisruliT. Tu

gaviTvaliswinebT am disciplinebis Tanamedrove mdgomareobas, es SeuZ-

lebelicaa. Cven SevecdebiT daveufloT mxolod funqcionaluri anali-

zisa da lebegis zomis Teoriis safuZvlebs.

aRsaniSnavia, rom lebegis integralis SemoRebis sxvadasxva meTodi

arsebobs. Cven klasikur gzas avirCevT, anu SemoviRebT lebegis zomas,

SeviswavliT mis Tvisebebs da Semdgom am ukanasknelze dayrdnobiT

avagebT lebegis integrals. simartivisaTvis lebegis integralis Teori-

as ganvixilavT namdvil ricxvTa simravleze.

3

§1. helderisa da minkovskis utolobebi

lema 1 (helderis utoloba). vTqvaT a1, a2,…, an da b1, b2,…, bn (sazoga-

dod kompleqsur) ricxvTa n-eulebia, xolo p>1. maSin yoveli naturalu-

ri n ricxvisaTvis 1/ 1/

1 1 1,

p qn n np q

k k k kk k k

a b a b= = =

⎛ ⎞ ⎛ ⎞≤ ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ ∑ ∑ (1.1)

sadac

1 1 1.p q

+ = (1.2)

damtkiceba. ganvixiloT funqcia g, romlisTvisac g(x)=x1/p-x/p, 0≤x<+∞. ad-vili saCvenebelia, rom g′(x)=x-1/q/p-x/p. radgan g′(x)>0, roca x∈(0,1), xolo

g′(x)<0, roca x∈(1,+∞), amitom g funqciis udidesi mniSvneloba (1,+∞) inter-valze miiRweva x=1 wertilze. amis gamo yoveli x-Tvis x1/p- x/p≤ 1-1/p=1/q; anu x1/p≤x/p+1/q. Tu vigulisxmebT, rom

p qk kx u v −= (k=1,2,…,n), maSin ukanaskneli

Sefasebidan, miviRebT

/ 1 1 .q p p qk k k ku v u v

p q− −≤ +

aqedan

/ 1 1 .q q p p qk k k ku v u v

p q− ≤ +

radgan (1.2)–is ZaliT q-q/p=1, amitom ukanaskneli utolobebis ajamviT mi-viRebT

1 1 1

1 1 .n n n

p qk k k k

k k ku v u v

p q= = =

≤ +∑ ∑ ∑ (1.3)

aqedan martivad miiReba helderis utoloba. marTlac, vigulisxmoT, rom

1/

1

ik pn

pk

k

aua

=

=⎛ ⎞⎜ ⎟⎝ ⎠∑

, xolo 1/

1

ik qn

qk

k

bvb

=

=⎛ ⎞⎜ ⎟⎝ ⎠∑

,

maSin (1.3)–dan gveqneba 1/ 1/

1 1

p qn np q

k kk k

a b− −

= =

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑

1

1 1 1.n

k kk

a bp q=

≤ + =∑

es ki niSnavs, rom Sesrulebulia helderis (1.1) utoloba.

(1.1) utoloba im SemTxvevaSi, roca p=2, miiRebs saxes 2 2

1 1 1

.n n n

k k k kk k k

a b a b= = =

≤ ⋅∑ ∑ ∑

am ukanasknel Tanafardobas koSisa da Svarcis utoloba ewodeba. lema 2 (minkovskis utoloba). Tu a1, a2,…, an da b1, b2,…, bn raRac ricx-

vebia da p≥1, maSin 1/ 1/ 1/

1 1 1,

p p pn n np p p

k k k kk k k

a b a b= = =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ≤ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠∑ ∑ ∑ (1.4)

(1.4) utolobas minkovskis utoloba ewodeba.

damtkiceba. roca p=1, maSin (1.4) utolobis samarTlianoba cxadia. ami-

tom vigulisxmebT, rom p>1. advili Sesamowmebelia, rom

4

1 1

1 1 1.

n n np p p

k k k k k k k kk k k

a b a a b b a b− −

= = =

+ ≤ + + +∑ ∑ ∑ (1.5)

helderis utolobis Tanaxmad

( )1/ 1/

1 1

1 1 1

p qn n np p p q

k k k k k kk k k

a a b a a b− −

= = =

⎧ ⎫ ⎧ ⎫+ ≤ ⋅ +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

∑ ∑ ∑

da

( )1/ 1/

1 1

1 1 1

.p qn n n

p p p qk k k k k k

k k k

b a b b a b− −

= = =

⎧ ⎫ ⎧ ⎫+ ≤ ⋅ +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

∑ ∑ ∑

radgan (p-1)q=p, amitom (1.5)-dan davaskvniT 1/ 1/ 1/

1 1 1 1

.p p qn n n n

p p p pk k k k k k

k k k k

a b a b a b= = = =

⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎧ ⎫⎪ ⎪+ ≤ + +⎨ ⎬⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎩ ⎭⎪ ⎪⎩ ⎭

∑ ∑ ∑ ∑

Tu ukanaskneli Tanafardobis yvela wevrs gavamravlebT

1/

1

qnp

k kk

a b−

=

⎧ ⎫+⎨ ⎬⎩ ⎭∑

ricxvze da gaviTvaliswinebT (1.2) tolobas, miviRebT (1.4) Sefasebas.

lema 3 (helderis ganzogadebuli utoloba). vTqvaT (ai) da (bi) ricxv-

Ta raime mimdevrobebia, xolo p>1. maSin 1/ 1/

1 1 1

,p q

p qk k k k

k k k

a b a b∞ ∞ ∞

= = =

⎛ ⎞ ⎛ ⎞≤ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑ ∑

sadac q ricxvi ganisazRvreba (1.2) toloba.

lema 4 (minkovskis ganzogadebuli utoloba). Tu (ai) da (bi) ricxvTa

nebismieri mimdevrobebia, xolo p≥1. maSin 1/ 1/ 1/

1 1 1

.p p p

p p pk k k k

k k k

a b a b∞ ∞ ∞

= = =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ ≤ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠∑ ∑ ∑ (1.6)

lema 3-isa da 4-is samarTlianoba martivad gamomdinareobs, Sesabami-

sad, lema 1-dan da lema 2-dan (mkiTxvels vTxovT Seamowmos ukanaskneli

winadadebis samarTlianoba).

axla ganvixiloT helderisa da minkovskis integraluri analogebi.

lema 5 (helderis integraluri utoloba). vTqvaT f da g funqciebi integrebadia rimanis azriT [a,b] Sualedze. amasTan vigulisxmoT, rom p>1 raime ricxvia. maSin

1/ 1/

( ) ( ) ( ) ( ) ,p qb b b

p q

a a a

f x g x dx f x dx g x dx⎛ ⎞ ⎛ ⎞

≤ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫ ∫ ∫ (1.7)

sadac q ricxvi akmayofilebs (1.2) pirobas. damtkiceba. im SemTxvevaSi, roca n=1 utolobidan (1.3) miiReba Sefase-

ba

1 1 1 11 1 .p qu v u up q

≤ +

amitom yvela x-Tvis [a,b] segmentidan adgili eqneba utolobas

1/ 1/

( ) ( )

( ) ( )p qb b

p q

a a

f x g x

f x dx g x dx

⋅ ≤⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫ ∫

5

( ) ( )1 1 .( ) ( )

p q

b bp q

a a

f x g xp q

f x dx g x dx≤ ⋅ + ⋅

∫ ∫

Tu am utolobis orive mxares vaintegrebT [a,b] segmentze, da gaviTvalis-

winebT (1.2)-s, miviRebT 1/ 1/

( ) ( ) ( ) ( )p qb b b

p q

a a a

f x dx g x dx f x g x dx− −

⎛ ⎞ ⎛ ⎞⋅ ≤⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫ ∫ ∫

1 1 1.p q

+ =

lema 6 (minkovskis integraluri utoloba). vigulisxmoT, rom f da g funqciebi integrebadia rimanis azriT [a,b] segmenze. maSin nebismieri p ricxvisaTvis (p≥1) adgili aqvs Sefasebas

1/ 1/ 1/

( ) ( ) ( ) ( ) ,p p pb b b

p p p

a a a

f x g x dx f x dx g x dx⎧ ⎫ ⎛ ⎞ ⎛ ⎞

+ ≤ +⎨ ⎬ ⎜ ⎟ ⎜ ⎟⎩ ⎭ ⎝ ⎠ ⎝ ⎠∫ ∫ ∫ (1.8)

damtkiceba. (1.5) utoloba n=1 SemTxveveSi miiRebs saxes 1 1

1 1 1 1 1 1 1 1 .p p pa b a a b b a b− −+ ≤ + + +

aqedan gveqneba

( ) ( )b

p

a

f x g x dx+∫ ≤ 1 1( ) ( ) ( ) ( ) ( ) ( )b b

p p

a a

f x f x g x dx g x f x g x dx− −+ + +∫ ∫ .

Tu ukanaskneli utolobis marjvena mxaris orive SesakrebisaTvis gamovi-

yenebT helderis integraluri utolobas da CavatarebT igive msjelobas,

rac – lema 2-is mtkicebisas, advilad davaskvniT, rom

1( ) ( ) ( )b

p

a

f x f x g x dx−+∫ ≤1/

( )pb

p

a

f x dx⎛ ⎞

⋅⎜ ⎟⎝ ⎠∫

1/( 1)( ) ( )

qbq p

a

f x g x dx−⎧ ⎫+⎨ ⎬

⎩ ⎭∫ ,

1( ) ( ) ( )b

p

a

g x f x g x dx−+∫ ≤1/

( )pb

p

a

g x dx⎛ ⎞

⋅⎜ ⎟⎝ ⎠∫

1/( 1)( ) ( )

qbq p

a

f x g x dx−⎧ ⎫+⎨ ⎬

⎩ ⎭∫ .

radgan (p-1)q=p, amitom

( ) ( )b

p

a

f x g x dx+∫ ≤1/ 1/

( ) ( )p pb b

p p

a a

f x dx g x dx⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪+ ×⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭∫ ∫

1 1/

( ) ( )pb

p

a

f x g x dx−

⎧ ⎫× +⎨ ⎬

⎩ ⎭∫ .

aqedan miiReba (1.8) utoloba.

§2. metrikuli sivrcis cneba. magaliTebi

analizis erT-erTi mTavari operacia zRvarze gadasvlis moqmedebaa. am

operacias safuZvlad udevs is faqti, rom wrfeze gansazRvrulia manZi-

li amave wrfis erTi wertilidan meoremde. analizis mTeli rigi funda-

menturi faqti ar aris dakavSirebuli namdvili ricxvebis algebrul bu-

nebasTan (anu TvisebasTan, rom namdvil ricxvTa simravle qmnis vels),

aramed emyareba mxolod manZilis cnebas. ganvazogadebT, ra Cvens warmod-

genas namdvil ricxvebze, rogorc iseT simravleze, romelzedac SemoRe-

6

bulia elementebs Soris manZilis cneba, bunebrivad mivdivarT metriku-

li sivrcis cnebamde – Tanamedrove maTematikis erT-erT yvelaze mniS-

vnelovan cnebamde.

gansazRvreba. vTqvaT X raime simravlea. vigulisxmoT, rom ρ arauar-yofiTi funqciaa, romelic gansazRvrulia X-isa da X-is dekartul nam-

ravlze (e.i. simravleze X×X) da akmayofilebs Semdeg pirobebs:

1. ρ (x,y)=0 maSin da mxolod maSin, roca x=y. 2. ρ (x,y)=ρ (y,x) (simetriis aqsioma), 3. ρ (x,z)≤ρ (x,y)+ρ (y,z) (samkuTxedis aqsioma).

ρ funqcias ewodeba metrika simravleze, xolo X simravles – (X,ρ)-met-rikuli sivrce. Tu X×X simravleze ganvixilavT aRniSnuli Tvisebis mqone

sxva ρ1 funqcias, maSin miviRebT sazogadod gansxvavebul metrikul (X,ρ1)-

sivrces.

ganvixiloT metrikul sivrceTa magaliTebi. zogierTi maTgani analiz-

Si asrulebs Zalzed mniSvnelovan rols.

1. vTqvaT X raime simravlea. vigulisxmoT, rom nebismieri x,y∈X ele-

mentebisaTvis

ρ (x,y)=0,

1,x = y,x y.

⎧⎨ ≠⎩

Tu

TuU (2.1)

advili saCvenebelia, rom amgvarad gansazRvruli ρ funqcia akmayofi-lebs metrikis samive aqsiomas (gTxovT SeamowmoT). miRebul sivrces izo-

lirebul wertilTa sivrces (diskretuli sivrces) uwodeben, xolo (2.1) tolobiT gansazRvrul ρ funqcias – diskretul metrikas.

2. namdvil ricxvTa R simravle funqciiT ρ (x,y)=|x-y|, x,y∈R, qmnis metri-kul sivrces (gTxovT SeamowmoT).

3. dalagebul namdvil ricxvTa n–eulebis simravle funqciiT

ρ (x,y)=1/

1

,pn

pk k

k

x y=

⎛ ⎞−⎜ ⎟⎝ ⎠∑

sadac da x=(x1, x2,…, xn), y=(y1, y2,…, yn), xolo p (p ≥1) nebismieri fiqsirebuli

ricxvia, Seadgens metrikul sivrces.Aam sivrces Cven aRvniSnavT simbolo-

Ti npR –iT. cxadia, rom metrikuli sivrcis 1 da 2 aqsiomebi sruldeba. Se-

vamowmoT me-3 aqsiomis samarTlianoba. Tu p =1, maSin me-3 aqsioma srulde-

ba (ratom?).Namitom vixilavT SemTxvevas p>1. vTqvaT x=(x1, x2,…, xn), y=(y1, y2, …, yn) da z =(z1, z2,…, zn) n

pR sivrcis sami wertilia. davuSvaT yk-xk=ak, zk-yk=bk,

maSin minkovskis utolobis (ix. lema 2) ZaliT gveqneba

ρ (x,z)=1/

1

pnp

k kk

x z=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ ≤ ( )

1/

1

pn pk k k k

k

x y y z=

⎛ ⎞− + − ≤⎜ ⎟⎝ ⎠∑

≤1/

1

pnp

k kk

x y=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ +

1/

1

pnp

k kk

y z=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ =ρ (x,y)+ρ (y,z).

cxadia, 11R sivrce emTxveva namdvil ricxvTa R simravles.

4. lpsivrce (p≥1). lp sivrcis elementebad ganixileba namdvil (SesaZle-

belia kompleqsur) ricxvTa yvela iseTi mimdevroba x=(xi)=(x1, x2,…, xn,…),

7

romlisTvisacA1

.pk

k

x∞

=

< +∞∑ lpsivrcis nebismieri x da y elementebisaTvis

ρ (x,y) gainmarteba ase:

ρ (x,y)=1/

1

,p

pk k

k

x z∞

=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ p>1. (2.2)

es ganmarteba koreqtulia im TvalsazrisiT, rom lp sivrcis yoveli x= =(xi) da y=(yi) elementisaTvis ρ (x,y) sasrulia. marTlac, minkovskis ganzo-

gadebuli utolobis ZaliT (ix. lema 4) davaskvniT, rom 1/ 1/ 1/

1 1 1.

p p pp p p

k k k kk k k

x y x y∞ ∞ ∞

= = =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− ≤ + < +∞⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠∑ ∑ ∑

aseve minkovskis ganzogadebuli utolobis safuZvelze lp sivrcis nebismi-

eri sami x=(xi), y=(yi) da z=(zi) elementisaTvis miviRebT

ρ (x,z)= 1/

1

pp

k kk

x z∞

=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ ≤ ( )

1/

1

pp

k k k kk

x y y z∞

=

⎛ ⎞− + − ≤⎜ ⎟⎝ ⎠∑

≤1/

1

pp

k kk

x y∞

=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑ +

1/

1

pp

k kk

y z∞

=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑ =ρ (x,y)+ρ (y,z).

amrigad, Sesrulebulia metrikis ganmartebis me-3 aqsioma. rac Seexeba

pirvelsa da meore aqsiomas, maTi samarTlianoba cxadia.

5. m-iT aRiniSneba SemosazRvrul x=(xn) mimdevrobaTa sivrce, e.i. am

sivrcis yoveli elementi SemosazRvruli mimdevrobaa (ra Tqma unda,

sxvadasxva x=(xn) mimdevroba, sazogadod, sxvadasxva mudmiviTaa Semosaz-

Rvruli). SemoviRoT aRniSvna

ρ (x,y)= sup .k kk N

x y∈

− (2.3)

cxadia, nebismieri sami SemosazRvruli x=(xn), y=(yn) da z=(zn) mimdevrobis-Tvis

ρ (x,y)= ( )sup supk k k k k kk N k N

x y x z z y∈ ∈

− ≤ − + − ≤

≤ sup supk k k kk N k N

x z z y∈ ∈

− + − =ρ (x,z)+ ρ (z,y),

anu daculia metrikis gansazRvrebis me-3 aqsioma. pirveli da meore aqsi-

omis samarTlianobis Semowmeba araviTar siZneles ar warmoadgens.

6. c0 sivrce aris nulisaken krebad mimdevrobaTa sivrce. am sivrceSi

metrika (2.3) formuliT moicema.

7. yvela mimdevrobaTa sivrce Sedgeba ricxvTa (sazogadod, kompleq-

sur ricxvTa) nebismieri x=(xn), y=(yn), z=(zn),… mimdevrobebisagan. vTqvaT

ρ (x,y)=1

1 .2 1

k kk

k k k

x yx y

∞

=

−+ −∑ (2.4)

am funqciisaTvis pirveli da meore aqsiomis WeSmaritebis Cveneba ara-

viTar siZneles ar warmoadgens. mesame aqsiomis samarTlianobis saCveneb-

lad ganvixiloT funqcia f(t)=t/(t+1), t≥0. radgan f(t)=1-1/(t+1), amitom f funq-cia zrdadia Sualedze [0,+∞). amasTan radgan nebismieri a da b ricxvebi-saTvis |a+b|≤| a|+|b|, amitom

| |1 1 | |

a b a ba b a b+ +

≤ =+ + + + 1 | | 1 | |

a ba b a b

+ ≤+ + + +

8

≤ .1 | 1

a ba b

++ +

ukanaskneli Sefasebis gaTvaliswinebiT nebismieri xn, yn da zn ricxvebisa-

Tvis gveqneba

1n n n n

n n n n

x z z yx z z y− + −

≤+ − + −

.1 1

n n n n

n n n n

x z z yx z z y− −

++ − + −

maSasadame, (2.4)-is safuZvelze davaskvniT, rom sruldeba mesame aqsioma.

8. uwyvet funqciaTa sivrce C[a,b] Sedgeba [a,b] segmentze gansazRvruli

yvela uwyveti funqciisagan. aseTi x da y funqciebisaTvis davuSvaT

ρ (x,y)=max ( ) ( ) .a t b

x t y t≤ ≤

−

ρ metrikaa. marTlac, pirveli ori aqsiomis samarTlianoba cxadia. me-

ore mxriv, [a,b] segmentze mocemuli nebismieri uwyveti x da y funqciisa-Tvis

ρ (x,y)=max ( ) ( )a t b

x t y t≤ ≤

− ≤ ( )max ( ) ( ) ( ) ( )a t b

x t z t z t y t≤ ≤

− + − ≤

≤ max ( ) ( ) max ( ) ( )a t b a t b

x t z t z t y t≤ ≤ ≤ ≤

− + − = ρ (x,z)+ ρ (z,y).

9. Lp[a,b] sivrce (p≥1). [a,b] segmentze gansazRvruli nebismieri uwyveti x da y funqciisaTvis vigulisxmoT, rom

ρ (x,y)=1/

( ) ( ) .pb

p

a

x t y t dt⎧ ⎫

−⎨ ⎬⎩ ⎭∫ (2.5)

SevamowmoT, rom ρ funqciisaTvis daculia metrikis samive aqsioma.

cxadia, ρ (x,y)≥0; Tu x=y, anu Tu nebismieri t-Tvis [a,b] segmentidan x(t)= =y(t), maSin ρ (x,y)=0. Tu ρ (x,y)=0, maSin (2.5)–is ZaliT

( ) ( )b

p

a

x t y t dt−∫ =0.

vTqvaT, miuxedavad amisa, [a,b] segmentis raime t0 wertilze x(t0)≠y(t0). maSin x da y funqciebis t0 wertilze uwyvetobis gamo arsebobs am wertilis ise-

Ti u(t0) midamo, romlis yvela t wertilzec

|x(t)-y(t)|≥|x(t0)-y(t0)|/2. u(t0) midamos sigrZe aRvniSnoT δ–Ti. gveqneba

ρ (x,y)=1/

( ) ( )pb

p

a

x t y t dt⎧ ⎫

−⎨ ⎬⎩ ⎭∫ ≥

0

1/

( )

( ) ( )p

p

u t

x t y t dt⎧ ⎫⎪ ⎪− ≥⎨ ⎬⎪ ⎪⎩ ⎭

∫

≥0

1/

0 0( )

1 ( ) ( )2

p

p

u t

x t y t dt⎧ ⎫⎪ ⎪−⎨ ⎬⎪ ⎪⎩ ⎭

∫ = 1/0 0| ( ) ( ) |p x t y tδ − /2>0,

rac Cvens daSvebas ewinaaRmdegeba.

meore aqsiomis samarTlianoba naTelia. minkovskis integraluri uto-

lobidan (ix. (1.6)) martivad gamomdinareobs mesame aqsioma. marTlac, [a,b] segmentze nebismieri sami uwyveti x, y da z funqciisTvis gvaqvs

ρ (x,y)=1/

( ) ( )pb

p

a

x t y t dt⎧ ⎫

−⎨ ⎬⎩ ⎭∫ = ( ) ( )

1/

( ) ( ) ( ) ( )pb

p

a

x t z t z t y t dt⎧ ⎫

− + −⎨ ⎬⎩ ⎭∫ ≤

≤1/

( ) ( )pb

p

a

x t z t dt⎧ ⎫

−⎨ ⎬⎩ ⎭∫ +

1/

( ) ( )pb

p

a

z t y t dt⎧ ⎫

−⎨ ⎬⎩ ⎭∫ =ρ (x,z)+ ρ (z,y).

9

10. vTqvaT (X,ρ)-metrikuli sivrcea, xolo M⊂X. cxadia, M simravle

igive ρ funqciiT warmoadgens metrikul sivrces.Mmas ewodeba (X,ρ)-metri-kuli sivrcis qvesivrce. am gziT SeiZleba SemoRebul iqnas uamravi met-

rikuli sivrce.

§3. metrikuli sivrcis elementTa krebadi

mimdevrobebi da maTi Tvisebebi.

separabeluri metrikuli sivrceebi

metrikuli (X,ρ) sivrceSi Ria birTvi B(x0,r) ewodeba simravles

B(x0,r)={x: x∈X, ρ(x, x0)< r}. x0-s ewodeba B(x0,r) birTvis centri, xolo r ricxvs – radiusi. Caketili

birTvi B[x0,r] ase ganimarteba: B[x0,r]={x: x∈X, ρ(x, x0)≤ r}.

r-radiusiani sfero, centriT x0 wertilSi vuwodoT simravles

S[x0,r]={x: x∈X, ρ(x, x0)=r}. Tu birTvis an sferos centri ar aris miTiTebuli, maSin maT, saTana-

dod, aRvniSnavT simboloebiT B(r) (B[r]), S[r]. sazogadod, metrikul (X,ρ) sivrceSi SenarCunebulia namdvil ricxvTa

R simravlis Tvisebebi (rogorc cnobilia, am sivrceSi moqmedebs metrika

ρ(x, y)=|x-y|). magram SesaZlebelia metrikul sivrces gaaCndes „arabunebrivi“

Tvisebac. arsebobs metrikuli (X,ρ) sivrce da am sivrcis iseTi Caketili

birTvebi B1 (radiusiT r1) da B2 (radiusiT r2), rom B1⊂ B2, xolo r1> r2. marT-

lac, vTqvaT X aris sibrtyis yvela im (x1,x2) wertilTa erToblioba, rom–lebiც akmayofileben pirobas:

2 21 2 9x x+ ≤ (anu aris wre centriT (0,0) wert-

ilSi da radiusiT 3) da ganvixiloT metrikuli sivrce (X,ρ), sadac ρ ga-nimarteba Semdegnairad:

ρ (x,y)=1/ 22

2

1

;k kk

x y=

⎛ ⎞−⎜ ⎟⎝ ⎠∑

e.i. ρ „Cveulebrivi“ metrikaa sibrtyeze. vTqvaT B2= X, xolo

B1={ }2 21 2 1 2( , ) , ( 2) 16 .x x X x x∈ − + ≤

cxadia, r1=4 xolo r2=3; amasTan, Zneli ar aris imis Semowmeba, rom B1⊂B2

(SeamowmeT). Ria B(x0,ε) sferos radiusiT ε da centriT x0 wertilSi vuwodebT x0 wer-

tilis ε-midamos da mas xSirad aRvniSnavT simboloTi Oε(x0). simravles ewodeba SemosazRvruli, Tu igi aris romelime birTvis qve-

simravle.

(X,ρ) metrikuli sivrcis x wertils ewodeba amave sivrcis M qvesimrav-lis Sexebis wertili, Tu x–is nebismieri ε-midamo Seicavs M simravlis

erT wertils mainc. M–is yvela Sexebis wertilTa simravles ewodeba am

simravlis Caketva da aRiniSneba simboloTi M (mas xSirad aRniSnaven

simboloTic [M]). amrigad, Cven ganvmarteT Caketvis operacia: igi aris as-

axva M→[M].

10

Teorema 3.1. Caketvis operacias aqvs Semdegi Tvisebebi:

1) M⊂[M], 2) [[M]]= [M], 3) Tu M1⊂ M2, maSin [M1]⊂ [M2], 4) [M1∪M2]=[M1]∪[M2].

damtkiceba. Cven davamtkicebT 4)-s. 1)-3) winadadebebis samarTlianobis

Semowmebas mkiTxvels vandobT.

Tu x∈[M1∪M2], maSin x∈[M1] an x∈[M2]; winaaRmdeg SemTxvevaSi iarsebebs x wertilis iseTi midamo, romelic ar Seicavs arc M1-is da arc M2 simrav-

lis arcerT wertils (ratom?); e.i. [M1∪M2]⊂[M1]∪[M2]. Mmeore mxriv, radgan M1⊂M1∪M2 da M2⊂M1∪M2, amitom 3) Tvisebis ZaliT [M1]⊂[M1∪M2] da [M2]⊂ ⊂[M1∪M2]; amrigad, [M1]∪[M2]⊂ [M1∪M2] da 4) Tviseba damtkicebulia.

x wertils ewodeba M⊂ X simravlis zRvruli (dagrovebis) wertili,

Tu x–is nebismieri midamo Seicavs M simravlis usasrulod bevr wer-

tils. magaliTad, Tu M aris [0,1] segmentis racionalur wertilTa sim-

ravle, anu M=[0,1]∩Q, maSin M simravlis zRvrul wertilTa simravle iq-neba [0,1] segmenti. x∈M⊂ X wertils ewodeba M simravlis izolirebuli

wertili, Tu arsebobs iseTi dadebiTi ε ricxvi, rom midamo Oε(x) ar Sei-cavs x-gan gansxvavebul M simravlis arcerT wertils. mkiTxvels vTxovT

aCvenos, rom simravlis yoveli Sexebis wertili aris am simravlis

zRvruli an izolirebuli wertili.

vTqvaT x1, x2,…, xn,… (X,ρ) metrikuli sivrcis elementebia. amboben, rom

(xn) mimdevroba krebadia am sivrcis x wertilisken, Tu x-is nebismieri

Oε(x)-midamo Seicavs (xn) mimdevrobis yvela wevrs dawyebuli garkveuli in-

deqsidan, anu rac igivea, Tu lim ( , ) 0.nnx xρ

→∞= advili Sesamowmebelia, rom:

a) Tu mimdevrobas zRvari gaaCnia, maSin igi erTaderTia; b) Tu (xn) mimdev-

roba krebadia x wertilisken, maSin misi nebismieri ( )knx qvemimdevroba

krebadia igive zRvrisken.

Teorema 3.2. Tu (xn) da (yn) (X,ρ) metrikuli sivrcis fundamenturi

mimdevrobebia, maSin (ρ(xn,yn)) mimdevroba krebadia. damtkiceba. Tu orjer visargeblebT samkuTxedis aqsiomiT, maSin gveq-

neba

ρ(xn, yn)≤ ρ(xn, ym)+ ρ(ym, yn)≤ ρ(xn, xm)+ ρ(xm, ym)+ ρ(ym, yn). aqedan

ρ(xn, yn)−ρ(xm, ym)≤ρ(xn, xm)+ ρ(ym, yn). Tu ukanasknel utolobaSi n–sa da m–s adgilebs SevucvliT da gaviTva-

liswinebT metrikis simetriulobis Tvisebas, miviRebT

ρ(xm, ym)−ρ(xn, yn)≤ ρ(xm, xn)+ ρ(yn, ym)=ρ(xn, xm)+ ρ(ym, yn). ase rom,

|ρ(xn, yn)−ρ(xm, ym)|≤ρ(xn, xm)+ ρ(ym, yn). am Sefasebidan (X,ρ) metrikuli sivrceSi (xn) da (yn) mimdevrobebis funda-

menturobidan gamomdinareobs (ρ(xn, yn)) ricxviTi mimdevrobis fundamentu-

roba da, maSasadame, krebadoba.

Teorema 3.3. imisaTvis, rom x∈X iyos M⊂X simravlis Sexebis wertili,

aucilebelia da sakmarisi arsebobdes M simravlis wertilTa iseTi (xn)

mimdevroba, romelic krebadia x–ken.

11

damtkiceba. Teoremis piroba aucilebelia, radgan Tu x wertili M

simravlis Sexebis wertilia, maSin nebismieri naturaluri n-Tvis O1/n(x) midamo Seicavs M simravlis erT elements mainc, e.i. ρ(xn,x)<1/n, anu (xn) mim-

devroba krebadia x wertilisken.

vTqvaT axla moiZebna iseTi (xn) mimdevroba, romelic krebadia x∈X wer-tilisken. maSin x wertilis nebismieri midamo Seicavs (xn) mimdevrobis (da,

maSasadame, M simravlis) erT wertils mainc. ase rom, x aris M simravlis

Sexebis wertili.

(X,ρ) metrikuli sivrcis M qvesimravles ewodeba Caketili simravle,

Tu is Seicavs yvela mis Sexebis wertils, anu [M]= M. sxvagvarad rom vTqvaT, simravles ewodeba Caketili, Tu is Seicavs yvela mis Sexebis

wertils. Teorema 2.1-is 2) Tvisebis ZaliT simravlis Caketva Caketili

simravlea. Yyoveli [a,b] segmenti Caketili simravlea. Caketili birTvi

B[x0,r] Caketili simravlea (aCveneT). kerZod, C[a,b] sivrcis yvela f funq-ciaTa simravle, romlebic akmayofileben pirobas |f(t)|≤K (t∈[a,b], K raime arauaryofiTi ricxvia) Caketili simravlea. am SemTxvevaSi x0 igivurad 0-is toli funqcia, xolo radiusi r=K (ratom? gaixseneT manZilis cneba

C[a,b] sivrceSi). metrikuli sivrcis yoveli sasruli qvesimravle Caketi-

lia (ratom?).

Teorema 3.4. Tu {xα} Caketil simravleTa sistemaa, maSin am sistemis

simravleTa TanakveTa ∩ xα Caketili simravlea. Caketili simravleebis sa-

sruli gaerTianeba Caketili simravlea. (mkiTxvels vTxovT daamtkicos

ukanaskneli debuleba)

sazogadod, Caketil simravleTa usasrulo gaerTianeba ar aris Cake-

tili. magaliTad, ganvixiloT F= nn F∞=1∪ , sadac Fn=[-1/n,1-1/n]. es simravle

ar aris Caketili (ratom? ra saxe aqvs F simravles?) . (X,ρ) metrikuli sivrcis x wertils ewodeba amave sivrcis M simravlis

Siga wertili, Tu arsebobs am simravlis iseTi Oε(x)-midamo, romelic mo-

Tavsebulia M-Si (Oε(x)⊂ M). simravles ewodeba Ria, Tu igi mxolod Siga

wertilebisagan Sedgeba. magaliTad, namdvil ricxvTa R sivrcis (a,b) in-tervali Ria simravlea. marTlac, Tu x∈(a,b), e.i. a<x<b, maSin Oε(x) simrav-le, sadac ε=min{x-a, b-x} mTlianad moTavsdeba (a,b) intervalSi (ratom? ra

saxe aqvs Oε(x) simravles?).

metrikuli sivrcis Ria sfero B(x0,r) Ria simravlea (ratom?).

savarjiSo. vTqvaT g∈C[a,b]. ganvixiloT simravle B={f : f∈C[a,b] da f(t)<g(t), ∀t∈[a,b]}. aCveneT, rom B Ria simravlea C[a,b] sivrceSi.B

Teorema 3.5. imisaTvis, rom (X,ρ) metrikuli sivrcis M simravle iyos

Ria aucilebeli da sakmarisia, rom misi damateba X \ M iyos Caketili.

(mkiTxvels vTxovT daamtkicos ukanaskneli debuleba)

radgan (X,ρ) metrikuli sivrcis carieli simravle da X simravle Cake-

tili simravleebia (ratom?), amitom ukanaskneli Teoremis ZaliT carieli

simravle da X simravle Ria simravleebicaa.

Teorema 3.6. Ria simravleTa nebismieri sistemis gaerTianeba Ria sim-

ravlea. Ria simravleTa nebismieri sasruli sistemis TanakveTa Ria sim-

ravlea.

ukanaskneli debulebis mtkiceba emyareba Teoremebs 2.3, 2.4 da simrav-leTaTvis oradobis princips:

12

\ ( \ )X A X Aα αα α

=∪ ∩ , \ ( \ )X A X Aα αα α

=∩ ∪ .

sazogadod, Ria simravleTa TanakveTa ar aris Ria. magaliTad, SeiZle-

ba ganvixiloT simravle 1

,nnG∞

=∩ sadac Gn=(-1/n, 1+1/n).

vTqvaT M aris (X,ρ) metrikuli sivrcis raime qvesimravle. vigulisx-

moT, rom x∈X. ganmartebiT manZili x wertilidan M simravlemde vuwo-

doT sidides d=inf ( , )M

xτ

ρ τ∈

. aRsaniSnavia, rom, Tu M⊂X aris Caketili qvesim-

ravle da x∉M, maSin d= inf ( , )τ M

xρ τ∈

>0. marTlac, Tu d=inf ( , )M

xτ

ρ τ∈

=0, maSin infi-

mumis ganmartebis Tanaxmad nebismieri naturaliri n ricxvisaTvis arse-

bobs iseTi xn∈M wertili, rom ( , )nx xρ <1/n. es niSnavs, rom x wertili aris

M simravlis Sexebis wertili. radgan M Caketili simravlea, amitom x∈M. es ki daSvebas ewinaaRmdegeba.

vTqvaT A da B metrikuli sivrcis ori simravlea. A simravles ewodeba

mkvrivi B-Si, Tu [A]⊃B. A simravles ewodeba yvelgan mkvrivi (an rac igi-

vea mkvrivi) metrikul sivrceSi (X,ρ), Tu misi Caketva [A]=X. magaliTad,

racionalur ricxvTa simravle yvelgan mkvrivia namdvil ricxvTa R sim-ravlSi (ratom?).E A simravles ewodeba arsad mkvrivi metrikul sivrceSi

(X,ρ), Tu igi ar aris mkvrivi am sivrcis arcerT birTvSi, e.i. nebismieri

birTvi Seicavs iseT birTvs, romlis TanakveTa A simravlesTan carieli

simravlea. magaliTad, naturalur ricxvTa simravle arsad mkvrivia nam-

dvil ricxvTa simravleSi (aCveneT).

metrikul sivrces ewodeba separabeluri sivrce, Tu am sivrceSi arse-

bobs yvelgan mkvrivi Tvladi qvesimravle. qvemoT am TvalsazrisiT ganvi-

xilavT ramdenime mniSvnelovan magaliTs.

1. izolirebul wertilTa sivrce (diskretuli sivrce) separabeluria

maSin da mxolod maSin, roca es sivrce Sedgeba araumetes Tvladi rao-

denoba wertilebisagan. marTlac, Tu diskretuli sivrce Tvladia, maSin

masSi mkvriv Tvlad simravled SeiZleba ganvixiloT TviT am sivrcis

yvela wertilTa simravle. vTqvaT axla diskretul X sivrceSi arsebobs Tvladi yvelgan mkvrivi M simravle (M⊂X). es niSnavs, rom am sivrcis nebismieri x wertili aris M simravlis Sexebis wertili, e.i. 1/2-radius-ian birTvSi centriT x wertilSi unda moTavsdes M simravlis erTi ma-

inc y wertili; anu ρ (x,y)<1/2. amitom diskretuli metrikis ganmartebis Za-

liT (ix. §2, magaliTi 1) ρ (x,y)=0. amrigad, x=y, e.i. x∈M. miviReT, rom X⊂ M, niSnavs, rom M=X.

2. raconalur ricxvTa simravle Tvladia da igi mkvriviaAnamdvil

ricxvTa R simravleSi (ix. §2, magaliTi 2) (ratom?). amitom namdvil ricxv-

Ta R simravle separabeluria.

3. racionalur ricxvTa n–eulebi – r =(r1, r2, …, rn) yvelgan mkvriv simra-

vles Seadgenen npR sivrceSi. marTlac, ganvixiloT nebismieri x=(x1, x2,…,

xn) ∈ npR da ε>0. radgan namdvil ricxvTa simravleSi racionalur ricxvTa

simravle mkvrivia, amitom yoveli i–Tvis (i=1,2,…) iarsebebs iseTi ricxvi ( )

irε, rom

( ) 1// .pi ix r nε ε− < vTqvaT

( ) ( ) ( )1 2( , ,..., )nr r rε ε ε ε=( )r , maSin miRebuli utolo-

bebisa da npR sivrceSi metrikis ganmartebis ZaliT (ix. §2, magaliTi 3)

13

ρ (x, ε( )r )<ε. radgan racionalur ricxvTa n–eulebis simravle Tvladia,

amitom sivrce npR separabeluria.

4. lpsivrce (p≥1) separabeluria. davasaxeloT lp sivrcis raime x= (x1, x2, …, xn) elementi da ε>0 ricxvi. radgan x∈lp, amitom moiZebneba iseTi n0= =n0(ε, x) naturaluri ricxvi, rom

0 1

( / 2) .p pi

i n

x ε∞

= +

<∑ (3.1)

meore mxriv, racionalur ricxvTa simravlis namdvil ricxvTa simravle-

Si simkvrivis gamo yoveli i-Tvis (i=1,2,…,n0) arsebobs iseTi racionaluri

ricxvi ( )

irε , rom

0

( )

1

n p

i ii

x r ε

=

− <∑ ( / 2) .pε (3.2)

ganvixiloT 0,nε( )r =0

( ) ( ) ( )1 2( , ,..., ,0,0,...)nr r rε ε ε . cxadia, 0,nε( )r ∈lp. minkovskis ganzoga-

debuli utolobisa (ix. (1.6)) da (3.1), (3.2) Sefasebebis safuZvelze mivi-

RebT

0,( , )nερ =( )x r0

0

1/1/( )

1 1

ppn p pi i i

i i nx r xε

∞

= = +

⎧ ⎫⎧ ⎫ ⎪ ⎪− +⎨ ⎬ ⎨ ⎬⎪ ⎪⎩ ⎭ ⎩ ⎭

∑ ∑ <2ε +

2ε =ε.

es ki niSnavs, rom 0,nε( )r tipis obieqtTa simravle { 0,nε( )r } mkvrivia lp sivr-

ceSi. radgan simravle { 0,nε( )r } Tvladia (ratom?), amitom lp sivrcis separa-

beluroba damtkicebulia.

5. SemosazRvrul mimdevrobaTa sivrce m ar aris separabeluri. uka-

nasknelis saCveneblad ganvixiloT ricxviT ξ =(ξ1, ξ2,… ) mimdevrobaTa {ξ} simravle, sadac TiToeuli ξ mimdevrobis nebismieri ξi wevri aris 0 an 1. radgan [0,1] segmentsa da {ξ} simravles Soris SeiZleba bieqciuri Tana-

dobis damyareba (ratom?), amitom {ξ} aris kontinuumis simZlavris simrav-

le. amasTan Tu visargeblebT m sivrceSi manZilis cnebiT (ix. (2.3)), da-vaskvniT, rom am simravlis nebismier or gansxvavebul elements Soris

manZili 1-is tolia. axla ganvixiloT birTvebi, romelTa centrebia {ξ} simravlis ξ mimdevrobebi, xolo radiusi - 1/2. Ees birTvebi ar Tanaikve-

Tebian (ratom?). cxadia, aseT birTvTa simravle kontinuumis simZlavri-

saa. davuSvaT, rom m sivrceSi arsebobs yvelgan mkvrivi simravle M. ma-Sin TiToeuli ganxiluli birTvi unda Seicavdes M simravlis werti-

lebs. radgan birTvebi ar TanaikveTebian, amitom M simravlis simZlavre

ar SeiZleba iyos kontinuumze naklebi da, maSasadame, ar SeiZleba iyos

Tvladi.

6. c0 sivrce aris separabeluri sivrce. vTqvaT θ aris stacionarul

(sazogadod, kompleqsur) ricxviT mimdevrobaTa simravle (rogorc cno-

bilia stacionaruli ewodeba mimdevrobas, Tu misi wevrebi 0-is tolia

garkveuli indeqsidan dawyebuli). amasTan vigulisxmoT, rom θ simravlis

TiToeuli mimdevrobis yoveli wevris namdvili da warmosaxviTi nawile-

bis koeficientebi racionaluri ricxvebia. θ simravle Tvladia (ratom?).

aviRoT c0 sivrcis (ix. §2, magaliTi 6) nebismieri x= (x1, x2,…, xn,…) element-

isa da ε>0 ricxvisTvis movZebnoT iseTi k0 ricxvi, rom |xk|<ε. meore mxriv,

14

movZebnoT θ simravlis iseTi elementi r(ε)= ( ) ( )1 2( , ,...,r rε ε

0

( ) , 0,0,...)krε , rom |xi - ( )

irε|

<ε, i=1,2,…,k0. es ki niSnavs, rom

( , )ερ =( )x r max{ }0 01

max ,maxi i ik k k kx r xε

≤ ≤ >− <ε.

maSasadame, c0 sivrce separabeluria.

7. yvela mimdevrobaTa s sivrce separabeluria. vaCvenoT, rom wina punq-

tSi ganxiluli θ simravle mkvrivia yvela mimdevrobaTa sivrceSi (ix. §2, magaliTi 7). iseve rogorc wina SemTxvevaSi, aviRoT ε>0 ricxvi da Sevar-CioT naturaluri k0 imgvarad, rom 0 11/ 2 .k ε− < ganvixiloT nebismieri mim-

devroba x= (x1, x2,…, xn,…). cxadia, moiZebneba iseTi ( )

irε (i=1,2,…,k0) ricxvebi

(sazogadod, kompleqsuri) racionaluri namdvili da warmosaxviTi nawi-

lebis koeficientebiT, rom |xi- ( )ir

ε|<ε. axla vTqvaT r(ε)= ( ) ( )

1 2( , ,...,r rε ε 0

( ) , 0,krε

0,...) ∈θ. maSin

( , )ερ =( )x r0

( )

( )1

12 1

ki i

kk i i

x r

x r

ε

ε=

−

+ −∑ +

0 1

12 1

ik

k k i

xx

∞

= +

≤+∑

0

01

1 1 .2 2 2 2 2

k

kkk

ε ε ε=

≤ ⋅ + < +∑

amrigad, ganxiluli sivrce separabeluria.

8. C[a,b] sivrce separabeluria. masSi mkvriv simravles Seadgens raci-

onalurkoeficientebian mravalwevrTa simravle. es simravle aRvniSnoT

M-iT. M simravlis TiToeuli elementi, cxadia, ganisazRvreba raciona-

lur ricxvTa sasruli dalagebuli sistemiT. amitom M simravle Tvla-

dia. vTqvaT f∈C[a,b]. maSin vaierStrasis Teoremis Tanaxmad nebismieri ε>0 ricxvisaTvis arsebobs iseTi mravalwevri Pn (ε), rom yoveli x∈[a,b] werti-lisaTvis

|Pn(ε)(x)-f(x)|< ε/2. (3.3) vTqvaT Pn(ε)(x)=a0(ε)+a1(ε)x+…+ an(ε)xn(ε) da SevarCioT iseTi racionaluri ric-

xvebi ri(ε) (i=1,2,…, n(ε)), rom |ai(ε)-ri(ε)|<ε/2δ, sadac δ= ( )( )

0max{| |,| |} .

nk

ka b

ε

=∑ Sevadgi-

noT mravalwevri ( ) ( ) ( ) ( )

( ) 0 1( ) ... nn nP x r r x r xε ε ε ε

ε = + + + . ( )nP ε ∈M. nebismieri x-Tvis

[a,b] segmentidan

( )( ) ( )

( )( )0 0

( ) ( ) ( ) ( ) ( ) ( ) max{| |,| |}n n

k knn k k k k

k k

P x P x a r x a r a bε ε

εε ε ε ε ε= =

− ≤ − ≤ − ≤∑ ∑

≤2εδ

( )( )

0max{| |,| |}

nk

ka b

ε

=∑ =ε/2.

Aamrigad, (3.3) utolobis ZaliT

AAAAAAAAAAA| f(x)- ( ) ( )nP xε |≤ ( )( ) ( )nf x P xε− + ( ) ( )( ) ( )n nP x P xε ε− <ε/2+ε/2=ε. (3.4)

AA9. Lp[a,b] sivrce (p≥1) separabeluria. marTlac, wina punqtSi nebismieri

ε>0 ricxvisaTvis avageT iseTi racinalurkoeficientebiani mravalwevri

( )nP ε , rom yoveli x-Tvis [a,b] segmentidan adgili aqvs (3.3) Sefasebas. Lp[a,b] sivrceSi metrikis ganmartebis ZaliT (ix. (2.5))

1/

( ) ( )( , ) ( )pb p

n n

a

f P f t P dtε ερ⎧ ⎫

= −⎨ ⎬⎩ ⎭∫ <ε(b-a). (3.5)

15

radgan racinalurkoeficientebian mravalwevrTa simravle Tvladia, ami-

tom (3.5) Sefasebidan davaskvniT, rom Lp[a,b] sivrce separabeluria.

10. metrikuli sivrcis qvesivrcis separabelurobidan, sazogadod, ar

gamomdinareobs TviT am sivrcis separabeluroba. MmagaliTad, c0 sivrce

separabeluria, xolo m sivrce ki ara. Teorema 3.7. separabeluri metrikuli sivrcis qvesivrce separabelu-

ria.

damtkiceba. vTqvaT (X,ρ) metrikuli sivrcis Qqvesivrcea (X′,ρ). amocana mdgomareobs SemdegSi. (X′,ρ) sivrceSi unda moiZebnos Tvladi da am qve-

sivrceSi mkvrivi simravle. radgan (X,ρ) sivrce separabeluria, amitom

masSi arsebobs Tvladi mkvrivi simravle. vTqvaT es simravlea {ξ1, ξ2,…, ξn,…}. davuSvaT x∈X′⊂ X. maSin nebismieri ε>0 ricxvisaTvis moiZebneba iseTi

ξn(ε), rom ρ(ξn(ε),x)<ε/3. SemoviRoT aRniSvna an(ε)= ( )inf ( , )ny Xyερ ξ

′∈. inf-is ganmartebis

ZaliT nebismieri naturaluri m ricxvisaTvis arsebobs iseTi yn(ε),m∈X′, rom ρ(ξn(ε),yn(ε),m)<an(ε)+1/m. m SevarCioT imgvarad, rom 1/m<ε/3. amis garda an(ε)= = ( )inf ( , )ny X

yερ ξ′∈

≤ ( )( , )n xερ ξ <ε/3. amitom ρ(ξn(ε), yn(ε),m)<ε/3+ε/3=2ε/3. amrigad, ρ(x, yn(ε),m)

≤ρ(x, ξn(ε))+ρ(ξn(ε), yn(ε),m)< ε/3+2ε/3=ε. maSasadame, X′-is nebismieri x wertili aris

Sexebis wertili X′-dan aRebuli {yn,m} wertilTa simravlisa. radgan {yn,m}

simravle Tvladia, amitom (X′,ρ) qvesivrcis separabeluroba damtkicebu-

lia.

§4. sruli metrikuli sivrceebi

vityviT, rom (xn) mimdevroba aris metrikuli (X,ρ) sivrcis mimdevroba,

Tu am mimdevrobis yoveli wevri xn∈X. metrikuli (X,ρ) sivrcis (xn) mimdevrobas ewodeba fundamenturi mimdev-

roba, Tu nebismieri ε>0 ricxvisTvis moiZebneba iseTi N(ε) ricxvi, rom

roca m da n naturaluri ricxvebisaTvis m,n > N(ε), gvaqvs ρ(xm, xn)<ε. martivia imis Cveneba, rom metrikuli sivrcis yoveli fundamenturi

mimdevroba SemosazRvrulia (aCveneT). aseve advili Sesamowmebelia, rom

Tu metrikuli sivrcis fundamenturi mimdevrobis raime qvemimdevroba

krebadia, maSin TviT es mimdevrobac krebadia (SeamowmeT).

metrikul sivrceSi yoveli krebadi mimdevroba aris fundamenturi.

marTlac, vTqvaT (X,ρ) metrikuli sivrcis (xn) mimdevrobisaTvis lim nnx x

→∞= ∈

∈X; e.i. nebismieri ε>0 ricxvisaTvis arsebobs iseTi N(ε) ricxvi, rom Tu

n>N(ε), maSin ρ(xn, x)<ε/2. aseve, Tu m>N(ε), maSin ρ(xm, x)<ε/2. amitom samkuTxe-

dis aqsiomis ZaliT, Tu m,n > N(ε), maSin ρ(xm, xn)≤ ρ(xm, x)+ ρ(x, xn)<ε/2+ε/2=ε,

rac (xn) mimdevrobis fundamenturobas niSnavs.

rogorc cnobilia, namdvil ricxvTa R simravles aqvs Tviseba: am

sivrceSi nebismieri fundamenturi mimdevroba krebadia. es Tviseba ar ga-

aCnia nebismier metrikul sivrces. magaliTad, ganvixiloT simravle R\{0} namdvil ricxvTa R simravlis metrikiT (ix. §2, magaliTi 2). mimdevroba

16

(1/n) fundamenturia R\{0}sivrceSi, anu nebismieri ε>0 ricxvisaTvis moiZeb-

neba iseTi N(ε) ricxvi, rom Tu m da n naturaluri ricxvebi metia N(ε) ricxvze, maSin |1/n-1/m|<ε. miuxedavad amisa R\{0}sivrceSi ar arsebobs iseTi

obieqti (ricxvi), romliskenac krebadi iqneba (1/n) mimdevroba (aseTi obi-

eqti rom arsebobdes, igi unda iyos 0-is toli, magram 0∉ R\{0}). Tu (X,ρ) metrikuli sivrce iseTia, rom am sivrcis yoveli fundamentu-

ri mimdevroba krebadia, maSin aseT metrikul sivrces sruli sivrce ewo-

deba. ganvixiloT §2-Si mocemuli metrikuli sivrcis magaliTebi sisrulis

TvalsazrisiT.

1. izolirebul wertilTa sivrce aris sruli sivrce. vTqvaT (xn) disk-

retuli sivrcis fundamenturi mimdevrobaa. amitom ε=1/2 ricxvisaTvis ar-

sebobs iseTi n0≡N(1/2) ricxvi, rom Tu n,m>n0, maSin ρ(xn, xm)<1/2; anu am sivr-ceSi metrikis ganmartebis ZaliT ρ(xn, xm)=0, e.i. xn=xm, Tu n,m>n0. es ki niS-navs, rom (xn) mimdevroba stacionaruli mimdevrobaa da, maSasadame, - kre-

badi.

2. namdvil ricxvTa R simravle srulia. radgan namdvil ricxvTa sim-

ravleSi adgili aqvs ricxviTi mimdevrobis koSis kriteriums, amitom

ukanaskneli winadadebis samarTlianoba eWvs ar iwvevs.

3. npR sivrce srulia. am sivrceSi ganvixiloT

( ) ( ) ( )1 2( , ,..., )m m m m

nx x x=( )x fun-

damenturi mimdevroba. es ki niSnavs yoveli i-Tvis (i=1,2,...,m) mimdevroba ( )( )m

ix aris fundamenturi ricxviTi mimdevroba (ratom?), maSasadame, - kre-

badi. vTqvaT ( ) (0)lim .mi im

x x→∞

= SemoviRoT aRniSvna (0) (0) (0) (0)

1 2( , ,..., )nx x x x= . gveqneba

( ) (0)lim m

mx x

→∞= (ratom?).

4. lpsivrce (p≥1) srulia. vTqvaT (x(n)), sadac ( ) ( ) ( )1 2( , ,..., ,...)n n n n

mx x x=( )x aris lp

sivrcis elementTa fundamenturi mimdevroba, anu yoveli ε>0 ricxvisa-Tvis moiZebneba iseTi N(ε) ricxvi, rom Tu m da n naturaluri ricxvebi

metia N(ε) ricxvze, maSin

ρ(x(n), x(m))=1/

( ) ( )

1

ppn m

k kk

x x∞

=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑ <ε/2. (4.1)

ukanasknelidan nebismieri naturaluri k-Tvis miviRebT ( ) ( )n mk kx x− <ε ; anu

yoveli fiqsirebuli naturaluri k ricxvisTvis ( )( )nkx ricxviTi mimdevro-

ba aris krebadi. vTqvaT ( ) (0)lim .nk kn

x x→∞

= ganvixiloT (0) (0) (0) (0)

1 2( , ,..., ,...)mx x x=x mim-

devroba. igi lp sivrcis elementia. marTlac, (4.1) Sefasebidan yoveli na-

turaluri r ricxvisTvis 1/

( ) ( )

1

pr pn mk k

k

x x=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ ≤

1/( ) ( )

1

ppn m

k kk

x x∞

=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ <ε/2.

Tu gadavalT zRvarze, roca m→∞, miviRebT 1/

( ) (0)

1

pr pnk k

kx x

=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑ ≤ε/2.

aqedan gveqneba

171/

( ) (0)

1

ppn

k kk

x x∞

=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ ≤ε/2<ε.

(4.2) minkovskis ganzogadoebuli utolobis ZaliT

1/(0)

1

pp

kk

x∞

=

⎛ ⎞≤⎜ ⎟

⎝ ⎠∑

1/(0) ( )

1

ppn

k kk

x x∞

=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑ +

1/( )

1

ppn

kk

x∞

=

⎛ ⎞<⎜ ⎟

⎝ ⎠∑ ε +

1/( )

1

.p

pnk

kx

∞

=

⎛ ⎞⎜ ⎟⎝ ⎠∑

radgan yoveli n-Tvis x(n)∈lp, amitom ukanasknelidan miviRebT 1/

(0)

1

;p

p

kk

x∞

=

⎛ ⎞ < +∞⎜ ⎟⎝ ⎠∑

e.i. (0)x ∈ lp. axla, (4.2)-dan davaskvniT, rom lp sruli sivrcea.

5. m sivrce srulia. aviRoT m sivrceSi nebismieri fundamenturi

mimdevroba (x(n)), sadac ( ) ( ) ( )1 2( , ,..., ,...)n n n n

mx x x=( )x ; anu nebismieri ε>0 ricxvisa-Tvis moiZebneba iseTi N(ε) ricxvi, rom roca m, n> N(ε), maSin

ρ(x(n), x(m))= ( ) ( )sup n mk k

kx x− <ε/2. (4.3)

aqedan yoveli fiqsirebuli k-Tvis ( ) ( )n mk kx x− <ε/2. es ki niSnavs, rom ricxvi-

Ti mimdevroba ( )( )nkx aris krebadi. vigulisxmoT, rom

( ) (0)lim nk kn

x x→∞

= da ganvi-

xiloT mimdevroba (0) =x (0) (0) (0)

1 2( , ,..., ,...)mx x x . jer vaCvenoT, rom es mimdevroba

SemosazRvrulia, anu ekuTvnis m sivrces. davasaxeloT naturaluri ric-

xvebi m0>N(ε) da k. radgan 0( )m m∈x , amitom es mimdevroba SemosazRvrulia

raRac M(m0) mudmiviT. meore mxriv, (4.3)-dan nebismieri m naturaluri ric-

xvisTvis gvaqvs

0 0( ) ( )( ) ( ) m mm mk k k kx x x x≤ − + ≤ 0( )( )sup mm

k kk

x x− + 0( )sup mk

kx <ε/2+ M(m0).

Tu am utolobaSi gadavalT zRvarze roca m→∞, miviRebT (0)kx ≤ ε/2+ M(m0),

rac ( )( )0kx mimdevrobis SemosazRvrulobas niSnavs, e.i.

(0) .m∈x radgan m da

n naturaluri ricxvebisTvis ( m, n> N(ε)) ( ) ( )n mk kx x− ≤ ( ) ( )sup n m

k kk

x x− <ε/2,

amitom ukanasknel utolobaSi zRvarze gadasvlis Sedegad gveqneba ( ) (0)nk kx x− ≤ε/2,

anu

ρ(x(n), x(0))=supk

( ) (0)nk kx x− ≤ε/2<ε,

roca m, n>N(ε). 6. c0 sivrce aris sruli sivrce. am winadadebis samarTlianobis dam-

tkiceba wina punqtis msgavsad xdeba (SeamowmeT).

7. yvela mimdevrobaTa s sivrce srulia. ganvixiloT s sivrcis funda-menturi mimdevroba

( ) ( ) ( )1 2( , ,..., ,...)n n n n

mx x x=( )x . am ukanasknelis gansazRvris Ta-

naxmad nebismieri ε>0 ricxvisaTvis moiZebneba iseTi N(ε) ricxvi, rom ro-ca m, n> N(ε), maSin

18

ρ(x(n), x(m))= ( ) ( )

1

12 1

n mk k

kk k k

x xx y

∞

=

−

+ −∑ <2ε ;

miTumetes, yoveli fiqsirebuli naturaluri k0-Tvis

0 0

0 0

( ) ( )

( ) ( )1

n mk k

n mk k

x x

x x

−

+ −< 0 12k ε− . (4.4)

SeiZleba vigulisxmoT, rom 02 .kε −< maSin (4.4) Sefasebidan miviRebT

0 0

( ) ( )n mk kx x− <

00

0

1

1

2 2 ,1 2

kk

k

ε εε

−

− <−

roca m, n> N(ε). maSasadame, ( )0

( )nkx ricxviTi mimdevroba fundamenturia, e.i.

aris krebadic. vTqvaT 0 0

( )lim .nk kn

x x→∞

= SevadginoT mimdevroba 1 2( , ,...)x x=x . va-

CvenoT, rom s sivrceSi ganxiluli ( )n( )x mimdevroba krebadia x-ken. davasa-

xeloT ε>0 ricxvi da movZebnoT iseTi naturaluri r0, rom 01/ 2 / 2.r ε< ra-

dgan ( )( )nkx mimdevroba yoveli fiqsirebuli k-Tvis krebadia xk-ken, amitom

sakmarisad didi n-Tvis SegviZlia vigulisxmoT, rom

0( )

( )1

12 1

nrk k

k nk k k

x x

x x=

−

+ −∑ <2ε.

maSin

ρ(x(n), x)< 0

( )

( )1

12 1

nrk k

k nk k k

x x

x y=

−

+ −∑ +

0 1

12k

k r

∞

= +∑ <

2ε+

2ε=ε.

8. C[a,b] sivrce srulia. vTqvaT (xn) uwyvet funqciaTa mimdevrobaa

C[a,b] sivrceSi, e.i. yoveli ε>0 ricxvisaTvis arsebobs iseTi N(ε) ricxvi, rom roca m, n> N(ε), maSin

ρ(xn, xm)=[ , ]

maxt a b∈

|xn(t)- xm(t)|< ε/2. (4.5)

kerZod, yoveli fiqsirebuli t-Tvis [a,b] segmentidan |xn(t)- xm(t)|< ε/2, (4.6)

roca m, n> N(ε). es niSnavs, rom fiqsirebuli t-Tvis ricxviTi mimdevroba (xn(t)) aris fundamenturi. ricxviTi mimdevrobis koSis kriteriumis ZaliT

(xn(t)) mimdevroba krebadia. cxadia, misi zRvari damokidebulia t-ze. aRvni-SnoT igi x(t)-Ti. TuU(4.6) utolobaSi gadavalT zRvarze, roca m→∞, maSin

miviRebT

|xn(t)- x(t)|≤ ε/2, roca n> N(ε). aqedan davaskvniT, rom

[ , ]maxt a b∈

|xn(t)- x (t)|<ε. (4.6)

es ki niSnavs, rom uwyvet funqciaTa (xn) mimdevroba Tanabrad krebadia x funqciisaken. amitom x uwyveti funqciaa [a,b] segmentze (x∈C[a,b]), xolo

(4.6) Sefaseba niSnavs C[a,b] sivrceSi (xn) mimdevrobis krebadobas.

9. Lp[a,b] sivrce (p≥1) ar aris sruli sivrce. am winadadebis damtkice-

bis simartivisaTvis vigulisxmoT, rom a= -1 da b=1. vaCvenoT, rom am sivr-

ceSi arsebobs uwyvet funqciaTa fundamenturi mimdevroba [-1,1] segmentze, romelic ar aris krebadi am sivrceSi (am sivrcis arcerTi elementisken).

yoveli naturaluri n–Tvis vigulisxmoT, rom

19

1, [1 ,1],( ) , [-1 ,1 ],

1,n

x /nx t nt x /n /n

x n

∈⎧⎪= ∈⎨⎪− ∈⎩

Tu

Tu

Tu [-1, -1/ ].

Lp[-1,1] sivrceSi es mimdevroba fundamenturia. marTlac, vTqvaT garkveu-

lobisaTvis m≥n (winaaRmdeg SemTxvevaSi msjeloba analogiuria). radgan xn(t)= xm(t)=0, roca t∈[-1,-1/n]∪[1/n,1], amitom integralis adiciurobisa da

minkovskis integraluri utolobis ZaliT

ρ(xn, xm)= 1/1

1

| ( ) ( ) |p

pn mx t x t dt

−

⎧ ⎫− ≤⎨ ⎬

⎩ ⎭∫

1/1/

1

| ( ) ( ) |pn

pn mx t x t dt

−

−

⎧ ⎫− +⎨ ⎬

⎩ ⎭∫

+1/ 1/1/ 1

1/ 1/

| ( ) ( ) | | ( ) ( ) |p pn

p pn m n m

n n

x t x t dt x t x t dt−

⎧ ⎫ ⎧ ⎫− + − =⎨ ⎬ ⎨ ⎬

⎩ ⎭ ⎩ ⎭∫ ∫

=1/1/

1/

| ( ) ( ) |pn

pn m

n

x t x t dt−

⎧ ⎫−⎨ ⎬

⎩ ⎭∫ ≤

1/1/

1/

pn

n

dt−

⎧ ⎫⎨ ⎬⎩ ⎭

∫ =1/2 .

p

n⎧ ⎫⎨ ⎬⎩ ⎭

meore mxriv, ar arsebobs iseTi uwyveti funqcia, romliskenac Lp[-1,1] siv-rceSi krebadi iqneba uwyvet funqciaTa ganxiluli (xn) mimdevroba. davu-

SvaT sawinaaRmdego, vTqvaT aseTi uwyveti x funqcia arsebobs. vigulis-

xmoT, rom x(0)>0. x funqciis 0 wertilSi uwyvetobis gamo arsebobs δ (0<δ< <1), rom roca t∈[-δ,δ], maSin x(t)>x(0)/2. ganvixiloT nebismieri naturaluri

ricxvi, romlisTvisac 1/n<δ/2. maSin aseTi n–ebisTvis

ρ(xn, x)=1/1

1

| ( ) ( ) |p

pnx t x t dt

−

⎧ ⎫− ≥⎨ ⎬

⎩ ⎭∫

1// 2

| ( ( )) ( ) |p

pnx t x t dt

δ

δ

−

−

⎧ ⎫−⎨ ⎬

⎩ ⎭∫ =

=1// 2

| ( 1) ( ) |p

px t dtδ

δ

−

−

⎧ ⎫− −⎨ ⎬

⎩ ⎭∫ >

1// 2 p

dtδ

δ

−

−

⎧ ⎫⎨ ⎬⎩ ⎭

∫ =1/

.2

pδ⎛ ⎞⎜ ⎟⎝ ⎠

miviReT winaaRmdegoba - ρ(xn, x)=o(1), roca n→∞. SemTxvevebi, roca x(0)=0 an x(0)<0 analogiurad ganixileba.

10. ganvixiloT racionalur ricxvTa Q simravle, romelSic metrika

gansazRvrulia tolobiT ρ(r1, r2)=|r1-r2|, arasruli sivrcea. marTlac, ganvi-

xiloT racionalur ricxvTa (rn) mimdevroba, sadac rn=(1+1/n)n. es mimdevro-ba fundamenturia, magram misi zRvari ar aris racionaluri ricxvi (ra-

tom?).

axla ganvixiloT sruli sivrcis zogierTi Tviseba.

Teorema 4.1. sruli metrikuli sivrcis Caketili qvesimravle aseve

sruli metrikuli sivrcea.

mkiTxvels vTxovT daamtkicos ukanaskneli debuleba.

Teorema 4.2. metrikuli (X,ρ) sivrcis sisrulisaTvis aucilebeli da

sakmarisia, rom am sivrcis erTmaneTSi Calagebul, Caketil birTvTa neb-

ismieri B[rn] mimdevrobisaTvis, sadac lim 0nnr

→∞= , TanakveTa

1[ ]nn

B r∞

=∩ ar iyos

carieli simravle.

damtkiceba. (aucilebloba) vTqvaT (X,ρ) metrikuli sivrce srulia, xo-

lo (B[rn]) aris erTmaneTSi Calagebul, Caketil birTvTa mimdevroba, ro-

melTaTvisac lim 0nnr

→∞= . ganvixiloT am birTvTa centrebis mimdevroba (xn).

es mimdevroba fundamenturia. marTlac, Tu n>m, maSin ρ(xm, xn)≤rm. radgan

20

(X,ρ) sivrce srulia, amitom (xn) mimdevroba aris krebadi. vTqvaT lim nnx

→∞=

= 0x . yoveli naturaluri m-Tvis B[rm]≡B[xm,rm] birTvi Seicavs (xn) mimdevro-

bis yvela wevrs dawyebuli m indeqsidan. amitom B[rm] birTvis Caketilo-

bis gamo x0∈B[rm]. aqedan gamomdinareobs, rom x0∈ 1[ ]mm

B r∞

=∩ .

(sakmarisoba) vTqvaT axla (X,ρ) metrikuli sivrce arasrulia; e.i. am

sivrceSi arsebobs erTi mainc fundamenturi mimdevroba, romelic ar ar-

is krebadi. avagoT Caketil, erTmaneTSi Calagebul birTvTa mimdevroba,

romelTa radiusTa mimdevroba 0-ken krebadia da amasTan 1

[ ]mmB r∞

=∩ =∅.

vTqvaT naxsenebi fundamenturi mimdevrobaa (xn), romelsac ar aqvs zRvari.

SevarCioT m1 indeqsi ise, rom, roca n>m1, gvqondes 1

( , ) 1/ 2.m nx xρ < ganvixi-

loT B[1mx ,1]. Semdeg aviRoT m2>m1 ise, rom, Tu n>m2, maSin Sesruldes

2

2( , ) 1/ 2m nx xρ < utoloba. amjerad ganvixilavT B[2mx ,1/2] birTvs. sazogadod,

Tu B[kmx ,1/2k-1] birTvi ukve agebulia, aviRebT iseT naturalur mk+1

ricxvs, rom Tu n>mk+1, maSin 1

1( , ) 1/ 2k

km nx xρ

+

+< da ganvixilavT B[1kmx

+,1/2k]

birTvs. ase amrigad, miviRebT Caketil birTvTa mimdevrobas, romelTa

radiusTa mimdevroba 0-ken krebadia. Zneli ar aris imis Cveneba, rom

birTvTa es mimdevroba erTmaneTSi Calagebul birTvTa mimdevrobaa (aCve-

neT), amasTan 1

1[ ,1/ 2 ]

k

kmk

B x∞ −

== ∅∩ . marTlac, davuSvaT sawinaaRmdego –

vTqvaT arsebobs x0∈ 11

[ ,1/ 2 ]k

kmk

B x∞ −

=∩ maSin yoveli k–Tvis 1[ ,1/ 2 ]

k

kmB x −

sfero

Seicavs (xn) mimdevrobis yvela wevrs dawyebuli mk indeqcidan, e.i. Tu

n>mk, maSin 2

0( , ) 1/ 2 .knx xρ −< aqedan gamomdinareobs, rom (xn) mimdevroba

krebadia x0 wertilisken, rac ewinaaRmdegeba daSvebas.

(X,ρ) metrikuli sivrcis raime simravles ewodeba pirveli kategoriis

simravle (X,ρ) sivrceSi, Tu igi warmoidgineba am sivrceSi arsad mkvrivi

simravleebis sasruli an Tvladi gaerTianebis saxiT. Tu simravle ar

aris pirveli kategoriis, maSin aseT simravles ewodeba meore kategori-

is simravle.

am gansazRvris Tanaxmad racionalur ricxvTa simravle namdvil

ricxvTa simravleSi aris pirveli kategoriis simravle (ratom?), xolo

namdvil ricxvTa simravle aris meore kategoriis. ufro metic, adgili

aqvs Semdeg debulebas.

Teorema 4.3. yoveli sruli metrikuli (X,ρ) sivrce aris meore kate-goriis.

damtkiceba. davuSvaT sawinaaRmdego, vTqvaT arsebobs sruli (X,ρ) met-rikuli sivrce, romelic aris pirveli kategoriis, e.i. adgili aqvs war-

modgenas

X=1

,nn

X∞

=∪

sadac yoveli Xn aris arsad mkvrivi (X,ρ) sivrceSi. radgan X1 arsad mkvri-

via, amitom arsebobs B[ 1x ,ε1] birTvi, romelic ar Seicavs X1 simravlis

arcerT wertils. vigulisxmoT, rom ε1<1. radgan X2 arsad mkvrivia, amitom

B[ 1x ,ε1] birTvi Seicavs B[ 2x ,ε2] birTvs, romelic ar Seicavs X2 simravlis

arcerT wertils. vigulisxmoT, rom ε2<1/2 da, sazogadod, Tu [ , ]k kB x ε

21

birTvi ukve agebulia, ganvixilavT 1 1[ , ]k kB x ε+ + birTvs, romlisTvisac

1 1[ , ]k kB x ε+ + ⊂ [ , ]k kB x ε , εk+1<1/(k+1) da 1 1[ , ]k kB x ε+ + birTvi ar Seicavs Xk+1 simrav-

lis wertilebs. cxadia, miviRebT Caketil, Calagebul birTvTa mimdevro-

bas, romelTa radiusTa mimdevrobac 0-ken krebadia, amasTan am birTvebis

centrTa (xk) mimdevroba fundamenturia (SeamowmeT), amitom (X,ρ) sivrcis sisrulis gamo krebadia am sivrcis raRac x0 elementisaken. meore mxriv,

x0 yoveli naturaluri k-Tvis ekuTvnis [ , ]k kB x ε sferos, rac niSnavs, rom

x0∉Xk. k–s nebismierobis gamo x0∉X=1

kk

X∞

=∪ . miRebuli winaaRmdegoba Teoremas

amtkicebs.

§5. metrikul sivrceTa uwyveti asaxvebi.

izometria. homeomorfizmi.

metrikuli sivrcis gasruleba

vTqvaT (X,ρ1) da (Y,ρ2) ori metrikuli sivrcea da f :X→Y ( f funqcia asa-

xavs X simravles YYsimravleSi (simravleze)). am asaxvas ewodeba uwyveti

(X,ρ1) metrikuli sivrcis x0 wertilze, Tu yoveli ε>0 ricxvisaTvis arse-

bobs iseTi δ>0 ricxvi, rom nebismieri x∈X elementisaTvis, romlisTvisac

ρ1(x, x0)<δ, sruldeba utoloba ρ2(f(x), f(x0))<δ. Tu es asaxva uwyvetia A⊂X sim-

ravlis nebismier wertilSi, maSin amboben, rom uwyvetia A simravleze.

SevniSnoT, rom TviT ρ1 funqcia, rogorc (x,y) wyvilis (x,y∈X) funqcia, aris uwyveti. ukanaskneli martivad gamomdinareobs Semdegi utolobidan

(ix. Teorema 3.2-is damtkiceba)

|ρ(x, y)−ρ(x0, y0)|≤ρ(x, x0)+ ρ(y, y0). rogorc cnobilia, Tu asaxva f :X→Y urTierTcalsaxaa (bieqciaa), maSin

arsebobs Seqceuli asaxvac f -1 :Y→X. Tu asaxva f :X→Y aris bieqcia da, amave dros, f da f -1

uwyveti funqciebia, saTanadod, X da Y simravleebze,

maSin aseT f asaxvas homeomorfuli asaxva ewodeba. aseT SemTxvevaSi vit-

yviT, rom arsebobs homeomorfizmi (X,ρ1) da (Y,ρ2) metrikul sivrceebs So-

ris. homeomorful metrikul sivrceTa magaliTad SeiZleba gamodges

ricxviTi X=(-∞,+∞) da Y=(-1,1) intervalebi “Cveulebrivi” metrikiT (ix. §2, magaliTi 2). aRniSnul simravleebs Soris homeomorful Tanadobas amya-

rebs asaxva x→ 12 arctgxπ − . ibadeba kiTxva: Tu asaxva f :X→Y aris bieqcia da uwyveti X simravleze, maSin misi Seqceuli f -1

aris Tu ara uwyveti Y-ze? zog SemTxvevaSi (magaliTad, roca f intervalze gansazRvruli namdvili,

uwyveti, urTiercalsaxa funqciaa) pasuxi dadebiTia, magram ara yovel-

Tvis. magaliTad, vTqvaT X=(-∞,+∞) diskretuli ρ1 metrikiT, xolo Y=(-∞,+∞) “Cveulebrivi” ρ2 metrikiT. vigulisxmoT, rom f igivuri asaxvaa, e.i. yoveli

x∈(-∞,+∞)-Tvis f(x)= x. cxadia, f bieqciaa. is uwyveticaa X simravleze. marT-

22

lac, vTqvaT x0 nebismieri wertilia (-∞,+∞) intervalidan. nebismieri ε>0 ricxvisTvis moiZebneba δ (0<δ<1, yoveli ε-Tvis δ-s rolSi SeiZleba vigu-

lisxmoT 1/2), rom roca ρ1(x, x0)< δ (diskretuli metrikis ganmartebis Za-

liT ρ1(x, x0)=0 da, maSasadame, x=x0), maSin ρ2(f(x), f(x0))=ρ2(x, x0)=|x-x0|=|x0-x0|=0<ε. f–is igivurobis ZaliT Seqceul funqcias aqvs igive saxe: f -1(x)=x, x∈Y=R. f -1 asaxva ar aris uwyveti arcerT x0∈Y wertilSi. marTlac, davasaxeloT

ε0=1/2 ricxvi da nebismieri δ>0 ricxvisaTvis aviRoT xδ ricxvi, iseTi rom

xδ ≠x0, amasTan vigulisxmoT, rom ρ2(xδ, x0)=|xδ-x0|<δ. radgan ρ2 diskretuli met-

rikaa, amitom am SemTxvevaSi miviRebT ρ1( f -1(xδ), f -1(x0)) =ρ1(xδ , x0)=1>1/2=ε0. f bieqcias (X,ρ1) da (Y,ρ2) metrikul sivrceebs Soris ewodeba izomet-

ria, Tu X simravlis nebismier x1 da x2 wertilebisaTvis ρ1(x1, x2)=

=ρ2(f(x1), f(x2)). aseT SemTxvevaSi (X,ρ1) da (Y,ρ2) sivrceebs ewodebaT izomet-

ruli.

(X,ρ1) da (Y,ρ2) metrikul sivrceebis izometruloba niSnavs, rom maT

elementebs Soris metruli kavSirebi erTidaigivea; gansxvavebuli SeiZ-

leba aRmoCndes mxolod maT elementTa buneba, rac metrikul sivrceTa

TvalsazrisiT ar aris arsebiTi. momavalSi erTmaneTis izometrul

metrikul sivrceebs (funqcionaluri analizis TvalsazrisiT) erTmaneTi-

sgan ar ganvasxvavebT.

wina paragrafSi naCvenebi iyo, rom yoveli metrikuli sivrce ar aris

sruli. vTqvaT (X1,ρ1) arasruli sivrcea. bunebrivad ibadeba kiTxva: arse-

bobs Tu ara iseTi sruli sivrce (X,ρ), romelic moicavs (X1,ρ1) sivrces;

am kiTxvaze pasuxis gasacemad dagvWirdeba Semdegi ganmarteba.

srul metrikul (X,ρ) sivrces ewodeba (X1,ρ1) metrikuli sivrcis gas-

ruleba, Tu: a) arsebobs (X,ρ) metrikuli sivrcis qvesivrce (X0,ρ), rome-lic mkvrivia (X,ρ) sivrceSi; b) (X1,ρ1) da (X0,ρ) metrikuli sivrceebi izo-

metrulia.

Teorema 5.1. (metrikuli sivrcis gasrulebis Sesaxeb). yoveli (X1,ρ1)

metrikuli sivrcis gasruleba (X,ρ) arsebobs. es gasruleba erTaderTia

izometriis sizustiT, romelic X1-is wertilebs uZravad tovebs.

gavaanalizoT Camoyalibebuli Teoremis Sinaarsi. jer erTi – debule-

ba amtkicebs, rom (X1,ρ1) metrikuli sivrcis gasruleba SeiZleba. meore –

Tu (X,ρ) da ( X ′ , ρ′ ) sivrceebi warmoadgenen (X1,ρ1) metrikuli sivrcis gas-

rulebas (vTqvaT am gasrulebisas X X⊂ da X X′ ′⊂ aris “Sesabamisi” X1-isa),

maSin isini izometrulia. mesame – izometria (X,ρ)-sa da ( X ′ , ρ′ )-s Soris iseTia, rom misi saSualebiT myardeba bieqciuri Tanadoba X -sa da X ′ –s Soris.

damtkiceba. jer davamtkicoT (X1,ρ1) metrikuli sivrcis gasrulebis Se-

saZlebloba. (X1,ρ1) sivrcis or fundamentur (xn) da (yn) mimdevrobas ewo-

deba tolfasi (ekvivalenturi), Tu lim ( , ) 0.n nnx yρ

→∞= Teorema 3.2-is ZaliT uka-

naskneli zRvari arsebobs da sasruli ricxvia. termini “ekvivalenturi”

gamarTlebulia, radgan es Tanadoba refleqsuri, simetriuli da tranzi-

tulia (SeamowmeT. gamoiyeneT metrikis ganmartebis aqsiomebi). rogorc

cnobilia, Tu raime simravlis elementebs Soris arsebobs refleqsuri,

simetriuli da tranzituli mimarTeba, maSin es simravle iyofa Tanauk-

veT qvesimravleebad (gadaamowmeT am winadadebis marTebuloba). aqedan

gamomdinareobs, rom yvela fundamentur mimdevrobaTa simravle, romle-

23

bic SeiZleba SevadginoT (X1,ρ1) sivrcis elementebisagan daiyofa ekviva-

lentur mimdevrobaTa klasebad. TiToeuli aseTi klasi aRvniSnoT simbo-

loebiT: x∗, y∗, z∗ da a.S.

ganvsazRvroT (X,ρ) metrikuli sivrce. am sivrcis elementad CavTva-

loT yvela SesaZlo ekvivalentur fundamentur mimdevrobaTa klasi, xo-

lo manZili maT Soris ase ganvmartoT. vTqvaT x∗ da y∗

ori aseTi klasia.

TiToeuli aseTi klasidan aviRoT TiTo fundamenturi mimdevroba. am Sem-

TxvevaSi vTqvaT (xn)∈x∗ da (yn)∈y∗. davuSvaT

ρ( x∗, y∗)= lim ( , )n nnx yρ

→∞. (5.1)

vaCvenoT am ganmartebis koreqtuloba. (5.1) zRvris arseboba uSualod

gamomdinareobs Teorema 3..2-dan. axla davamtkicoT, rom (5.1) zRvari ar aris damokidebuli (xn)∈x∗ da (yn)∈y∗ fundamentur mimdevrobaTa arCevaze.

marTlac, vTqvaT (xn), ( nx ′ )∈x∗ da (yn), ( ny ′ )∈y∗. Teorema 3..2-is mtkicebisas

Catarebuli msjelobis analogiurad miviRebT

|ρ(xn, yn)−ρ( nx ′ , ny ′ )|≤ρ(xn, nx ′ )+ ρ( yn, ny ′ ),

saidanac imis gamo, rom (xn) ∼ ( nx ′ ) da (yn) ∼ ( ny ′ ), miviRebT limn→∞

ρ( nx ′ , ny ′ )=

= limn→∞

ρ(xn, yn).

axla vaCvenoT, rom (X,ρ) sivrceSi Sesrulebulia metrikis samive aqsi-

oma.

pirveli aqsioma. radgan ρ(xn, yn)≥0, amitom (5.1) ganmartebidan naTelia,

rom ρ( x∗, y∗)≥0. amis garda, Tu ρ( x∗, y∗)=0, maSin limn→∞

ρ(xn, yn)=0, e.i. (xn) ∼ (yn),

anu x∗ da y∗ klasebi erTmaneTs emTxveva. piriqiT, Tu x∗=y∗, maSin am klase-

bidan aRebuli nebismieri (xn) da (yn) mimdevroba erTmaneTis ekvivalen-

turia, e.i. lim ( , )n nnx yρ

→∞=0. amrigad, ganmartebis ZaliT ρ( x∗, y∗)=0.

meore aqsiomis samarTlianoba cxadia (ix. (5.1)). mesame aqsioma. radgan (X1,ρ1) metrikuli sivrceSi sruldeba samkuTxe-

dis aqsioma, amitom

ρ(xn, zn)≤ρ(xn, yn)+ρ(yn, zn). am utolobaSi gadavalT ra zRvarze, roca n→∞, miviRebT (ix. (5.1))

ρ( x∗, z∗)≤ ρ( x∗, y∗)+ ρ( y∗, z∗). axla davasaxeloT (X1,ρ1) metrikuli sivrcis nebismieri x elementi da

SevadginoT stacionaruli mimdevroba (x), romlis yvela wevri x–is to-lia. am mimdevrobis Sesabamisi klasi aRvniSnoT x∗

-iT da aseT x∗-Ta sim-

ravle aRvniSnoT X0-iT. ganvixiloT Sesabamisi asaxva x→ x∗ (X1–isa X0-Si).

es asaxva bieqciaa (ratom?). amasTan Tu x→ x∗ da y→ y∗ , maSin (ix. (5.1))

ρ1(x, y)=ρ( x∗, y∗). amrigad, (X0,ρ) aris (X,ρ) sivrcis qvesivrce, amasTan (X1,ρ1)

da (X0,ρ) sivrceebi izometrulia.

vaCvenoT, rom (X0,ρ) sivrce mkvrivia (X,ρ) sivrceSi. marTlac, vTqvaT x∗

(X,ρ) sivrcis raime wertilia, xolo ε aris raRac dadebiTi ricxvi. avi-

RoT x∗-dan raime fundamenturi mimdevroba (xn). vTqvaT N(ε) iseTia, rom

ρ1(xn, xm)<ε/2, roca m,n> N(ε). bunebrivia, nx∗-iT aRvniSnoT stacionaruli mim-

devroba, romlis yvela wevri xn–is tolia. maSin, roca n> N(ε) gveqneba ρ( nx∗ , x∗)= 1lim ( , )n mm

x xρ→∞

≤ ε/2< ε.

24

miviReT, rom x∗–is nebismieri ε-midamo Seicavs wertils X0-dan, anu (X0,ρ)

sivrce mkvrivia (X,ρ) sivrceSi. axla vaCvenoT (X,ρ) sivrcis sisrule. aviRoT (X,ρ) sivrcis nebismieri

fundamenturi mimdevroba ( )( )nx∗

, e.i. nebismieri ε>0 ricxvisTvis moiZebne-

ba iseTi N(ε), rom

( ) ( )( ),n m

x xρ ∗ ∗ <ε/4, (5.2)

roca n,m>N(ε). yovel ( )n

x∗ klasidan aviRoT erT-erTi fundamenturi ( )( )n

kx

mimdevroba. am mimdevrobis fundamenturobis gamo moiZebneba iseTi natu-

raluri kn ricxvi, rom

( )( ) ( )1 , 1/

n

n nr kx x nρ < , roca r> kn. (ratom?) (5.3)

axla ganvixiloT mimdevroba ( )( )n

nkx (X1,ρ1) sivrcidan. vaCvenoT, rom es mim-

devroba fundamenturia. (5.3)–is gamoyenebiT, roca r>max (kn, km) gveqneba

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1, , , ,

n m n m

n m n n n m m mk k k r r r r kx x x x x x x xρ ρ ρ ρ< + + <

<1/n+ ( )( ) ( )1 ,n m

r rx xρ +1/m. amis garda, (5.2)-dan (5.1)–is ZaliT miviRebT

( ) ( )( ),n m

x xρ ∗ ∗ = ( )( ) ( )1lim ,n m

r rrx xρ

→∞<ε/4, roca n,m> N(ε).

amitom, moiZebneba iseTi n0=n0(m,n), rom Tu n,m>N(ε) da r>n0, maSin ( )( ) ( )

1 ,n mr rx xρ <ε/3.

maSasadame,

( )( ) ( )1 ,

n m

n mk kx xρ < 1/n+ε/3+1/m.

Tu axla vigulisxmebT, rom max{1/n,1/m}<ε/3, maSin

( )( ) ( )1 ,

n m

n mk kx xρ < ε,

rac ( )( )n

nkx mimdevrobis fundamenturobas niSnavs.

x∗-iT aRvniSnoT is klasi (x∗∈X), romelic Seicavs ( )( )

n

nkx mimdevro-

bas.�vaCvenoT, rom ( )limnn

x∗

→∞= x∗. gvaqvs�

( )( ),n

x xρ ∗ ∗= ( )( ) ( )

1lim ,r

n rr kr

x xρ→∞

.

cxadia,�

( )( ) ( )1 ,

r

n rr kx xρ ≤ ( )( ) ( )

1 ,n

n nr kx xρ + ( )( ) ( )

1 ,n r

n rk kx xρ .

Tu ukanasknel utolobaSi gadavalT zRvarze (gaanalizeT, ratom arse-

bobs es zRvrebi?) da gaviTvaliswinebT (5.3)–s, miviRebT

( )( ),n

x xρ ∗ ∗ ≤ ( ) ( )( ) ( ) ( ) ( )1 1lim , lim ,

n n r

n n n rr k k kr r

x x x xρ ρ→∞ →∞

+ <1/n+ ( )( ) ( )1lim ,

n r

n rk kr

x xρ→∞

. (5.4)

radgan ( )( )n

nkx mimdevroba fundamenturia, amitom sakmarisad didi n-ebis-

Tvis SeiZleba vigulisxmoT, rom

( )( ) ( )1lim ,

n r

n rk kr

x xρ→∞

<ε,

e.i. (ix. (5.4)) (X,ρ) sivrce srulia.

25

axla vaCvenoT, rom (X1,ρ1) sivrcis gasruleba erTaderTia izometriis

sizustiT, romelic X1-is wertilebs uZravad tovebs.

Cven ukve avageT erTi sruli sivrce (X,ρ), romelic aris (X1,ρ1) sivrcis

gasruleba. amasTan X1–sa da X0⊂ X-s Soris arsebobs izometria. davuSvaT

aris meore metrikuli sivrce, romelSic mkvrivia 0'X ⊂ X ′ simravle. amas-

Tan 0'X izometrulia X1–sa. aviRoT X ′ -dan nebismieri x′ elementi. radgan

am sivrceSi 0'X mkvrivia, amitom arsebobs 0

'X simravlis elementTa iseTi

mimdevroba ('nx ), romelic krebadia ρ ′ metrikiT x′-ken. aqedan gamomdinare

('nx ) mimdevroba aris fundamenturi. radgan 0

'X izometrulia X1-sa da X1,

Tavis mxriv, izometrulia X0-isa, amitom 0'X izometrulia X0–isa. es imas

niSnavs, rom ('nx ) fundamenturi mimdevroba X0–Si gansazRvravs Sesabamis

fundamentur mimdevrobas (xn). es ki, (X,ρ) sivrcis agebis ZaliT, gansaz-

Rvravs X sivrcis raRac x elements. piriqiT, vTqvaT gvaqvs X sivrcis

raime x elementi (klasi). aviRoT am klasidan fundamenturi mimdevroba

(xn). es mimdevroba Sedgeba X1-is elementebisagan da gansazRvravs (X1-isa

da 0'X -is izometrulobis gamo) fundamentur mimdevrobas 0

'X -Si. radgan

X ′ srulia, amitom igi krebadia am sivrcis raRac x′ elementisken. x-s Se-vusabamoT x′. es Sesabamisoba bieqciuria. amasTan

' '1( , ) lim ( , ) lim ( , ) ( ', ').n n n nn n

x y x y x y x yρ ρ ρ ρ→∞ →∞

′ ′= = =

ganvixiloT metrikuli sivrcis gasrulebis ori magaliTi.

1. aviRoT racionalur ricxvTa simravle Q “Cveulebrivi ”metrikiT,

anu r1 da r2 ricxvebisaTvis ρ(r1,r2)=|r1-r2|. racionalur ricxvTa simravlis

((Q, ρ) sivrcis) gasrulebas namdvil ricxvTa simravle ewodeba.

racionalur ricxvTa simravlis gasrulebis amocana gadaWril iqna

me-19 saukunis meore naxevarSi dedekindis, kantorisa da vaierStrasis

mier. maT aages namdvil ricxvTa formaluri Teoriebi, romlebic formiT

sxvadasxvaa, SinaarsiT – tolfasi.

2. rogorc ukve vnaxeT, Lp[a,b] sivrce (p≥1) ar aris sruli (ix. §4, maga-liTi 9). Uukanaskneli damtkicebuli Teoremis Tanaxmad arsebobs am sivr-

cis gasruleba, romelsac aRvniSnavT simboloTi Lp[a,b]. es sivrce Sed-geba lebegis azriT zomadi, p(p≥1) xarisxSi jamebadi funqciebisagan:

( )b

p

a

f x dx < +∞∫

(integrali aiReba lebegis azriT), amasTan Lp[a,b] sivrcis ori f da g fun-qciebisaTvis ρ( f, g) ganimarteba (2.5) tolobiT, sadac, sazogadod, integ-

rali aiReba lebegis azriT.

ukanasknel punqtSi mocemuli winadadebebis dasabuTebas Cven ar ganvi-

xilavT.

26

§6. kumSviTi asaxvis principi; misi gamoyeneba

diferencialur da integralur

gantolebaTa amoxsnisaTvis

mimdevrobiTi miaxlovebis meTodi farTod gamoiyenaba sxvadasxva fun-

qcionaluri gantolebebis (diferencialuri da integraluri gantolebe-

bis, sasruli da usasrulo algebruli sistemebis da sxvaTa) amoxsnis

arsebobisa da erTaderTobis damtkicebaSi. am meTodiT garkveul piro-

bebSi SegviZlia davamtkicoT amoxsnis arseboba da erTaderToba, amasTan

misi saSualebiT SesaZlebelia aigos saZiebeli amoxsnis miaxlovebani sa-

Wiro sizustiT.

mimdevrobiTi miaxlovebis meTodis ganzogadoebebs nebismieri sruli

metrikuli sivrcisaTvis warmoadgenen debulebebi operatoris uZravi

wertilis arsebobisa da erTaderTobis Sesaxeb. maT Soris yvelaze mar-

tivia e.w. kumSviTi asaxvis principi (banaxisa da kaCopolis Teorma). vid-

re am Teoremas ganvixilavdeT, SevCerdeT or aucilebel gansazRvrebaze.

(X,ρ) metrikuli sivrcea, xolo operatori A : X→X. A operators ewo-deba kumSviTi asaxva, Tu arsebobs iseTi ricxvi α (0≤α <1), rom X-is nebis-mieri x da y elementebisaTvis sruldeba utoloba

ρ(Ax,Ay)≤ αρ(x,y). (6.1) yoveli kumSviTi asaxva uwyvetia (ratom?).

x∈X wertils ewodeba A operatoris (A : X→X) uZravi wertili, Tu Ax=x. sxvagvarad rom vTqvaT, A asaxvis uZravi wertili aris Ax=x gantolebis

amoxsna.

Teorema 6.1 (banaxisa da kaCopolis). srul (X,ρ) metrikul sivrceze

gansazRvrul yovel kumSviT operators gaaCnia uZravi wertili.

damtkiceba. jer vaCvenoT uZravi wertilis arseboba. vTqvaT x0 X-is ne-bismieri wertilia. davuSvaT

x1=Ax0, x2=Ax1=A(Ax0)=A(2)x0, x3=Ax2=A(A(2)x0)= A(3)x0, … , xn=Axn-1= A(n)x0.

vaCvenoT, rom (xn) mimdevroba fundamenturia. marTlac garkveulobisaTvis

vigulisxmoT, rom m≥n. maSin ρ(xn, xm)=ρ( A(n)x0, A(m)x0)≤α ρ( A(n-1)x0, A(m-1)x0)≤

≤α2ρ( A(n-2)x0, A(m-2)x0)≤...≤ αnρ(x0, xm-n). (6.2) ramdenjerme gamoviyenebT ra samkuTxedis aqsiomas, gveqneba

ρ(x0, xm-n)≤ ρ(x0, x1)+ρ(x1, xm-n)≤ρ(x0,x1)+ρ(x1, x2)+ρ(x2, xm-n)≤ ≤ ρ(x0,x1)+ρ(x1, x2)+...+ρ( xm-n-1, xm-n)=

≤ρ(x0,x1)+ρ( Ax0, Ax1)+ρ( A(2)x0, A(2)x1)+...+ρ( A(m-n-1)x0, A(m-n-1)x1)≤ ≤ρ(x0,x1)(1+α + α2+...+ α m-n-1)≤ρ(x0,x1)(1− α )-1

.

amitom (6.2)-is ZaliT

ρ(xn,xm)≤ αnρ(x0,x1)(1−α )-1. (6.3)

radgan α erTze naklebi arauaryofiTi ricxvia, amitom gaviTvaliswinebT

ra (6.3)–s da utolobas m≥n, gveqneba lim ( , ) 0,m nn

x xρ→∞

=

e.i. nebismieri ε>0 ricxvisaTvis arsebobs N(ε) ricxvi, rom roca n> N(ε) (da, maSasadame, m-ic metia N(ε)-ze), miviRebT ρ(xn, xm)< ε . amrigad, (xn) mimdev-

27

roba fundamenturia. (X,ρ) metrikul sivrcis sisrulis gamo (xn) mimdevro-ba krebadia. vTqvaT

lim ( , ) 0.nnx xρ

→∞=

radgan A kumSvis operatoria, amitom 1lim ( , ) lim ( , ) 0.n nn n

Ax Ax x Axρ ρ +→∞ →∞= =

amrigad, (xn) mimdevroba krebadia rogorc Ax–ken, aseve x–kenac. amitom Ax= =x.

uZravi wertilis arseboba damtkicebulia. axla davamtkicoT erTader-

Toba. vTqvaT Ax=x da Ay=y. maSin (6.1)-is ZaliT

ρ(x,y)=ρ(Ax,Ay)≤αρ(x,y). aqedan ρ(x,y)(1−α)≤0; da radgan 0≤α <1, amitom ρ(x,y)=0, e.i. x=y.

aRsaniSnavia, rom TeoremaSi moTxovna 0≤α<1 arsebiTia. anu, Tu Teore-

ma 6.1-is yvela pirobas ucvlelad davtovebT garda (6.1)-isa, xolo (6.1)-s SevcvliT utolobiT ρ(Ax,Ay)<ρ(x,y), maSin, SesaZlebelia A operators ar gaaCndes uZravi wertili. marTlac, ganvixiloT namdvil ricxvTa simrav-

le “Cveulebrivi” metrikiT, xolo A operatori ase ganvmartoT: Ax= π/2+x- -arctgx. advili SesamCnevia, rom A operators ar gaaCnia uZravi wertili

(ratom?), amasTan lagranJis Teoremis ZaliT

ρ(Ax,Ay)=|Ax−Ay|=|A′(ξ)||x−y|= 2 11 (1 )ξ −− + |x−y|=

= 2 2 1(1 )ξ ξ −+ ρ(x,y)<ρ(x,y); ξ wertili moTavsebulia x–sa da y–s Soris (ξ≠x, y).

axla ganvixiloT banaxisa da kaCopolis zogierTi gamoyeneba.

1. vTqvaT f : [a,b]→[a,b] da vigulisxmoT, rom f funqcia akmayofilebs

lipSicis pirobas, anu [a,b] segmentis nebismieri t1 da t2 wertilebisaTvis

|f(t1)- f(t2)|≤M|t1- t2|. vigulisxmoT, rom M<1. cxadia, am pirobebSi f funqcia aris kumSvis ope-ratori da amitom banaxisa da kaCopolis Teoremis Tanaxmad arsebobs er-

TaderTi t∈[a,b] wertili, romelic warmoadgens f (t)=t gantolebis fesvs.

amis garda, es fesvi aris mimdevrobis t0, t1=f (t1), t2=f (t1),… zRvari. 2. koSis amocana. vTqvaT mocemuli gvaqvs diferencialuri gantole-

ba

( , ),dy f x ydx

= (6.4)

sawyisi pirobebiT

y(x0)=y0. (6.5) vigulisxmoT agreTve, rom f uwyvetia raime G brtyel areze, romelic

(x0, y0) wertils Seicavs; amasTan igulisxmeba, rom am areSi f funqcia y–is mimarT akmayofilebs lipSicis pirobas

|f(x,y1)- f(x,y2)|≤M| y1-y2|. Cven vaCvenebT, rom samarTliania pikaris Teorema: moiZebneba iseTi da-

debiTi d ricxvi, rom segmentze [x0-d, x0+d] arsebobs (6.4) gantolebis erTa-

derTi amoxsna (ϕ funqcia), romelic akmayofilebs (6.5) sawyis pirobas - ϕ(x0)= y0.

(6.4) gantoleba (6.5) sawyisi pirobiT ekvivalenturia Semdegi integra-

luri gantolebis

28

ϕ(x)= y0+0

( , ( ))x

x

f t t dtϕ∫ (ratom?). (6.6)

G areze f funqciis uwyvetobis gamo arsebobs (x0,y0) wertilis iseTi G′⊂G

midamo, romelzedac

|f(x,y)|≤K, sadac K raime arauaryofiTi ricxvia. SevarCioT d dadebiTi ricxviKise,

rom

a) (x,y)∈ G′, Tu |x-x0|≤ d da |y-y0|≤ Kd; b) Md<1.

aRvniSnoT C0–iT im uwyvet ϕ funqciaTa simravle, romlebic gansazRvru-

lia [x0-d, x0+d] segmentze da romlebic akmayofilebs pirobas |ϕ(x)-y0|≤Kd. amasTan ganvixiloT (C0,ρ) metrikuli sivrce, sadac ρ moicema tolobiT

(ix, §2, punqti 8) ρ(ϕ1,ϕ2)=

0 01 2[ , ]

max ( ) ( )x x d x d

x xϕ ϕ∈ − +

− .

rogorc cnobilia, uwyvet funqciaTa C[x0-d, x0+d] sivrce srulia (ix. §4, punqti 8), amasTan (C0,ρ) metrikuli sivrce warmoadgens C[x0-d, x0+d] sivrcis Caketil qvesivrces (anu C0

aris C[x0-d, x0+d] simravlis Caketili qvesimrav-

le) (ratom?). amitom (ix. Teorema 4.1) (C0,ρ) aris sruli metrikuli sivrce.

ganvixiloT asaxva ψ=Aϕ , romelic ganisazRvreba tolobiT

ψ(x)=y0+0

( , ( )) ,x

x

f t t dtϕ∫

sadac x∈[x0-d, x0+d]. es asaxva C0 simravles asaxavs Tavis TavSi, amave dros

warmoadgens kumSvis operators. marTlac, vTqvaT ϕ∈C0 da x∈[x0-d, x0+d]. ma-

Sin

|ψ(x)-y0|=

0

( , ( ))x

x

f t t dtϕ ≤∫ Kd (ratom?),

da, maSasadame, A(C0)⊂C0. amis garda

1 2( ) ( )x xϕ ϕ− ≤0

1 2( , ( )) ( , ( ))x

x

f t t f t t dtϕ ϕ− ≤∫

≤M0

1 2( )) ( ))x

x

t t dtϕ ϕ− ≤∫ Md0 0

1 2[ , ]max ( ) ( )

x x d x dx xϕ ϕ

∈ − +− = Mdρ(ϕ1,ϕ2).

radgan Md<1, amitom A operatori aris kumSvis operatori. aqedan gamom-dinareobs, rom ϕ=Aϕ gantolebas (anu, rac igivea, (6.6) gantolebas) gaaCnia

erTaderTi amoxsna (C0,ρ) sivrceSi. 3. fredgolmis gantoleba. axla fredgolmis meore gvaris wrfivi

integraluri gantolebis amoxsnis arsebobisa da erTaderTobis dadgeni-

saTvis gamoviyenoT kumSviTi asaxvis principi. aRniSnul integralur gan-

tolebas aqvs saxe

f(x)=λ ( , ) ( ) ( ),b

a

K x y f y dy xϕ+∫ (6.7)

sadac K (e.w. birTvi) da ϕ mocemuli funqciebia, f saZebni funqciaa, xolo

λ nebismieri parametria.

29

qvemoT Cven vnaxavT, rom kumSviTi asaxvis principis gamoyeneba SeiZle-

ba mxolod parametris sakmarisad mcire mniSvnelobebisaTvis.

vigulisxmoT, rom K funqcia uwyvetia kvadratze [a,b]×[a,b]= [a,b]2, xolo

ϕ funqcia uwyvetia [a,b] segmentze. maSasadame, arsebobs iseTi arauaryofi-

Ti M ricxvi, rom yoveli (x,y)∈[a,b]2-Tvis | K(x,y)|≤M. ganvixiloT g=Af, rome-

lic moicema tolobiT

g(x)=λ ( , ) ( ) ( ).b

a

K x y f y dy xϕ+∫

A operatori srul C[a,b]sivrces asaxavs Tavis TavSi (ratom?). amis garda,

gvaqvs

ρ(g1,g2)= 1 2[ , ]max ( ) ( )x a b

g x g x∈

− ≤ 1 2[ , ]max | | ( , ) ( ) ( )

b

x a ba

K x y f y f y dyλ∈

−∫ ≤

≤|λ|M(b-a) 1 2[ , ]max ( ) ( )x a b

f x f x∈

− = |λ|M(b-a)ρ(f1,f2).

maSasadame, Tu |λ|M(b-a)<1 anu, Tu |λ|< M -1(b-a)-1, maSin A kumSvis operatoria.

banaxisa da kaCopolis Teoremis ZaliT vaskvniT, rom Tu |λ|<M -1(b-a)-1, ma-

Sin fredgolmis gantolebas gaaCnia erTaderTi uwyveti amonaxsni.

rogorc banaxisa da kaCopolis Teoremis mtkicebisas iqna naCvenebi,

Ax=x operatoruli gantolebis amoxsna miiReba rogorc mimdevrobiTi

miaxlovebis Sedegi, e.i. miiReba rogorc (xn)=(Axn-1) mimdevrobis zRvari.

Cvens SemTxvevaSi fredgolmis integraluri gantolebis amoxsnis mim-

devrobiT miaxlovebas iZleva mimdevroba 0( )n nf ∞= , sadac

1( ) ( , ) ( ) ( ),b

n na

f x K x y f y dy xλ ϕ−= +∫

xolo f0–is rolSi SeiZleba ganvixiloT [a,b] segmentze nebismieri uwyve-ti funqcia.

§7. kompaqturi simravleebi metrikul sivrceSi

rogorc cnobilia, namdvil ricxvTa yoveli SemosazRvruli simrav-

lis nebismieri mimdevrobidan SeiZleba krebadi qvemimdevrobis gamoyofa.

es Tviseba aqvs n-ganzomilebiani Rn sivrcis SemosazRvrul simravlesac.

igi gamoiyeneba maTematikuri analizis mTeli rigi fundamenturi debu-

lebis damtkicebisas (magaliTad, vaierStrasis, koSis da kantoris Teo-

remebSi). rogorc irkveva, es Tviseba mniSvnelovania, sazogadod, metri-

kuli sivrceebisTvisac.

metrikuli (X,ρ) sivrcis K simravles ewodeba kompaqturi simravle, Tu

am simravlis nebismier usasrulo qvesimravles gaaCnia erTi zRvariTi

wertili mainc. Tu yoveli aseTi zRvruli wertili ekuTvnis K-s, maSin K simravles ewodeba Tavis TavSi kompaqturi; xolo Tu aseTi wertilebi X simravles ekuTvnis, maSin K-s ewodeba X-Si kompaqturi.

namdvil ricxvTa R simravle (“Cveulebrivi” metrikiT) ar aris kompaq-

turi simravle, radgan mis usasrulo M={1,2,3,…,n,…} qvesimravles

zRvruli wertili ar gaaCnia. cxadia, (a,b), (a,b], [a,b) da [a,b] intervalebi

30

kompaqturi simravleebia, xolo [a,b] segmenti Tavis TavSi kompaqturicaa.

advilia imis Cveneba, rom K simravlis Tavis TavSi kompaqturobisaTvis

aucilebelia da sakmarisi, rom igi iyos Caketili da kompaqturi X-Si (aCveneT).

Tu (X,ρ) metrikuli sivrcis yoveli qvesimravles gaaCnia zRvruli wer-

tili, romelic ekuTvnis X simravles, maSin (X,ρ) metrikuli sivrces ewo-

deba kompaqturi, an, rogorv mas xSirad uwodeben, - kompaqti.

kompaqti sruli metrikuli sivrcea. marTlac, ganvixiloT metrikuli

sivrcis fundamenturi mimdevroba. gvaqvs ori SemTxveva: a) am mimdevrobis

mniSvnelobaTa simravle sasrulia; b) mimdevrobis mniSvnelobaTa simrav-

le usasruloa. a) SemTxvevaSi am fundamenturi mimdevrobidan, cxadia, ga-

moiyofa stacionaruli da, maSasadame, - krebadi qvemimdevroba. rogorc

cnobilia (ix. §4), Tu metrikuli sivrcis fundamenturi mimdevrobis raime

qvemimdevroba krebadia, maSin TviT es mimdevrobac krebadia. b) SemTxveva-

Si radgan mimdevrobis mniSvnelobaTa simravle usasruloa, amitom mas

(rogorc kompaqtis usasrulo qvesimravles) eqneba erTi zRvariTi werti-

li mainc, saidanac gamomdinareobs ganxiluli fundamenturi mimdevrobis

krebadi qvemimdevrobis arseboba. maSasadame, aRebuli fundamenturi mim-

devroba krebadia, e.i. sivrce srulia.

bolcanosa da vaierStrasis Teoremis ZaliT namdvil ricxvTa R sim-ravlis nebismieri SemosazRvruli qvesimravle kompaqturia (ratom?). Rn

sivrce, namdvil ricxvTa R simravlis analogiurad, arakompaqturia, xo-

lo misi nebismieri SemosazRvruli qvesimravle - kompaqturi (ratom?).

cxadia, Tu Rn sivrcis SemosazRvruli qvesimravle aris Caketili, maSin

es qvesimravle iqneba Tavis TavSi kompaqturi.

C[a,b] sivrce ar aris kompaqturi. ufro metic, am sivrceSi arsebobs SemosazRvruli arakompaqturi simravle. amis saCveneblad davamtkicoT

Semdegi

Teorema 7.1. vTqvaT K aris (X,ρ) metrikuli sivrcis Tavis TavSi kom-

paqturi simravle, xolo f : K→R uwyveti asaxvaa. maSin a) f asaxva Semosaz-Rvrulia K-ze; b) f asaxva aRwevs Tavis zeda da qveda sazRvrebs K simrav-leze.

damticeba. a) vaCvenoT, rom f asaxva SemosazRvrulia K simravleze. da-

vuSvaT sawinaaRmdego, vTqvaT f asaxva ar aris SemosazRvruli, e.i. moiZeb-

neba K simravlis elementTa iseTi (xn) mimdevroba, rom |f(xn)|>n. radgan K simravle Tavis TavSi kompaqturia, amitom (xn) mimdevroba Seicavs krebad

( )knx qvemimdevrobas. vTqvaT 0lim .

knkx x K

→∞= ∈ maSin ( ) ,

kn kf x n> e.i. lim ( )knk

f x→∞

=

.= +∞ meore mxriv, f asaxvis uwyvetobis gamo lim ( )knk

f x→∞

= 0( ) .f x amrigad,

0( ) .f x = +∞ miviReT winaaRmdegoba.

b) vTqvaT sup ( ).x K

f xβ∈

= es niSnavs, rom nebismieri x∈K wertilisaTvis

f(x)≤β da yoveli ε>0 ricxvisTvis arsebobs xε ∈K, rom f(xε)>β -ε. amrigad, ar-sebobs mimdevroba (xn), iseTi rom β -1/n< f(xn)≤β. K simravlis Tavis TavSi

kompaqturobis gamo (xn) mimdevroba Seicavs ( )knx qvemimdevrobas, romelic

krebadia x0 wertilisaken. gvaqvs β -1/nk< ( )knf x ≤β. amrigad, lim ( ) .

knkf x β

→∞= meo-

re mxriv, f asaxvis uwyvetobis gamo lim ( )knk

f x→∞

= 0( ).f x e.i. 0( ) .f x β= analogi-

31

urad SeiZleba vaCvenoT, rom Tu inf ( ),x K

f xα∈

= maSin moiZebneba iseTi y0∈K,

rom 0( ) .f y α=

Teorema 7.2. C[a,b] sivrceSi arsebobs SemosazRvruli arakompaqturi

simravle.

damticeba. simartivisaTvis ganvixiloT C[0,1] sivrce. vTqvaT M aris [0,1] segmentze uwyvet t→x(t) funqciaTa simravle, romelTaTvisac x(0)=0, x(1)=1 da

[0,1]max ( ) 1.t

x t∈

≤ cxadia, M simravle SemosazRvrulia C[0,1] sivrceSi. igi

Caketilicaa (ratom?). ganvixiloT asaxva 1

2

0

( ) ( ) .f x x t dt= ∫

vaCvenoT, rom es asaxva uwyvetia M simravleze. aviRoT M simravlis raime

y0 elementi (t→ y0(t), t∈[0,1]). ganvixiloT

f(x)- f(y0)=1

2 20

0

[ ( ) ( )] ,x t y t dt−∫

saidanac

|f(x)- f(y0)|≤1 1

2 20 0 0

0 0

( ) ( ) ( ) ( ) ( ) ( )x t y t dt x t y t x t y t dt− = − + ≤∫ ∫

≤1 1

0 0 0[0,1]0 0

2 ( ) ( ) 2 max ( ) ( ) 2 ( , ).t

x t y t dt x t y t dt x yρ∈

− ≤ − =∫ ∫

ukanasknelidan cxadia, rom Tu ρ(x,y0)→0, maSin f(x)→ f(x0); e.i. f asaxva uwyve-tia y0-ze. y0-is nebismierobis gamo f uwyvetia M simravleze. vaCvenoT, rom f ver aRwevs M-ze qveda sazRvars. marTlac jer gamovTvaloT f–is infimumi M–ze. f asaxvis ganmartebis ZaliT inf ( ) 0.

x Mf x

∈≥ vaCvenoT, rom inf ( ) 0.

x Mf x

∈= M

simravlidan ganvixiloT x(t)=t n funqciaTa mimdevroba. eWvs ar iwvevs, rom

asaxva t→ t n ekuTvnis M–s ([0,1]

max | | 1n

tt

∈≤ ). advili gamosaTvlelia, rom

12

0

1( ) .1

nf x t dtn

= =+∫

aqedan cxadia, rom inf ( ) 0.x M

f x∈

= meore mxriv, Tu funqcia t→ x(t) akmayofi-

lebs pirobas x(1)=1, es ukve niSnavs, rom 1

2

0

( ) 0x t dt >∫ . amrigad, M simravlis

nebismieri x funqciisaTvis f(x)>0. maSasadame, f asaxva ver aRwevs infimums. 7.1 Teoremis ZaliT M ar SeiZleba iyos kompaqturi.

vTqvaT M simravle (X,ρ) metrikuli sivrcis raime qvesimravlea, xolo

ε raime dadebiTi ricxvia. (X,ρ) sivrcis A simravles ewodeba M simravlis

ε-bade, Tu nebismieri x∈M wertilisaTvis arsebobs iseTi y∈A wertili,

rom ρ(x,y)≤ ε. aRsaniSnavia, rom am ganmartebaSi ar moiTxoveba piroba A⊂M.

magaliTad, mTelkoordinatebian wertilTa simravle qmnis R2-is (sibr-

tyis) 1/ 2 -bades.

M simravles ewodeba savsebiT SemosazRvruli, Tu am simravlisaTvis

da nebismieri ε>0 ricxvisaTvis arsebobs sasruli ε-bade (e.i. arsebobs

sasruli simravle, romelis ε-bades qmnis M simravlisTvis). yoveli sav-

sebiT SemosazRvruli simravle SemosazRvrulicaa (ratom?), magram yove-

32

li SemosazRvruli simravle ar aris savsebiT SemosazRvruli. magali-

TisaTvis ganvixiloT l2 sivrce (ix. §2, punqti 4) da am sivrceSi elementTa

mimdevroba (en), sadac e1=(1,0,0,…), e2=(0,1,0,0,…),… . am mimdevrobis yvela

elementi moTavsebulia birTvSi B[0,1], sadac 0= =(0,0,0,…) (ratom?). amis garda, am mimdevrobis nebismier or gansxvavebul elements Soris manZili

2 -is tolia (ratom? gaixseneT metrikis ganmarteba l2-Si). aqedan vaskvniT,

rom { en} simravlisaTvis da, miTumetes, mTeli l2 sivrcisaTvis ar arse-

bobs sasruli ε-bade (ε< 2 / 2 ). Rn sivrceSi savsebiT SemosazRvrulobis da SemosazRvrulobis cnebebi

ekvivalenturia. marTlac, vTqvaT M simravle SemosazRvrulia Rn –Si, e.i.

arsebobs iseTi n-ganzomilebiani kubi, romelSic moTavsdeba M simravlis

yvela wertili. Tu aseT kubs davyofT tol kubebad, romelTa wiboebi

ε-s ar aRemateba, maSin miRebuli kubebis wveroebi Seadgenen gamosaval

kubSi da, maSasadame, M simravleSi / 2n ε⋅ –bades (ratom?). radgan sivrcis

ganzomileba – n fiqsirebulia, xolo ε SeiZleba nebismierad vcvaloT

(ε>0), amitom Camoyalibebuli winadadeba damtkicebulia.

samarTliania Semdegi

Teorema 7.3 (hauzdorfi). (X,ρ) metrikuli sivrcis K qvesimravlis

kompaqturobisaTvis aucilebelia, xolo X-is sisrulis SemTxvevaSi sakma-

risicaa, rom nebismieri ε–Tvis arsebobdes K simravlis sasruli ε-bade. damtkiceba. (aucilebloba) vTqvaT K kompaqturia da x1∈K. Tu yoveli

x1∈K-Tvis ρ(x, x1)≤ ε, maSin x1–is saxiT sasruli ε-bade ukve agebulia. Tu es

ar sruldeba, maSin arsebobs iseTi wertili x2∈K, rom ρ(x1, x2)>ε. Tu K-s nebismieri x wertilisTvis ρ(x, x1)≤ ε an ρ(x, x2)≤ ε, maSin sasruli ε-bade uk-ve agebulia. Tu es ar sruldeba, maSin iarsebebs iseTi x1∈K wertili,

rom ρ(x1, x3)>ε da ρ( x2, x2)>ε. Tu ase gavagrZelebT, maSin n–ur nabijze mivi-

RebT x1, x2,…, xn wertilebs, romelTaTvisac ρ(xi, xj)>ε, roca i≠j. SesaZlebe-

lia ori SemTxveva: raime sasruli k∗ nabijis Semdeg es procesi Sewydeba

an igi usasrulod gagrZeldeba. pirvel SemTxvevaSi nebismieri x∈K ele-

mentisaTvis Sesruldeba erT-erTi Semdegi utolobaTagan ρ(x, xj)<ε, sadac j=1,2,…, k∗

. am SemTxvevaSi x1, x2,…,k

x ∗ wertilebi qmnian sasrul ε-bades K-Si. meore SemTxvevaSi miiReba K simravlis usasrulo mimdevroba (xn), romlis

wevrebisTvisac ρ(xi, xj)>ε (i≠j). ukanaskneli utolobebidan gamomdinareobs,

rom am mimdevrobis mniSvnelobaTa simravles (romelic usasruloa) ar

gaaCnia zRvariTi wertili (ratom?), radgan K simravle kompaqturia, ami-

tom miviReT winaaRmdegoba.

(sakmarisoba) axla vigulisxmoT, rom (X,ρ) metrikuli sivrce srulia.

amasTan, davuSvaT, rom arsebobs K simravlis sasruli ε-bade. ganvixiloT

(εn) ricxviTi mimdevroba (εn>0), romlisTvisac lim 0.nnε

→∞= yoveli εn-Tvis ava-

goT K simravlis εn-bade: { }( ) ( ) ( )1 2, ,...,

n

n n nkx x x . aviRoT nebismieri usasrulo sim-

ravle T⊂K. yovel 1

(1) (1) (1)1 2, ,..., kx x x wertilze “SemovweroT” Caketili birTvi,

radiusiT ε1. maSin T–s yoveli wertili Tavsdeba erT-erT birTvSi. rad-

gan birTvTa raodenoba sasrulia, amitom erT-erT birTvSi moTavsdeba T simravlis usasrulo qvesimravle. T–s es usasrulo qvesimravle aRvniS-

noT T1–iT. axla aviRoT 2

(2) (2) (2)1 2, ,..., kx x x wertilebi da yoveli maTganis gar-

33

Semo “SemovweroT” Caketili birTvi, radiusiT ε2. iseve, rogorc zemoT,

moiZebneba usasrulo T2⊂T1 simravle, romelic moTavsdeba erT-erT age-

bul ε2-radiusian birTvSi. Tu ase gavagrZelebT miviRebT mimdevrobas

T1⊃T2⊃T3⊃…⊃Tn⊃… ., romlis elementebic T–s qvesimravleebia, amasTan Ca-

ketili birTvi, romlis radiusia εn, moicavs Tn–s. amrigad, manZili Tn–is

nebismier or wertils Soris ar aRemateba 2εn–s. aviRoT ξ1∈T1. Semdeg gan-vixiloT wertili ξ2∈T2, ise, rom ξ2≠ξ1. es SesaZlebelia, radgan T2 Seicavs

usasrulod bevr wertils. sazogadod, yoveli naturaluri n–Tvis avi-

RebT ξn∈Tn wertils; amasTan, ξn gansxvavebulia ξ1, ξ2,…, ξn-1 wertilebisa-

gan. ganvixiloT simravle Tω ={ξ1, ξ2,…, ξn,…}. mimdevroba (ξn) aris funda-

menturi. marTlac, radgan ξn∈Tn da ξn+p∈Tn+p⊂ Tn, amitom ρ(ξn+p, ξn)<2εn→0, n→∞.

radgan (X,ρ) metrikuli sivrce srulia, amitom (ξn) mimdevroba krebadia

amave sivrcis raRac ξ wertilisaken. amrigad, K–s nebismier usasrulo T qvesimravles gaaCnia erTi mainc zRvariTi wertili.

am Teoremidan gamomdinareobs

Sedegi. kompakturi (X,ρ) metrikuli sivrce separabeluria.

damtkiceba. aviRoT dadebiTi usasrulod mcire mimdevroba (εn). yoveli

n-Tvis avagoT sasruli εn-bade – Nn={ }( ) ( ) ( )1 2, ,...,

n

n n nkx x x . vTqvaT

1

.nn

N N∞

=

=∪ cxa-

dia, N Tvladi simravlea. amis garda, N mkvrivia X–Si (ratom?). axla ganvixiloT Tavis TavSi kompaqturi simravleebis erTi mniSvne-

lovani Tviseba, romelsac farTo gamoyeneba aqvs.

vTqvaT mocemuli gvaqvs Ria simravleTa sistema {Gα} (X,ρ) metrikuli

sivrcidan. vityviT, rom {Gα} sistema faravs M⊂X simravles, Tu pirobi-

dan x∈M gamomdinareobs, rom x aris {Gα} sistemidan aRebuli romeliRac

Gα simravlis elementi.

Teorema 7.4 (kantori). vTqvaT (Kn) aris metrikuli sivrcis Tavis Tav-

Si kompaqtur simravleTa iseTi mimdevroba, rom Kn+1⊂Kn, n=1,2,… . maSin

K=1 iiK∞

=∩ ar aris carieli.

damtkiceba. yoveli Ki simravlidan avirCioT erTi xi elementi. Sevadgi-

noT mimdevroba (xi). cxadia, {xi}⊂ K1. radgan K1 kompaqturi simravlea, ami-

tom (xi) mimdevrobidan SeiZleba krebadi ( )kix qvemimdevrobis gamoyofa.

vTqvaT x0= limkik

x→∞

. radgan yoveli naturaluri n-Tvis, dawyebuli k (ik>n)

nomridan ( )kix qvemimdevrobis yoveli wevri ekuTvnis Kn simravles da Kn

Caketili simravlea (ratom?), amitom x0∈Kn. n-is nebismierobis gamo x0∈

∈1

ii

K∞

=∩ .

Teorema 7.5 (lebegisa da borelis lema). imisaTvis, rom (X,ρ) metri-kuli sivrcis Caketili F simravle iyos Tavis TavSi kompaqturi aucile-

belia da sakmarisi, rom F simravlis Ria simravleTa yoveli sistemiT

dafarvidan gamoiyos sasruli dafarva.

damtkiceba. (aucilebloba) vTqvaT {Gα} Ria simravleTa raime sistemaa,

romelic Tavis TavSi kompaqtur F simravles faravs (anu, rogorc xSi-

34

rad amboben, {Gα} aris F simravlis Ria dafarva). vaCvenoT, rom {Gα} sis-

temidan SeiZleba gamoiyos sasruli qvesistema, romelic kvlav faravs

F–s. davuSvaT sawinaaRmdego – vigulisxmoT, rom {Gα}-s arcerTi sasruli

qvesistema ar faravs F–s. aviRoT 0-ken krebadi dadebiT ricxvTa (εn) mim-

devroba. vTqvaT 1

(1) (1) (1)1 2, ,..., kx x x aris F simravlis ε1-bade (7.4 Teoremis ZaliT

aseTi 1

(1) (1) (1)1 2, ,..., kx x x sistema moiZebneba). maSin, cxadia, F= 1

1,k

iiF

=∪ sadac Fi=

= (1)1,iB x ε⎡ ⎤⎣ ⎦ ∩ F. radgan F Tavis TavSi kompaqturi simravlea, xolo

(1)1,iB x ε⎡ ⎤⎣ ⎦

- Caketili, amitom Fi kompaqturia Tavis TavSi (ratom?). amis garda misi

diametri - dimFi=,sup ( , )

ix y Fx yρ

∈≤2ε1. Tu F ar SeiZleba daifaros {Gα} sistemis

arcerTi sasruli qvesistemiT, maSin erTi mainc Fi ar SeiZleba daifaros

{Gα} sistemis arcerTi sasruli qvesistemiT. vTqvaT es simravlea 1i

F . Tu

analogiurad vimsjelebT, 1i

F -dan gamovyofT Tavis TavSi kompaqtur iseT

1 2i iF simravles, romlis diametric ar aRemateba ε2–s da romelic ar SeiZ-

leba daifaros {Gα}-s arcerTi sasruli qvesistemiT, da a. S. Cven mivi-

RebT Caketil, Calagebul kompaqtur simravleTa mimdevrobas 1i

F ⊃1 2i iF ⊃…

⊃1 2 ... ni i iF ⊃..., romelTa diametrTa mimdevroba 0-ken krebadia. Teorema 7.4-is

Tanaxmad arsebobs iseTi x0, romelic yvela 1 2 ... ni i iF simravles ekuTvnis. rad-

gan {Gα} sistema F–is dafarvaa, amitom x0∈0

Gα (0

{ }G Gα α∈ ). 0

Gα Ria simravlea,

amitom arsebobs iseTi B(x0,ε) birTvi, romelic moTavsebulia 0

Gα simravle-

Si. davasaxeloT nebismieri dadebiTi ε ricxvi da avirCioT n imdenad di-di, rom dim

1 2 ... ni i iF <ε (amis gakeTeba SesaZlebelia, radgan lim 0nnε

→∞= ). maSin,

cxadia, 1 2 ... ni i iF ⊂ B(x0,ε)⊂

0Gα . amrigad, miviReT, rom erTi mxriv,

1 2 ... ni i iF -is da-

farva ar SeiZleba 0

Gα -s arcerTi sasruli qvesistemiT, meore mxriv, ki

igi am {Gα} sistemis erTi 0

Gα elementiT davfareT. miRebuli winaaRmdego-

ba Teoremis aucilebel nawils amtkicebs.

(sakmarisoba) vTqvaT F simravle Caketilia da misi yoveli Ria dafar-

vidan SeiZleba sasruli Ria dafarvis gamoyofa. vigulisxmoT, rom M ar-is F-is qvesimravle, romelsac arcerTi zRvariTi wertili ar gaaCnia. ma-

Sin x∈F wertilisaTvis iarsebebs birTvi B(x,εx), romelic M simravlis

erT x wertils Tu Seicavs. aseTi birTvebis erToblioba warmoadgens M

simravlis dafarvas. misgan gamovyoT sasruli dafarva 1

( , )n

i ii

B x ε=∪ . radgan

M⊂1

( , )n

i ii

B x ε=∪ da TiToeuli birTvi M simravlis mxolod erT elements Tu

Seicavs, amitom M sasruli simravlea. amrigad, Tu F–is nebismier qvesim-ravles zRvruli wertilebi ar aqvs, maSin igi sasruli simravlea; e.i.

F–is yoveli usasrulo qvesimravle Seicavs zRvrul wertils, rac F–is Caketilobis gamo mis Tavis TavSi kompaqturobas niSnavs.

sazogadod, metrikuli sivrceebisaTvis Cven ukve SeviswavleT am siv-

rceebSi kompaqturobis pirobebi. aRsaniSnavia, rom konkretuli sivrce-

ebisaTvis kompaqturobis pirobebi sxvagvaradac SeiZleba Camoyalibdes.

35

qvemoT Cven ganvixilavT kompaqturobis kriteriums mxolod C[0,1] sivr-cisTvis. amasTan dakavSirebiT CamovayaliboT ori ganmarteba.

C[0,1] sivrcis SemosazRvruli M simravlis funqciaTa sistemas ewode-

ba Tanabrad SemosazRvruli; anu vityviT, rom M simravlis funqciaTa

sistema aris Tanabrad SemosazRvruli, Tu arsebobs iseTi dadebiTi mud-

mivi C, rom yoveli x-Tvis M simravlidan da nebismieri t wertilisaTvis

segmentidan |x(t)|≤C. funqciaTa sistemas M⊂C[0,1] ewodeba Tanabarxarisxovnad uwyveti, Tu

nebismieri x funqciisaTvis M-dan da yoveli dadebiTi ε ricxvisaTvis ar-

sebobs iseTi δ=δ(ε), rom nebismieri t1 da t1 wertilisaTvis [0,1] segmentidan, romelTaTvisac |t1-t2|<δ, gvaqvs | x(t1)-x(t2)|.


Teorema 7.6 (arcela). aucilebeli da sakmarisi piroba imisaTvis,

rom simravle K⊂C[0,1] iyos kompaqturi, mdgomareobs SemdegSi: funqciaTa

sistema K simravlidan unda iyos Tanabrad SemosazRvruli da Tanabarxa-

risxovnad uwyveti.

am debulebas Cven aq ar davamtkicebT.

§8. topologiuri sivrceebi. uwyveti asaxva

metrikuli sivrcis ZiriTadi cnebebi (zRvruli wertili, Sexebis wer-

tili, simravlis Caketva da a.S.) SemoRebul iqna midamos cnebaze dayrd-

nobiT an, rac arsebiTad igivea, Ria simravlis cnebaze dayrdnobiT. Ta-

vis mxriv, es ukanaskneli cnebebi (midamo, Ria simravle) ganmartebuli

iyo metrikis daxmarebiT. SesaZlebelia Cven sul sxva gzas davadgeT –

simravleze ar SemovitanoT metrika, aramed Ria simravleebi SemoviRoT

uSualod aqsiomaturad. es gza uzrunvelyofs moqmedebis ufro met Ta-

visuflebas. am gzas Cven mivyavarT Semdeg ganmartebamde.

vTqvaT X nebismieri simravlea. X simravleze topologia ewodeba X–is qvesimravleTa raime τ sistemas, Tu igi akmayofilebs Semdeg sam pirobas:

1. TviT X da carieli simravle ∅ ekuTvnian τ sistemas. 2. τ sistemis Gα simravleebis nebismieri (sasruli an usasrulo) ga-

erTianeba Gαα∪ kvlav ekuTvnis τ sistemas.

3. τ sistemis Gα simravleebis yoveli sasruli TanakveTa 1

n

kk

G=∩ kvlav τ

sistemas ekuTvnis. wyvils (X,τ) ewodeba topologiuri sivrce. τ topologiis TiToeul Gα

simravles Ria simravle ewodeba. Ria G∈τ simravlis damatebiT simravles

X\G ewodeba Caketili. am ganmartebidan gamomdinareobs, rom carieli sim-

ravle ∅ da X Caketili simravleebia (ratom?) da (simravleTaTvis ora-

dobis principis gamoyenebiT miviRebT): Caketil simravleTa TanakveTa Ca-

ketilia. sasruli raodenoba Caketil simravleTa gaerTianeba Caketili

simravlea (ratom?).

36

x∈X wertilis midamo ewodeba nebismier Ria simravles, romelic Sei-

cavs x wertils. vTqvaT M⊂X. vityviT, rom x∈M aris M simravlis Sexebis

wertili, Tu x wertilis nebismieri midamo Gx∈τ Seicavs M simravlis

erT wertils mainc. M simravlis zRvruli wertili da izolirebuli

wertili zustad imgvarad ganimarteba, rogorc metrikul sivrceSi Sesa-

bamisi cnebebi. aseve analogiurad ganisazRvreba simravlis Caketva .M

ganvixiloT topologiuri sivrcis magaliTebi.

1. metrikuli sivrcis Ria simravleebi akmayofileben topologiuri

sivrcis ganmartebis samive aqsiomas. amitom yoveli metrikuli sivrce

topologiuricaa.

2. vTqvaT X raime simravlea. τ sistemis elementad (Ria simravled) Cav-

TvaloT X-is nebismieri simravle, e.i. τ sistemis rolSi ganvixiloT X-is yvela qvesimravleTa sistema. cxadia, τ daakmayofilebs samive aqsiomas

(ratom?). am topologias diskretul topologias uwodeben.

3. vTqvaT X raRac simravlea. davuSvaT τ ={X,∅}. maSin or elementiani

τ sistemac daakmayofilebs samive aqsiomas (ratom?).

vTqvaT X Sedgeba ori a da b elementisagan. amasTan Ria simravleebad

CavTvaloT {a,b}, ∅, b. am SemTxvevaSi samive aqsioma Sesrulebulia (ra-

tom?). am sivrceSi Caketili simravleebi Semdegia: X, ∅, a (ratom?). amasTan

{b} simravlis Caketva mTeli X={a,b} simravlea (ratom?).

vTqvaT erTi da igive X simravleze mocemulia ori topologiaτ1 da τ2.

amrigad, gvaqvs ori topologiuri sivrce (X, τ1) da (X, τ2). vityviT, rom τ1

topologia ufro Zlieria vidre τ2 (an τ2 topologia ufro sustia vidre τ1), Tu τ1 sistema Seicavs τ2–s.

Teorema 8.1. vTqvaT X-ze mocemulia τα (α∈A) topologiaTa sistema. ma-

Sin τ=A

αα

τ∈∩ topologiaa, romelic sustia nebismier τα topologiaze.

damtkiceba. yvela τα Seicavs ∅ da X–s, amitom τ=A

αα

τ∈∩ ⊃{X,∅}. vTqvaT Gβ

(β∈B) ekuTvnis τ-s, e.i. – nebismier τα –s. radgan τα topologiaa, amitom

B

Gβ αβ

τ∈

∈∪ (α nebismieria A-dan). amrigad, B

Gββ

τ∈

∈∪ . analogiurad vimsje-

lebT me-3 aqsiomis Sesamowmebladac (SeamowmeT).

rogorc vnaxeT, X simravleze topologiis mocema niSnavs X-ze Ria

simravleTa τ sistemis mocemas, romelic sam aqsiomas akmayofilebs. mag-

ram konkretul SemTxvevaSi sakmarisia mTeli τ sistemis magivrad ganvixi-loT τ–s raime qvesistema, romelic calsaxad gansazRvravs τ sistemas. ma-galiTad, metrikul sivrceSi jer ganvixileT Ria birTvis cneba da Sem-

deg SemoviReT Ria simravlis cneba. metrikul sivrceSi Ria simravlis

ganmartebis ZaliT, simravle Riaa maSin da mxolod maSin, roca es sim-

ravle SeiZleba warmodgenil iqnas Ria birTvTa gaerTianebis saxiT (ra-

tom?). ukanaskneli msjeloba gvikarnaxebs topologiur sivrceSi Semovi-

RoT Semdegi ganmarteba.

vTqvaT mocemulia (X,τ) topologiuri sivrce. τ topologiis τ ∗ qvesis-

temas (τ ∗⊂τ) ewodeba τ topologiis baza (xSirad amboben - (X,τ) topolo-

giuri sivrcis baza), Tu τ –s nebismieri elementi (e.i. nebismieri Ria sim-

ravle (X,τ) topologiuri sivrcidan) SeiZleba warmodgenil iqnas ro-

gorc τ ∗–is elementebis sasruli an usasrulo gaerTianeba. magaliTad,

37

metrikul sivrceSi yvela birTvTa simravle nebismieri centrebiTa da

radiusebiT qmnian metrikuli sivrcis bazas. kerZod, wrfeze yvela

intervalTa simravle qmnis bazas wrfeze. wrfeze bazas qmnis raci

onalur boloebian intervalTa simravlec ki. marTlac, wrfeze yoveli

intervali SeiZleba racionalurboloebiani intervalebis gaerTianebiT

warmovadginoT (ratom?), xolo, Tavis mxriv, yoveli Ria simravle wrfeze

warmoidgineba Ria intervalebis gaerTianebis saxiT (ratom?).

Teorema 8.2. imisaTvis, rom τ ∗⊂τ iyos τ topologiis baza aucilebe-

li da sakmarisia, rom yoveli Ria G∈τ simravlisaTvis da nebismieri x∈G

wertilisaTvis arsebobdes simravle Gx∈τ ∗, iseTi rom x∈Gx⊂G.

damtkiceba. vTqvaT Teoremis piroba sruldeba, maSin, cxadia,

G=x G

x∈∪ ⊂ x

x G

G∈∪ ⊂

x G

G∈∪ =G.

amrigad, G= xx G

G∈∪ . radgan Gx∈τ ∗, amitom τ ∗

aris τ topologiis baza. piri-

qiT, Tu τ ∗ aris τ topologiis baza, maSin yoveli G∈τ warmoidgineba τ ∗-is

raRac elementTa gaerTianebiT. ase rom, yoveli x∈G miekuTvneba am gaer-

Tianebis romeliRac Gx simravles (Gx∈τ ∗). amis garda, cxadia, Gx⊂G. vityviT, rom topologiuri sivrce aris Tvladi baziT, Tu am sivrce-

Si arsebobs Tvladi baza. Tu (X,τ) topologiuri sivrce aris Tvladi ba-

ziT, maSin igi aris separabeluri, e.i. moiZebneba Tvladi yvelgan mkvrivi

simravle (anu Tvladi simravle, romlis Caketva emTxveva X-s). marTlac,

vTqvaT {Gn}, n=1,2,…, aris Tvladi baza. yovel Gn-Si avirCioT erTi xn el-

ementi. radgan nebismieri G Ria simravle (da, maSasadame nebismieri mida-

moc) warmoidgineba rogorc raRac Gn-ebis gaerTianeba, amitom Tvladi

{xn} simravle mkvrivia X-Si. amrigad, X-Si moTavsdeba erTi xn wertili ma-

inc. es {xn} simravlis yvelgan mkvrivobas niSnavs.

metrikuli sivrceebisaTvis adgili aqvs sapirispiro debulebasac: Tu

metrikuli sivrce (X,ρ) separabeluria, maSin masSi arsebobs Tvladi ba-

za. marTlac, (X,ρ) metrikul sivrceSi aseT bazas qmnis B(xn,1/m) birTvTa

sistema, sadac {xn} raRac mkvrivi simravlea X-Si, xolo m da n nebismieri naturaluri ricxvebia (ratom?).

topologiuri sivrcis separabelurobidan, sazogadod, ar gamomdina-

reobs, rom es sivrce aris Tvladi baziT (Sesabamis magaliTs Cven aq ar

ganvixilavT).

metrikuli sivrceebidan topologiuri sivrceebisaTvis martivad gada-

itaneba mimdevrobis krebadobis cneba. kerZod, (X,τ) topologiuri sivr-

cis (xn) mimdevrobas ewodeba krebadi x elementisaken (x∈X), Tu x-is nebis-mieri midamo Seicavs am mimdevrobis yvela wevrs garkveuli indeqsidan

dawyebuli.

mimdevrobis kerebadobis cneba topologiuri sivrceebisaTvis ar as-

rulebs iseT fundamentur rols, rogorsac - metrikuli sivrceebisTvis.

es imis gamo xdeba, rom (X,ρ) metrikuli sivrcis x wertili amave sivrcis

M simravlis Sexebis wertilia maSin da mxolod maSin, roca M simravleSi arsebobs mimdevroba, romelic krebadia x–ken. sazogadod,

nebismieri topologiuri sivrcis SemTxvevaSi ukanasknel faqts adgili

ar aqvs. iqidan, rom topologiur sivrceSi x aris M-is Sexebis wertili,

ar gamomdinareobs M simravleSi iseTi mimdevrobis arseboba, romelic

krebadi iqneba x–ken. magaliTisaTvis ganvixiloT X=[0,1]. Ria

38

simravleebad, cariel simravlesTan erTad, CavTvaloT [0,1] segmentis iseTi qvesimravleebi, romlebic miiRebian [0,1] segmentidan araumetes

Tvladi wertilebis amogdebiT. advili Sesamowmebelia, rom aseT qvesim-

ravleTa sistema X=[0,1] simravleze topologias qmnis (SeamowmeT). amri-

gad, miviRebT topologiur sivrces. aseT sivrceSi krebadni iqnebian

mxolod stacionaruli mimdevrobebi. marTlac, vTqvaT lim .nnx x

→∞= ganvixi-

loT x–is iseTi midamo, romelSic ar iqneba moTavsebuli (xn) mimdevrobis

arcerTi wevri, romlebic x–gan gansxvavebulia (magaliTad, ganvixilavT

simravles [0,1]\{xn}∪{x}). Tu es mimdevroba ar aris stacionaruli (e.i. Tu

am mimdevrobis wevrebi garkveuli indeqsidan dawyebuli x–s ar emTxveva),

maSin (xn) mimdevrobis krebadobis ganmartebis ZaliT, es mimdevroba ar iq-

neba krebadi x–ken. axla M simravlis rolSi ganvixiloT intervali (0,1], xolo x–is rolSi – 0. cxadia, rom x aris (0,1] intervalis Sexebis werti-

li (ratom? gaixseneT midamos ganmarteba am topologiur sivrceSi),

magram M=(0,1] simravlidan aRebuli arcerTi mimdevroba ar aris krebadi

0-ken. marTlac, vTqvaT aseTi (xn) mimdevroba moiZebna. maSin [0,1]\{xn} sim-

ravle, romelic waroadgens 0-is erT-erT midamos, ar Seicavs (xn) mimdev-

robis arcerT wevrs. Uukanaskneli (xn) mimdevrobis 0-ken krebadobas uar-

yofs.

topologiur sivrceSi krebad mimdevrobebs sxva “arabunebrivi” Tvise-

bac gaaCnia. rogorc cnobilia, Tu mimdevroba krebadia metrikul sivrce-

Si, maSin mas erTaderTi zRvari gaaCnia. sazogadod, topologiur sivrce-

Si es ase ar aris. magaliTad, ganvixiloT nebismieri aracarieli simrav-

le X. am simravleSi SemoviRoT topologia: τ={X,∅}. maSin am sivrceSi ne-

bismier (xn) mimdevrobas eqneba zRvari da es zRvari X-is nebismieri ele-

menti iqneba. marTlac, ganvixiloT nebismieri wertili x∈X. mis yovel mi-

damoSi (Cvens SemTxvevaSi aseTi midamo erTaderTia da igi emTxveva X-s) moTavsdeba (X,τ) topologiuri sivrcis nebismieri mimdevrobis yvela wev-

ri.

axla ganvixiloT zogierTi sakiTxi, romlebic topologiur sivrceTa

asaxvebs ukavSirdeba.

vTqvaT mocemuli gvaqvs ori topologiuri sivrce (X,τx) da (y,τy). vigu-

lisxmoT, rom f : X→Y, x0∈X da y0=f(x0)∈Y. vityviT, rom f uwyvetia x0 wer-

tilze, Tu y0–is nebismieri 0yU midamosaTvis (

0y yU τ∈ ) arsebobs x0–is iseTi

0xV midamo (0x xV τ∈ ), rom f(

0xV )⊂0yU . asaxvas f : X→Y ewodeba uwyveti X-ze (an

ubralod, uwyveti), Tu is uwyvetia nebismier x∈X wertilze. advili dasa-

naxia, rom ukanaskneli gansazRvra metrikuli sivrceebisTvis ukve cno-

bil uwyveti asaxvis cnebas emTxveva (ratom?).

Camoyalibebuli ganmarteba “lokaluri” xasiaTisaa. aRsaniSnavia, rom

es cneba Ria simravleTa terminebSi SeiZleba gamoiTqvas.

Teorema 8.3. imisaTvis, rom asaxva f topologiuri (X,τx) sivrcisa

(Y,τy) topologiur sivrceSi iyos uwyveti aucilebeli da sakmarisia, rom

(Y,τy) sivrcis yoveli Ria G simravlisaTvis f -1(G) iyos topologiuri (X,τx)

sivrcis Ria simravle.

damtkiceba. (aucilebloba) vTqvaT f uwyveti asaxvaa, xolo G Ria sim-

ravlea (Y,τy) topologiuri sivrcidan. davamtkicoT, rom f -1(G) Ria simrav-

lea (x,τx) sivrceSi. vTqvaT x∈ f -1(G) da y=f(x). maSin y∈G. amrigad, G simrav-

39

le y wertilis midamoa. f asaxvis x wertilSi uwyvetobis gamo moiZebneba x wertilis Vx midamo, rom f(Vx)⊂G, saidanac miviRebT Vx⊂ f -1

(G). maSasadame,

f -1(G) simravlidan aRebuli nebismieri x wertilisTvis arsebobs misi ise-

Ti Vx midamo, rom Vx⊂ f -1(G), es ki niSnavs, rom (y,τy) sivrcidan aRebuli ne-

bismieri Ria G simravlisaTvis f -1(G) Riaa.

(sakmarisoba) vTqvaT (Y,τy) topologiuri sivrcis nebismieri Ria G sim-

ravlisaTvis f -1(G) Riaa (x,τx) sivrceSi. aviRoT nebismieri wertili x∈X da

y=f(x)–is nebismieri Uy midamo (Uy∈τy). maSin Teoremis pirobis Tanaxmad

f -1(Uy) Ria simravlea. amis garda, radgan f(x)∈Uy, amitom x∈ f -1

(Uy). Semovi-

RoT aRniSvna f -1(Uy)≡Vx. cxadia, f(f -1

(Uy))⊂Uy, e.i. f(Vx)⊂Uy. amrigad, Tu y=f(x) da Uy aris y wertilis midamo, maSin moiZebneba x wertilis iseTi Vx mi-

damo, rom f(Vx)⊂Uy. ukanaskneli f asaxvis uwyvetobas niSnavs x wertilSi.

radgan x nebismieri wertilia X–dan, amitom f uwyveti asaxvaa. SevniSnoT, rom Tu X da Y nebismieri simravleebia, f : X→Y da Y simrav-

leze mocemulia raime τy topologia, maSin am topologiis winasaxe f asa-xvis dros (anu erToblioba yvela f -1

(G) simravleebisa, sadac G∈τy ) qmnis X simravleze topologias. marTlac, sakmarisia gavixsenoT, rom samarT-

liania tolobebi 1 1( ) ( )

A A

f G f Gα αα α

− −

∈ ∈

=∪ ∪ da 1 1( ) ( ).

A A

f G f Gα αα α

− −

∈ ∈

=∩ ∩

miRebul topologias aRvniSnavT simboloTi f -1(τ) (SeamowmeT, rom f -1(τ) topologiaa).

vTqvaT axla mocemulia (X,τx) da (y,τy) topologiuri sivrceebi da

f : X→Y. maSin 8.3 Teoremis ZaliT (y,τy) sivrcis yoveli Ria G simravlisa-

Tvis f -1(G) Ria simravlea (X,τx) sivrceSi, e.i. f -1

(τy)⊂τx, rac niSnavs, rom f -1

(τy) topologia ufro sustia vidre τx. piriqiT, Tu f -1(τy) ufro sustia

vidre τx, maSin f -1(τy)⊂τx; e.i. τy-is yoveli Ria simravlisaTvis f -1

(G)∈τx, anu

f -1(G) Ria simravlea (X,τx) sivrceSi. ukanaskneli f funqciis uwyvetobas

niSnavs. maSasadame, samarTliania Semdegi

Teorema 8.4. vTqvaT mocemuli gvaqvs (X,τx) da (Y,τy) topologiuri

sivrceebi. asaxva f : X→Y uwyvetia maSin da mxolod maSin, roca f -1(τy) uf-

ro susti topologiaa vidre τx.

Teorema 8.3-dan da im faqtidan, rom damatebiTi simravlis winasaxe ar-

is winasaxis damateba, gamomdinareobs

Teorema 8.5. imisaTvis, rom f asaxva (X,τx) topologiuri sivrcisa (Y,τy)

topologiur sivrceSi iyos uwyveti aucilebeli da sakmarisia, rom (Y,τy)

sivrcis yoveli Caketili simravlis winasaxe iyos Caketili (X,τx) topo-

logiur sivrceSi.

(mkiTxvels vTxovT daamtkicos es Teorema.)

40

§9. wrfivi sivrcis cneba. faqtor-sivrce. wrfivi funqcionali; misi geometriuli

azri. amozneqili simravleebi da

funqcionalebi

maTTvis, visac Seswavlili aqvs algebris elementaruli kursi, kar-

gadaa cnobili wrfivi sivrcis cneba. miuxedavad amisa, Cven kidev erTxel

SevexebiT mas.

aracariel L simravles ewodeba wrfivi sivrce (veqtoruli sivrce),

Tu igi akmayofilebs Semdeg pirobebs:

1. L simravlis nebismieri x da y elementTaTvis (x,y∈L) calsaxad gani-

sazRvreba mesame elementi z∈L, romelsac x da y elementTa jams vuwo-

debT da ase aRvniSnavT x+y; amasTan am sivrcis nebismieri sami x,y,z ele-

mentisaTvis a) x+y= y+x, b) x+(y+z)=(x+y)+z, g) L-Si arsebobs iseTi elementi

0, rom nebismieri x∈L elementisaTvis x+0=x, d) nebismieri x∈L elemen-

tisaTvis arsebobs elementi -x, iseTi, rom x+(-x)=0. 2. nebismieri α ricxvisaTvis da x∈L elementisaTvis gansazRvrulia

namravli x elementisa α ricxvze - α x, amasTan α da β ricxvebisaTvis: a)

α(β x)=(αβ)x, b) 1⋅ x=x, g) (α+β)x=α x+ β x, d) α(x+y)= α x+αy. imis mixedviT, Tu rogori ricxvebi (namdvili Tu kompleqsuri) gamoi-

yeneba mocemuli sivrcisaTvis, miviRebT, Sesabamisad, namdvil an kompleq-

sur wrfiv sivrceebs.

ganvixiloT wrfiv sivrceTa magaliTebi.

1. namdvil ricxvTa R sivrce wrfivi sivrcea. ukanaskneli advili Ses-

amCnevia, Tu gaviTvaliswinebT namdvil ricxvTa Sekrebisa da gamravlebis

Tvisebebs.

2. npR sivrce (p≥1) wrfivi sivrcea, Tu ki Sekrebisa da ricxvze gamrav-

lebis operaciebs ase ganvmartavT: (x1,x2,...,xn)+(y1,y2,...,yn)=(x1+y1, x2+y2,..., xn+yn),

α(x1,x2,...,xn)=(α x1,α x2,...,α xn) (SeamowmeT, rom am SemTxvevaSi npR wrfivi

sivrcea).

3. lp (p≥1) wrfivi sivrcea, Tu ki Sekrebisa da ricxvze gamravlebis op-

eraciebs ase SemoviRebT: (x1,x2,...,xn,...)+(y1,y2,...,yn,...)=(x1+ y1, x2+ y2,...,xn+ yn,...), α(x1, x2,...,xn,...)=(αx1,αx2,...,αxn,...). Tu x,y∈lp, maSin nebismieri α da β ricxvebi-saTvis

α x +β x∈lp (SeamowmeT, gamoiyeneT minkovskis ganzogadebuli utolo-ba (ix. (1.6))).

4. Tu elementTa Sekrebisa da ricxvze gamravlebis operaciebs iseve

ganvmartavT, rogorc lp sivrcis SemTxvevaSi, maSin yvela ricxviT mimdev-

robaTa sivrce (§2, punqti 7) wrfiv sivrced gadaiqceva. 5. m da c0 sivrceebi wrfivi sivrceebia. am faqtis damtkiceba lp (p≥1)

sivrceebisaTvis Catarebuli msjelobis analogiuria.

6. C[a,b] sivrce wrfivi sivrcea. [a,b] segmentze ori uwyveti f da g funq-ciis jami, rogorc kargadaa cnobili, iseTi f +g funqciaa, rom nebismieri x∈[a,b] ricxvisaTvis (f+g)(x)=f(x)+g(x); amasTan αf , sadac α skalaria, xolo

f∈C[a,b], aris iseTi funqcia, rom yoveli x∈[a,b] ricxvisaTvis (αf)(x)=αf(x) (SeamowmeT, rom amgvarad ganmartebuli operaciebis Sedegad C[a,b] sivrce wrfiv sivrced gadaiqceva).

41

7. Lp[a,b] sivrce (p≥1) wrfivi sivrcea (SeamowmeT. gamoiyeneT minkovskis

integraluri utoloba (ix. (1.8))). vityviT, rom wrfivi Lx da Ly sivrceebi izomorfulia (maT Soris arse-

bobs izomorfizmi), Tu Lx da Ly simravleebs Soris arsebobs iseTi bieq-

cia ϕ (ϕ : Lx → Ly), romelic iqneba adiciuri (yoveli x1, x2∈Lx wertilis-

Tvis ϕ(x1+x2)=ϕ(x1)+ϕ(x2)) da erTgvarovani (nebismieri α ricxvisTvis da yo-

veli x∈Lx elementisTvis αϕ(x)=ϕ (αx)). L wrfivi sivrcis x, y,..., ω elementebs ewodebaT wrfivad damoukidebe-li,

Tu ar arsebobs iseTi ricxvebi α, β,..., λ, romelTagan erTi mainc gansxva-

vebulia 0-gan da αx+βy+...+λω=0. wrfiv L sivrces ewodeba n-ganzomilebiani, Tu am sivrceSi moiZebneba

n cali wrfivad damoukidebeli elementi (veqtori), xolo am sivrcis ne-

bismieri n+1 cali elementi aris wrfivad damokidebuli (e.i. ar aris

wrfivad damoukidebeli).

wrfiv L sivrces ewodeba usasruloganzomilebiani, Tu igi arcerTi

naturaluri n_Tvis ar aris n-ganzomilebiani, anu Tu nebismieri dasaxe-

lebuli naturaluri n ricxvisTvis am sivrceSi moiZebneba n cali wrfi-

vad damoukidebeli veqtorTa sistema.

L′ simravles ewodeba L wrfivi sivrcis qvesivrce, Tu L′ ⊂ L da aris wrfivi mravalsaxeoba (anu, Tu x,y∈ L′ , maSin nebismieri α da β ricxvebi-saTvis αx+βy∈ L′ ; ra Tqma unda, α da β namdvili ricxvebia an kompleqs-

uri, imisda mixedviT, Tu rogori skalarebis mimarT aris L wrfivi sivr-ce). magaliTad, namdvil ricxvTa R sivrce aris n

pR (n≥2, p=1) sivrcis (anu kargad cnobili Rn sivrcis) qvesivrce. C[a,b] sivrcis qvesivrces warmoad-gens [a,b] segmentze gansazRvrul mravalwevrTa simravle (ratom?). m sivr-cis qvesivrcea c0, xolo c0 sivrcis qvesivrcea lp (p≥1) (ratom?).

vTqvaT {xα} aris wrfivi L sivrcis aracarieli qvesimravle. maSin L sivrceSi arsebobs umciresi qvesivrce, romelic {xα} simravles Seicavs.

L-Si erTi qvesivrce mainc arsebobs, romelic {xα} simravles Seicavs;

magaliTad, - L sivrce. amis garda, cxadia, rom L-is Lγ (γ∈Γ) qvesivrceTa

TanakveTa kvlav L-is qvesivrcea. marTlac, vTqvaT L′= Lγγ∈Γ∩ da x,y∈ L′ . maSin

imis gamo, rom yoveli Lγ aris qvesivrce, miviRebT, rom nebismieri α,β∈R ricxvebisaTvis αx+βy∈Lγ. maSasadame, αx+βy∈ L′ . axla aviRoT L sivrcis yve-la qvesivrce, romelic {xα} simravles Seicavs da ganvixiloT aseT qve-

sivrceTa TanakveTa. cxadia, L sivrcis nebismieri qvesivrce, romelic

{xα} simravles Seicavs, agreTve moicavs ganxilul TanakveTasac. amrigad,

miiReba L sivrcis minimaluri qvesivrce, romelic {xα} simravles Seicavs.

aseT minimalur qvesivrces vuwodebT L sivrcis qvesivrces, romelic war-

moSobilia {xα} sistemiT (simravliT), an sxvagvarad, - vuwodebT {xα} sis-

temis garss.

axla SemoviRoT faqtor-sivrcis cneba. vTqvaT L aris wrfivi sivrce, xolo L′ - misi raRac qvesivrce. vityviT, rom L sivrcis ori x da y ele-

menti tolfasia, Tu x-y∈ .L′ advili Sesamowmebelia, rom es Tanadoba ref-

leqsuri, simetriuli da tranzitulia. es imas niSnavs, am Tanadobis mi-

marT L sivrce daiyofa TanaukveT klasebad. am klasebs momijnave kla-

sebs vuwodebT L′ qvesivrcis mimarT. aseT klasTa erTobliobas aRvniS-

42

navT simboloTi L/ L′ da mas vuwodebT L sivrcis faqtor-sivrces ( L′ qve-sivrcis mimarT). am sivrceSi SesaZlebelia Sekrebisa da skalarze gamrav-

lebis operaciebis SemoReba. vTqvaT ξ da η ori momijnave klasia, anu

ξ,η∈L/ L′ . maTgan aviRoT TiTo elementi x da y (x∈ξ da y∈η). radgan x+y∈L, amitom is erT-erTi momijnave klasis elementia. ξ+η vuwodoT im momijna-

ve klass, romelic Seicavs x+y elements. Tu α raime ricxvia, maSin α ri-cxvisa da ξ klasis namravli αξ vuwodoT im klass, romelic Seicavs

αx–s (x∈ξ). advili Sesamowmebelia, rom ξ+η–isa da αξ–is ganmarteba ar aris damokidebuli ξ da η klasebidan x da y elementebis arCevaze (ra-

tom?). aseve, ar aris Zneli saCvenebeli, rom es operaciebi akmayofileben

wrfivi sivrcis ganmartebaSi mocemul yvela pirobas. amrigad, yoveli

faqtor-sivrce SeiZleba ganvixiloT rogorc wrfivi sivrce.

axla davubrundeT wrfivi sivrcis ganzomilebis sakiTxs da ganvixi-

loT sasrulganzomilebiani da usasruloganzomilebiani wrfivi sivrce-

ebi.

1. namdvil ricxvTa R sivrce aris erTganzomilebiani sivrce. am sivr-

ceSi nebismieri aranulovani elementi qmnis wrfivad damoukidebel sis-

temas, xolo nebismieri ori elementi (ori ricxvi) ukve wrfivad damoki-

debulia.

2. npR sivrcis (p≥1) ganzomileba n-is tolia. marTlac, am sivrceSi ar-

sebobs n wrfivad damoukidebel elementTa (veqtorTa) Semdegi sistema:

(1,0,0,…,0), (0,1,0,…,0),…, (0,0,0,…,1) (aCveneT, rom es sistema aris wrfivad da-

moukidebeli), xolo nebismieri n+1 raodenoba veqtorebisa wrfivad da-mokidebelia (ratom?).

3. lp (p≥1), yvela ricxviT mimdevrobaTa sivrce, m da c0 sivrce usasru-

loganzomilebiani sivrceebia. marTlac, am sivrceebSi Semdegi usasru-

lo simravlidan aRebuli nebismieri sasruli qvesimravle wrfivad damo-

ukidebel sistemas qmnis: {(1,0,0,...,0,...), (0,1,0,...,0,...),..., (0,0,0,...,1,...)} (aCve-neT). 4. C[a,b] da Lp[a,b], p≥1, usasruloganzomilebiani wrfivi sivrceebia.

marTlac, funqciaTa t, t2, t3,..., tn,... sistemis nebismieri sasruli qvesistema

[a,b] segmentze wrfivad damoukidebelia. marTlac, vTqvaT α1,α2,...,αn ricx-

vTa raime sistemisaTvis da nebismieri t_Tvis (t∈[a,b]) α1t+α2t2+...+αn tn=0. (9.1)

vigulisxmoT, rom erTi mainc i0-Tvis 0 00, 1 .i i nα ≠ ≤ ≤ maSin radgan (gausis

Teoremis Tanaxmad) yovel k-rigis mravalwevrs aqvs k cali fesvi, amitom

(9.1) gantolebis fesvTa raodenoba ar SeiZleba aRematebodes n–s. meore mxriv [a,b] segmentis nebismieri wertili aris (9.1) gantolebis fesvi.

5. Tu L wrfivi sivrce n-ganzomilebiania, xolo misi qvesivrce ′L aris

k-ganzomilebiani, maSin L/ L′faqtor-sivrcis ganzomileba aris n-k. marT-lac, vTqvaT e1, e2, …, en aris bazisi L sivrceSi. radgan L′ aris k–ganzomi-lebiani, amitom bazisis veqtorTa Soris moiZebneba k cali veqtori (zo-

gadobis SeuzRudavad SeiZleba vigulisxmoT, rom es veqtorebia e1, e2, …, ek), rom L′ sivrcis nebismieri elementi warmoidgineba maTi wrfivi kombi-

naciis saxiT. maSasadame, e1, e2, ..., ek∈ L′ . es imas niSnavs, rom L sivrcis Ti-

Toeuli x veqtorisTvis gveqneba

x=x0+αk+1ek+1+αk+2ek+2+…+αnen, sadac x0∈ .L′ aqedan cxadia, rom koeficientTa TiToeuli αk+1 , αk+2 ,..., αn si-

stema gansazRvravs gansxvavebul momijnave klass (ratom?), sxva sityveb-

43

iT rom vTqvaT, wrfivad damoukidebel veqtorTa ek+1, ek+2,..., en sistema gan-

sazRvravs L/ L′faqtor-sivrcis TiToeul elements (TiToeul momijnave

klass). amrigad, L/ L′faqtor-sivrcis ganzomilebaa n- k. axla ganvixiloT (gavixsenoT) wrfivi operatoris cneba.

vTqvaT X da Y ori wrfivi sivrcea erTi da igive velis mimarT. wrfivi

A operatori ewodeba asaxvas A : X→Y, romelic nebismieri α da β skalare-

bisaTvis da x1, x2 ∈X elementebisaTvis akmayofilebs pirobas

A(αx1+β x2)= αA(x1)+βA(x2)≡αAx1+βAx2. ganvixiloT wrfivi operatoris ramdenime magaliTi. 1. vTqvaT L wrfivi sivrcea. davuSvaT nebismieri x∈L-Tvis Ax=x. cxadia,

aseTi operatori wrfivi operatoria (A : L→L). mas igivur operators uwo-deben da aRniSnaven I simboloTi.

2. vTqvaT L wrfivi sivrcea. vigulisxmoT, rom yoveli x∈L-Tvis Ax=0. maSin A operators 0-van operators uwodeben da aRniSnaven simboloTi 0.

3. ganvixiloT A wrfivi operatori, romelic n–ganzomilebian wrfiv

En sivrces asaxavs m–ganzomilebian wrfiv Em

sivrceSi. vigulisxmoT, rom

En sivrcis bazisia e1, e2, …, en veqtorTa sistema, xolo Em

-Si baziss qmnis

1 2, ,..., me e e′ ′ ′ sistema. gvaqvs x=1

n

i ii

x e=∑ (x∈En). A operatoris wrfivobis gamo Ax=

=1

n

i ii

x Ae=∑ . meore mxriv, Aei∈Em. amitom Aei=

1

m

k i kk

a e=

′∑ . amrigad, A operatori ga-

nisazRvreba amave operatoris mniSvnelobebiT bazisis ei (i=1,2,…,n) veqto-rebze. Aei mniSvnelobebi ki - ||aki|| matricis elementebiT. maSasadame,

Ax=1 1

.n m

ki i ki k

a x e= =

′∑∑

4. vTqvaT ϕ0∈C[a,b] fiqsirebuli funqciaa da ϕ∈C[a,b]. ganvsazRvroT A operatori tolobiT ψ(t)=ϕ0(t)⋅ϕ(t) (t∈[a,b]), anu Aϕ→ϕ0ϕ, e.i. A: C[a,b]→ C[a,b]. igi wrfivi operatoria. marTlac,

A(αϕ1+βϕ2)=ϕ0.(αϕ1+βϕ2)=αϕ0

.ϕ1+βϕ0.ϕ2=αAϕ1+βAϕ2.

Tu A operatoris mniSvnelobaTa simravle namdvili an kompleqsur

ricxvTa simravlis qvesimravlea, maSin aseT operators uwodeben funq-

cionals (Sesabamisad, - namdvil an kompleqsur funqcionals).

ganvixiloT wrfivi funqcionalis zogierTi magaliTi.

1. vTqvaT Rn aris n-ganzomilebiani sivrce, romlis elementebic arian

x=(x1,x2,…,xn), sadac xi∈R, i=1,2,…,n. vTqvaT a1,a2,…,an ricxvTa n–eulia. ganv-

sazRvroT f funqcionali Semdegnairad. Rn sivrcis nebismieri elementisa-

Tvis davuSvaT f(x)=1

n

i ii

a x=∑ . uanaskneli funqcionali aris wrfivi (ratom?).

2. vTqvaT ϕ0 uwyveti funqciaa [a,b] segmentze, anu ϕ0∈C[a,b]. davuSvaT ne-

bismieri ϕ∈C[a,b] funqciisaTvis

F(ϕ)= 0( ) ( ) .b

a

t t dtϕ ϕ∫

F funqcionalis wrfivoba uSualod gamomdinareobs integralis ZiriTa-

di Tvisebebidan (ratom?).

3. igive C[a,b] sivrceSi davuSvaT

0 0( ) ( )t tδ ϕ ϕ= (t0∈[a,b], ϕ∈C[a,b]).

44

advili saCvenebelia, rom 0t

δ aris wrfivi funqcionali (aCveneT).

4. ganvixoloT wrfivi funqcionalis magaliTi lp (p≥1) sivrceze.

vTqvaT x=(x1,x2,…,xk, xk+1,…)∈ lp da k raime fiqsirebuli naturaluri ricxvia.

fk funqcionali (k=1,2,…) ase ganvmartoT: fk(x)=xk. igi wrfivia (aCveneT). ana-

logiurad SeiZleba ganvmartoT wrfivi funqcionalebi c0 da m sivrceeb-ze.

wrfivi sivrceebis ganxilvisas mTel rig mniSvnelovan sakiTxebs saf-

uZvlad udevs amozneqilobis cneba. am cnebas aqvs naTeli geometriuli

interpretacia; amave dros igi SeiZleba ganimartos analizuradac.

vTqvaT L raime namdvili wrfivi sivrcea, xolo x da y aris am sivrcis ori elementi. L sivrcis Caketili monakveTi, romelic x da y wertilebs

aerTebs, vuwodoT L sivrcis yvela im elementTa simravles, romelTac

aqvT saxe αx+βy, α,β≥0, α+β=1. M⊂L simravles ewodeba amozneqili, Tu igi nebismier x da y wertile-

bTan erTad Seicavs maT SemaerTebel monakveTsac.

E (E⊂L) simravlis birTvi (mas aRvniSnavT simboloTi J(E)) ewodeba am simravlis yvela im x wertilTa erTobliobas, romelTaTvisac L sivrcis nebismieri y elementisaTvis arsebobs iseTi ricxvi ε≡ε(y)>0, rom roca

|t|<ε, maSin x+ty∈E. amozneqil simravles, romlis birTvic ar aris carieli, ewodeba amo-

zneqili sxeuli.

ganvixiloT ori magaliTi.

1. samganzomilebian ariTmetikul sivrceSi (R3-Si) kubi, birTvi da tet-

raedri warmoadgenen amozneqil simravleebs. am sivrceSi monakveTi, sibr-

tye da samkuTxedi amozneqili simravleebia, magram ar arian amozneqili

sxeulebi (ratom?). kubi, birTvi da tetraedri R3-Si amozneqili sxeule-

bia.

2. C[a,b] sivrceSi ganvixiloT yvela im f funqciaTa E simravle, romle-

bic akmayofileben pirobas |f(t)|≤1, t∈[a,b]. es simravle aris amozneqili.

marTlac, vTqvaT |f(t)|≤1 da |g(t)|≤1 (t∈[a,b]) da α+β=1, α,β≥0. maSin |αf(t)+β g(t)|≤|α||f(t)|+|β|| g(t)|≤α+β=1.

axla vaCvenoT, rom ganxiluli E simravlis birTvs Seadgens yvela im uw-

yvet f funqciaTa simravle, romlebic akmayofileben pirobas |f(t)|≤1, t∈[a,b]. marTlac, vTqvaT f∈C[a,b] da, |f(t)|<1, t∈[a,b]. radgan vaierStrasis Teoremis

Tanaxmad segmentze uwyveti funqcia aRwevs Tavis zeda sazRvars, amitom

yoveli f funqciisaTvis iarsebebs arauaryofiTi ricxvi C≡C(f)<1, rom yo-veli t∈[a,b] wertilisTvis |f(t)| ≤C(f). axla ganvixiloT nebismieri g funqcia C[a,b] sivrcidan. cxadia, g funqcia SemosazRvrulia. vTqvaT |g(t)| ≤M(g), t∈ ∈[a,b], da ricxvi ε≡ε(g) ase SevarCioT: 0<ε<(1-C(f))/M(g). radgan t∈[a,b], amitom gveqneba

|f(t)+εg(t)|≤ |f(t)|+ε|g(t)|≤C(f)+1 ( )( )

C fM g− M(g)=1,

rac niSnavs, rom yvela f funqciaTa simravle C[a,b] sivrcidan, romelTa-

Tvisac |f(t)|<1, warmoadgens birTvs yvela im funqciaTa simravlisaTvis,

romelTaTvisac |f(t)|≤1, roca t∈[a,b]. Tu M amozneqili simravlea, maSin misi birTvi J(M) aseve amozneqili

simravlea. marTlac, vTqvaT x,y∈J(M) da z=αx+βy, α,β≥0, α+β=1. vaCvenoT, rom

45

z∈J(M). J(M) simravlis ganmartebis ZaliT L sivrcis nebismieri a elementi-

saTvis iarsebebs dadebiTi ricxvebi ε1=ε1(a) da ε2=ε2(a), rom roca |t1|<ε1 da

|t2|<ε2, maSin x+t1a da x+t2a ekuTvnis M simravles (viTvaliswinebT, rom x,y∈ J(M)). advili SesamCnevia, rom z+ta=α(x+ta)+β(y+ta). amis garda, Tu vigulis-

xmebT, rom |t|<ε≡ε(a)=min(ε1(a), ε2(a)), maSin x+ta da y+ta wertilebi miekuTvneba

M simravles. radgan M amozneqili simravlea, amitom, α(x+ta)+β(y+ta)=z+ +ta∈M roca |t|<ε. maSasadame, z∈J(M).

amozneqil simravleTa nebismieri TanakveTa amozneqili simravlea (aC-

veneT).

amozneqili simravlis cnebasTan mWidrodaa dakavSirebuli amozneqili

erTgvarovani funqcionalis cneba. L wrfiv sivrceze gansazRvrul P funq-cionals ewodeba amozneqili, Tu P(αx+(1-α)y)≤αP(x)+(1-α)P(y) (x,y∈L, 0≤α≤1). funqcionals ewodeba dadebiTad erTgvarovani, Tu nebismieri x∈L–Tvis

da nebismieri α>0 ricxvisTvis P(αx)= αP(x). yoveli amozneqili, dadebiTad

erTgvarovani funqcionalisTvis samarTliania utoloba P(x+y)≤ P(x)+P(y). marTlac,

P(x+y)=22

x yP +⎛ ⎞⎜ ⎟⎝ ⎠

=22 2x yP ⎛ ⎞+⎜ ⎟

⎝ ⎠≤ P(x)+P(y).

xSirad, dadebiTad erTgvarovan, amozneqil funqcionals ubralod

erTgvarovan amozneqil funqcionals uwodeben. ganvixiloT aseTi funq-

cionalebis sami magaliTi.

magaliTi 1. yoveli wrfivi funqcionali aris erTgvarovani, amozneqi-

li funqcionali. erTgvarovani, amozneqili funqcionali iqneba agreTve

P(x)= |f(x)|, sadac f wrfivi funqcionalia (ratom?).

magaliTi 2. ganvixiloT sivrce 2nR sivrce. rogorc cnobilia, manZili

am sivrcis nebismier or x=(x1,x2,…,xn) da y=(y1,y2,…,yn) elements Soris ase

ganimarteba ρ(x,y)=1/ 2

2

1

( ) .n

i ii

x y=

⎛ ⎞−⎜ ⎟⎝ ⎠∑ manZili x wertilidan amave sivrcis 0=(0,0,

…,0) elementamde iqneba ρ(x,0)=1/ 2

2

1

| | .n

ii

x=

⎛ ⎞⎜ ⎟⎝ ⎠∑ ρ(x,0) aRvniSnoT ||x||-iT (qvemoT

Cven vnaxavT, rom am aRniSvnas gamarTleba eqneba). asaxva x→ρ(x,0)≡||x|| iq-neba erTgvarovani, amozneqili funqcionali (ratom?).

magaliTi 3. vTqvaT x=(x1,x2,…,xn,…) aris SemosazRvrul mimdevrobaTa m sivrcis elementi. maSin funqcionali P(x)= sup n

nx iqneba erTgvarovani da

amozneqili funqcionali (SeamowmeT).

§10. hanisa da banaxis Teorema

vTqvaT L namdvili wrfivi sivrcea, xolo L0 – misi qvesivrce. vigulis-

xmoT agreTve, rom L0 qvesivrceze gansazRvrulia raime wrfivi funqciona-

li f0. L sivrceze gansazRvrul f funqcionals ewodeba f0 funqcionalis

gagrZeleba, Tu f(x)= f0(x), roca x∈L0.

vityviT, rom A⊂L simravleze f0 funqcionali daqvemdebarebulia P funq-cionalze, Tu A simravlis yoveli x wertilisaTvis f0(x)≤P(x).

46

analizSi xSirad gvxvdeba funqcionalis gagrZelebis amocana. am mi-

marTulebiT yvelaze mniSvnelovania

Teorema 10.1 (hani da banaxi). vTqvaT P aris namdvil wrfiv L sivrceze gansazRvruli erTgvarovani, amozneqili funqcionali, xolo L0 am sivr-cis raRac qvesivrcea, romelzedac mocemulia P funqcionalze daqvemde-

barebuli f0 wrfivi funqcionali. maSin f0 ise SeiZleba gagrZeldes L siv-rceze gansazRvrul wrfiv f funqcionalamde, rom yoveli x wertilisa-

Tvis L sivrcidan f(x)≤P(x). damtkiceba. Tu L0=L, maSin dasamtkicebelic araferia (ratom?). amitom

vTqvaT, rom L0≠L. am SemTxvevaSi vaCvenoT, rom f0 funqcionali ise SeiZle-

ba gagrZeldes L0-ze ufro farTo L′ qvesivrceze (L0⊂L′⊂L) gansazRvrul

wrfiv f ′ funqcionalamde, rom f ′(x)≤ P(x), roca x∈L′. marTlac, vTqvaT z ar-is L sivrcis nebismieri fiqsirebuli elementi, romelic ar ekuTvnis L0

qvesivrces. vTqvaT L′≡L(z) aris tz+x wertilTa erToblioba (L′={tz+x}), sa-dac t nebismieri namdvili ricxvia, xolo x aseve nebismieradaa aRebuli L0

qvesivrcidan. L′ aris L sivrcis wrfivi qvesivrce. marTlac, L sivrcis

wrfivobis gamo, pirobidan z, x∈L gamomdinareobs , rom tz+x∈L. meore mxriv, Tu t1, t2, α, β nebismieri namdvili ricxvebia, xolo x1, x2∈L, maSin

α(t1z+x1)+β(t2z+x2)=(α t1+β t2)z+(αx1+βx2). radgan L0 aris L–is wrfivi qvesivrce, amitom αx1+βx2∈L0. amrigad, (αt1+

+β t2)z+(αx1+βx2) elements aqvs saxe tz+x. L′ qvesivrce “moWimulia” L0 qvesivrceze da z wertilze. ukanasknelis

gaTvaliswinebiT SemoviRoT aRniSvna L′={L0, z}. f′ funqcionali L′ qvesivrceze ase ganvmartoT: f′(tz+x)=tf′(z)+f0(x); cxadia,

f′(z)=f0(x), Tu x∈L0. amitom f′(tz+x)=tf ′(z)+f0(x). ganmartebuli funqcionali wrfivi

iqneba (ratom? SeamowmeT). ukanaskneli msjeloba samarTliania imisda

miuxedavad, Tu rogoraa ganmartebuli f ′(z) funqcionali z wertilze. ami-

tom f ′(z) mniSvneloba SeiZleba nebismierad aviRoT da Semdeg Cveni miznis

Sesabamisad SevarCioT. SemoviRoT aRniSvna f′(z)=C. C nebismieri ricxvia, amasTan

f ′(tz+x)=tC+ f0(x). (10.1) axla C-s nebismierobiT visargeblebT da mas ise SevarCevT, rom

f ′(tz+x)=tC+ f0(x)≤P(tz+x); (10.2) e.i. C ise unda SeirCes, rom L′ qvesivrceze f ′ funqcionali daqvemdebarebu-

li iyos P funqcionalze. vaCvenoT, rom es SesaZlebelia. Tu gamoviye-

nebT L0 qvesivrceze f0 funqcionalis wrfivobas da P funqcionalis dade-

biTad erTgvarovnebas, maSin dadebiTi t ricxvebisaTvis davaskvniT, rom

(10.2) Sefaseba Sesruldeba, Tu

f0(x/t)+C≤ P(x/t+z) ⇔ C≤P(x/t+z)- f0(x/t). (10.3) Tu igive msjelobas, amjerad uaryofiTi t_Tvis, CavatarebT, maSin mivi-

RebT

f0(x/(-t))-C≤ P(-z-x/t) ⇔ - f0(x/t)-C≤ P(-z-x/t) ⇔ C≥-P(-x/t-z)- f0(x/t). (10.4) amrigad, (10.2) utoloba dadebiTi t–Tvis tolfasia (10.3) utolobis, xo-

lo uaryofiTi t–Tvis eqvivalenturia (10.4)-is. Cveni uaxloesi mizania C mudmivis imgvarad SerCeva, rom saTanado t–Tvis Sesruldes (10.3) da (10.4) utolobebi. vTqvaT y′ da y′′ L0–is nebismieri elementebia. maSin, imis gamo,

rom L0 qvesivrceze f0 funqcionali daqvemdebarebulia P-ze, miviRebT f0(y′′-

47

y′)≤ P(y′′-y′), anu f0(y′′-y′)≤ P(y′′+z-y′-z). radgan P dadebiTad erTgvarovani amoz-

neqili funqciaa, amitom f0(y′′-y′)≤P(y′′+z)+P(-y′-z). f0 funqcionalis wrfivobis

gamo f0(y′′)- f0(y′)≤P(y′′+z)+P(-y′-z). aqedan - f0(y′′)+P(y′′+z)≥- f0(y′)- P(-y′-z). (10.5)

vTqvaT

C′′= 0inf ( ( ) ( ))y

f y p y z′′

′′ ′′− + + , C′= 0sup( ( ) ( )).y

f y p y z′

′ ′− − − −

radgan (10.5) utolobis marjvena mxare ar aris damokidebuli y′′–ze, xolo marcxena mxare - y′-ze, amitom C′≤ C′′. ricxvi C avirCioT nebismie-

rad, ise rom man daakmayofilos utoloba C′≤ C ≤C′′ (Tu C′<C′′, maSin ase-Ti C-s arCevis uamravi SesaZlebloba arsebobs. erT-erTi C-Tvis ganvmar-

toT f′ funqcionali (10.1) tolobiT. aseTi C-Tvis Sesruldeba utolobebi (10.3) da (10.4) (Sesamowmeblad sakmarisia dadebiTi t ricxvebisaTvis y′′-is nacvlad vigulisxmoT x/t, xolo uaryofiTi t-Tvis - y′ SevcvaloT x/t-Ti).

maTgan gamomdinareobs utoloba SerCeuli C-Tvis.

Tu zemoT Catarebul msjelobas SevajamebT, maSin davaskvniT, rom sa-

marTliania Semdegi: Tu f0 funqcionali L0⊂L qvesivrceze daqvemdebarebu-lia P funqcionalisadmi, maSin f0 SeiZleba gavagrZeloT L0-ze ufro far-

To L′ qvesivrceze (L0⊂L′⊂L) gansazRvrul f′ funqcionalamde, ise, rom yo-

veli x∈L′ wertilisaTvis f ′(x) ≤P(x). axla vigulisxmoT, rom L warmoSobilia Tvladi {x1, x2,…, xn,…} simrav-

liT; e.i. L-is minimaluri qvesivrce (da, maSasadame, yoveli qvesivrce), ro-

melic Seicavs {x1, x2,…,xn,…} sistemas, emTxveva L sivrces. “movWimoT” L0

qvesivrceze da x1 wertilze L(1)≡ {L0, x1}qvesivrce. Semdeg, vigulisxmoT,

rom L(2)≡{L(1), x2}. sazogadod, Tu naturaluri k-Tvis L(k ) qvesivrce ukve age-

bulia, davuSvaT L(k +1)≡{L(k ), xk+1}. cxadia, L(1)⊂ L(2)⊂…⊂L(k ) ⊂…. damtkicebulis

ZaliT, f0 funqcionalis gagrZeleba SeiZleba L(1) qvesivrceze gansazRvrul

f1 wrfiv funqcionalamde, romlisTvisac f0(x)≤P(x), x∈L(1). Semdeg, f1-s L(2)

qve-

sivrceze gavagrZelebT f2 wrfiv funqcionalamde, ise, rom f2(x)≤ P(x), x∈L(2).

Tu naturaluri k-Tvis fk funqcionali L(k) qvesivrceze ukve agebulia, ma-

Sin fk funqcionals L(k+1) qvesivrceze gavagrZelebT fk+1 wrfiv funqcional-

amde pirobiT fk+1(x)≤P(x), x∈L(k+1).AL sivrcis yoveli x wertili miekuTvneba

romeliRac L(k )–s. marTlac, L sivrcis yoveli x elementi unda warmoad-

gendes {x1, x2,…, xn,…} sistemis raime sasruli 1,nx

2,nx …,

inx (n1, n2,…, ni) qve-

sistemis elementebis wrfiv kombinacias. ( )inL qvesivrce (misi agebis Tanax-

mad) Seicavs am sasrul qvesistemasac. radgan ( )inL wrfivia, amitom igi Sei-

cavs am sasruli qvesistemis nebismier wrfiv kombinaciasac. maSasadame,

Seicavs x wertilsac. ase rom, ( )

1.k

kL L∞

==∪ L0-ze f funqcionali ase gan-

vmartoT f(x)= fk(x), Tu x∈L(k). f funqcionali gansazRvruli iqneba mTel L sivrceze da am sivrcis yovel x′ wertilSi akmayofilebs pirobas f(x′)≤ ≤P(x′). marTlac, Tu x′∈L, maSin arsebobs naturaluri k ricxvi, rom x′∈L(k)

,

e.i. f(x′)= fk(x′). radgan L(k) qvesivrceze fk(x)≤ P(x), amitom f(x′)≤ P(x′). f funqcio-

nali aris wrfivi. marTlac, x,y∈L. garkveulobisaTvis vigulisxmoT, rom

1kx L∈ , xolo 2ky L∈ da k1≤ k2. radgan 1 2( ) ( ) ,k kL L⊂ amitom

2, .kx y L∈ nebismieri

namdvili α da β ricxvebisaTvis gvaqvs f(αx+βy)=

2( )kf x yα β+ =

2 2( ) ( )k kf x f yα β+ = ( ) ( )f x f yα β+ .

48

Teorema ganxilul SemTxvevaSi damtkicebulia. Teoremis mtkiceba zo-

gad SemTxvevaSi emyareba e.w. amorCevis (cermelos) aqsiomas. am SemTxvevas

Cven ar ganvixilavT.

aqamde Cven vswavlobdiT topologiur sivrceebs da maT kerZo SemTxve-

vas – metrikul sivrceebs. meore mxriv, ganxiluli iyo wrfivi sivrceebi,

romlebSic Semotanili iyo metrika (daasaxeleT aseTi sivrceebis maga-

liTebi). am TvalsazrisiT mniSvnelovania e.w. normirebuli sivrceebi. as-

eTi sivrceebis mniSvnelovani nawili Seswavlil iyo poloneli maTema-

tikosis banaxis SromebSi.

vTqvaT L wrfivi sivrcea. L sivrceze gansazRvrul erTgvarovan, amozne-

qil P funqcionals ewodeba norma, Tu igi akmayofilebs pirobas: a) P(x)= =0, mxolod maSin, roca x=0; b) P(αx)=|α|P(x) nebismieri α–Tvis.

Tu gavixsenebT erTgvarovani, amozneqili funqcionalis ganmartebas,

martivad davrwmundebiT, rom wrfiv sivrceze gansazRvrul erTgvarovan,

amozneqil funqcionals ewodeba norma, Tu igi akmayofilebs pirobebs:

1. P(x)≥0, amasTan P(x)=0 mxolod maSin, roca x=0; 2. P(x+y)≤ P(x)+P(y), x, y∈L; 3. P(αx)=|α|P(x) nebismieri α ricxvisTvis.

normirebuli sivrce ewodeba iseT wrfiv sivrces, romelzec SemoRebu-

lia norma. x∈L elementis normas, rogorc wesi, aRniSnaven simboloTi ||x||, xolo TviT normas ase Caweren || ||⋅ . yuradReba unda mieqces erT gare-

moebas: || ||⋅ norma garkveuli tipis funqcionalia (igi aris asaxva), xolo x elementis ||x|| norma aris am arauaryofiTi || ||⋅ funqcionalis mniSvnelo-

ba x elementze.

srul normirebul sivrces banaxis sivrce ewodeba. mas xSirad B sivr-cesac uwodeben.

yoveli normirebuli sivrce aris metrikuli. marTlac, am normirebu-

li sivrcis nebismier x da y elements Soris manZili SeiZleba ase gani-

martos: ρ(x,y)=||x-y||. advili Sesamowmebelia, rom ρ akmayofilebs metrikis

samive aqsiomas (SeamowmeT).

1. namdvil ricxvTa simravle normirebul sivrced iqceva, Tu x-is ||x|| normas ase ganvmartavT: Tu x∈R, maSin ||x||≡|x| (SeamowmeT, rom es asaxva x→

→||x|| aris norma). 2. n

pR (p≥1) normirebuli sivrcea. marTlac, ricxvTa x=(x1,x2,…,xn) n-eul-

isTvis davuSvaT ||x||=1/

1

| |pn

pi

i

x=

⎧ ⎫⎨ ⎬⎩ ⎭∑ (SeamowmeT, rom x→||x|| normaa). amasTan

ρ(x,y)=||x-y||=1/

1

| |pn

pi i

i

x y=

⎧ ⎫−⎨ ⎬⎩ ⎭∑ . es is metrikaa, romelic Cven ganmartebuli

gvqonda npR (p≥1) sivrceSi (ix. §2, punqti 3).

3. lp (p≥1) sivrceSi x=(x1, x2,…, xn,…) elementis norma asea SemoRebuli:

||x||=1/

1

| |p

pi

i

x∞

=

⎧ ⎫⎨ ⎬⎩ ⎭∑ . igi normis ganmartebis samive Tvisebas akmayofilebs.

4. m da c0 sivrceebi normirebuli sivrceebia. marTlac, x=(x1, x2,…, xn,…) mimdevrobisaTvis sakmarisia davuSvaT ||x||= sup | |n

nx .

49

5. C[a,b] sivrce normirebuli sivrcea. [a,b] segmentze uwyveti f funq-ciisaTvis miRebulia ||f||=

[ , ]max | ( ) | .t a b

f t∈

me-9 paragrafSi Cven SemoviReT ori wrfivi Lx da Ly sivrcis izomor-

fizmis cneba. pirobiTad bieqcia, romelic axorcielebda am izomorfizms

aRniSnuli iyo ϕ simboloTi (ϕ : Lx→Ly). vTqvaT axla Lx da Ly sivrceebi

normirebuli sivrceebia. vityviT, rom normirebuli Lx da Ly sivrceebi

izomorfulia, Tu isini izomorfulia rogorc wrfivi sivrceebi, da ama-

ve dros, asaxva ϕ da misi Seqceuli ϕ −1 uwyvetia, saTanadod, Lx da Ly siv-

rceebze.

Teorema 10.2. nebismieri n–ganzomilebiani wrfivi normirebuli En siv-

rce izomorfulia 2nR sivrcis.

damtkiceba. vTqvaT En sivrcis bazisia e1, e2,…, en, xolo x∗ am sivrcis nebismieri elementia. maSin x∗ elementi calsaxad warmoidgineba saxiT

x∗= x1e1+x2e2+…+ xnen.

x∗-s SevuTanadoT 2nR sivrcis x=(x1, x2,…, xn) wertili. cxadia, es Tanadoba

(aRvniSnoT igi ϕ-Ti; e.i. ϕ(x∗)=x) iqneba izomorfizmi En-sa da 2nR sivrce-

ebs Soris. vaCvenoT, rom es da misi Seqceuli uwyvetia, saTanadod, En da

2nR sivrceebze. En sivrcis nebismieri x∗ elementisTvis koSisa da Svarcis

utolobis Tanaxmd gvaqvs

||x∗||=1

n

i ii

x e=∑ ≤

1

| |n

i ii

x e=∑ ≤

1/ 22

1

n

ii

e=

⎛ ⎞⋅⎜ ⎟

⎝ ⎠∑

1/ 22

1

n

ii

x=

⎛ ⎞⎜ ⎟⎝ ⎠∑ .

SemoviRoT aRniSvna C≡1/ 2

2

1

n

ii

e=

⎛ ⎞⎜ ⎟⎝ ⎠∑ . maSin ukanaskneli Sefasebidan miviRebT

||x∗||≤C||x||. (10.6) Tu y∗ En sivrcis nebismieri elementia, xolo ϕ(y∗)=y, maSin (10.6)-dan mivi-RebT

||x∗-y∗||≤C||x-y||. (10.7) ukanaskneli niSnavs, rom ϕ −1

asaxva aris uwyveti 2nR sivrceze.

axla vaCvenoT, rom arsebobs iseTi dadebiTi mudmivi C0, rom

C0||x-y||≤||x∗-y∗||. (10.8)

2nR sivrcis erTeulovan S[0,1] sferoze aviRoT nebismieri x wertili (e.i.

2

1

1n

ii

x=

=∑ ) da am sferoze f funqcia ganvsazRvroT ase:

f(x)=||x∗||=1

n

i ii

x e=∑ .

radgan erTi xi (i=1,2,…,n) mainc gansxvavdeba 0-gan, amitom e1, e2,…, en sistem-

is wrfivad damoukideblobis gamo f(x)>0. gvaqvs (ix. (10.7)) | f(x)- f(y)|= ∗ ∗−x y ≤||x∗-y∗||≤C||x-y||.

amrigad, f funqcia 2nR sivrcis S[0,1] sferoze uwyvetia. radgan S[0,1] kompaq-

tia, amitom 7.1 Teoremis Tanaxmad f funqcia S[0,1] kompaqtze aRwevs Tavis

qveda sazRvars. vTqvaT C0=[ ,1]

min ( )S

f∈ 0x

x . maSasadame,

50

f(x)=1

n

i ii

x e=∑ ≥C0.

aqedan nebismieri x∈ 2nR –Tvis

f(x)=||x∗||=||x||⋅1

ni

ii

x e=∑ x

≥C0||x||,

anu

||x∗||≥C0||x||, (10.9) saidanac davaskvniT, rom adgili aqvs (10.8) Sefasebas. ukanaskneli niS-

navs, rom ϕ funqcia uwyvetia En sivrceze.

(10.6) da (10.9) Sefasebebidan gamomdinareobs, rom arsebobs Ensivrcesa

da 2nR –s Soris iseTi bieqciuri Tanadoba ( x ↔x), rom am sivrceTa Sesaba-

misi elementebisaTvis ( x da x) adgili aqvs utolobaTa sistemas

2 21 2 .n n

nR E RC x x C x≤ ≤ (10.10)

Tu EnsivrceSi ganvixilavT raime normas 2

. nR

∗, misTvis samarTliani iq-

neba (10.10)-is analogiuri Sefaseba

2 21 2n n

nR E RC x x C x∗∗ ∗≤ ≤ , 1C∗ , 1C∗ >0.

(10.10) da ukanaskneli Sefasebidan ki davaskvniT iseTi dadebiTi C1, C2

mudmivebis arsebobas, rom

1 2 .n n nE E E

C x x C x∗≤ ≤

Aam SemTxvevaSi amboben, rom normebi nE

⋅ da nE

∗⋅ ekvivalenturia.

cxadia, 2nR sivrceSi yoveli SemosazRvruli simravle aris kompaqturi

(ratom?). ukanaskneli winadadebisa da Teorema 10.2-is safuZvelze marti-

vad davaskvniT, rom igive samarTliania nebismieri sasrulganzomilebiani

sivrcisaTvis (aCveneT): sasrulganzomilebiani sivrcis yoveli Semosaz-

Rvruli simravle aris kompaqturi.

normirebul sivrceebSi gansakuTrebuli mniSvneloba aqvs Caketil

wrfiv qvesivrceebs (ix. §9), anu iseT qvesivrceebs, romelTac ekuTvnis

yvela maTi Sexebis wertili. Teorema 10.2-is Tanaxmad sasrulganzomile-

bian sivrceSi yvela qvesivrce Caketilia (ratom? gaiTvaliswineT, rom

2nR sivrceSi yoveli mravalsaxeoba Caketilia). usasruloganzomilebian

SemTxvevaSi es ase ar aris. magaliTad, m sivrceSi yvela im mimdevrobaTa

simravle, romelTa wevrTa mxolod sasruli raodenoba gansxvavebulia

0-gan, qmnis qvesivrces (ratom?). magram igi ar aris Caketili ||x||= sup | |nn

x

normiT. marTlac, m sivrceSi ganvixiloT Semdeg (x(n)) mimdevroba, sadac x(n)=(1,1/2, …,1/n,0,0,…). vTqvaT x=(1,1/2, …,1/n, 1/(n+1), …). maSin, cxadia,

|| x(n)- x||=||(0,0,…,0,-1/(n+1),- 1/(n+2),…)||=1

sup | 1/ |k n

k≥ +

− =1/(n+1).

amrigad, (x(n)) mimdevroba krebadia amave sivrcis x elementisken. magram x ar ekuTvnis ganxilul qvesivrces, radgan misi yvela wevri gansxvavebu-

lia nulisagan.

51

§11. evkliduri sivrce. orTogonaluri bazisi.

beselis utoloba. sruli evkliduri sivrce.

risisa da fiSeris Teorema

Cven ukve gavecaniT normirebul sivrceebs. wrfiv sivrceebSi normis

Semotanis erT-erTi yvelaze cnobili meTodia maTSi skalaruli namrav-

lis ganmarteba.

vTqvaT L wrfivi sivrcea, xolo x da y am sivrcis raRac elementebia.

yovel x da y wyvils calsaxad SevuTanadoT ricxvi (x, y) da mas vuwodoT

am elementTa skalaruli namravli, Tu daculia Semdegi pirobebi:

1. ( , ) ( , );x y x y=

2. (x1+x2, y)=(x1, y)+(x2, y), x1, x2∈L; 3. yoveli λ ricxvisTvis (λx, y)=λ(x, y); 4. (x, x)≥0, amasTan (x, x)=0 mxolod x=0-Tvis.

wrfiv sivrces masSi ganmartebuli skalaruli namravliT ewodeba ev-

kliduri sivrce. evkliduri sivrce normirebuli sivrcea, radgan masSi

norma SeiZleba Semdegnairad ganimartos:

||x||= ( , )x x .

1 – 4 Tvisebis gamo normis yvela aqsioma sruldeba. jer erTi, L sivrcis yoveli x elementisTvis da nebismieri α ricxvisTvis cxadia, rom ||x||≥0 da ||αx|| =|α|||x||. vaCvenoT, rom || x+y||≤||x||+||y||. amisaTvis ganvixiloT funqcia ϕ, ϕ(λ)= =(λx+y,λx+y)≥0. skalaruli namravlis Tvisebebis gamoyenebis Sedegad mivi-

RebT

|λ|2(x, x)+λ(x, y)+λ (y,x)+(y, y)≥0. Tu λ=-(x, y)/(y, y), maSin ukanaskneli utolobidan miviRebT

(x, x)-|(x, y)|2/(y, y)≥0, saidanac davaskvniT

|(x, y)|≤||x||⋅||y||. aqedan gvaqvs

|| x+y||2=(x+y,x+y)=(x, x)+(x, y)+(y,x)+(y, y)≤||x||2+2||x||⋅||y||+||y||2=(||x||+||y||)2, anu

|| x+y||≤||x||+||y||. E evkliduri sivrcis or x da y elements ewodeba orTogonaluri, Tu

(x, y)=0. E sivrcis aranulovan elementTa {xα} sistemas ewodeba orTogonaluri

sistema, Tu am sistemis nebismieri ori xα da xβ elementisTvis (xα ,xβ)=0,

α≠β. orTogonaluri sistemis yoveli sasruli qvesistema wrfivad damouki-

debelia. marTlac, vTqvaT {xα} orTogonaluri sistemaa da

a11

xα+a2

2xα

+…+ann

xα=0.

vigulisxmoT, rom 1≤i≤n. maSin 0=(0,

ixα

)=(a11

xα+a2

2xα

+…+ann

xα,

ixα

)=ai(i

xα,

ixα

).

magram 0i

xα ≠ , e.i. (i

xα,

ixα

)= 20.

ixα ≠ amrigad, ai=0 (i=1,2,…,n).

{xα} orTogonaluri sistemas ewodeba orTonormirebuli, Tu

52

0, ,( , )

1, .x xα β

α βα β

≠⎧= ⎨ ≠⎩

roca

roca

advili saCvenebelia, Tu {xα} orTogonaluri sistemaa, maSin { xα /|| xα||} aris orTonormirebuli.

orTogonalur {xα} sistemas ewodeba sruli sistema (bazisi) E sivrce-Si, Tu E–s umciresi Caketili qvesivrce, romelic {xα} sistemas Seicavs,

emTxveva E sivrces. aqve SevniSnoT, rom evklidur sivrceSi SeiZleba sau-

bari qvesivrcis Caketilobis Sesaxeb, radgan am sivrceSi SemoRebulia

norma (da, maSasadame, metrikac). bazisis cneba SeiZleba sxva terminebi-

Tac Camoyalibdes. vTqvaT {xα} orTogonaluri sistemaa. ganvixilavT {xα}

sistemis yoveli sasruli qvesistemis nebismier wrfiv kombinacias: a11

xα+

+a22

xα+…+an

nxα . TiToeuli aseTi wrfivi kombinacia E sivrcis wrfivobis

gamo ekuTvnis E–s. Tu aseT wrfiv kombinaciaTa Caketva (E–Si SemoRebuli

normis azriT) emTxveva E–s, maSin {xα} sistemas ewodeba sruli orTogo-

nalur sistema (bazisi). Tu {xα} sistemis elementebi damatebiT akmayofi-

leben pirobas ||xα||=1, maSin {xα} sistemas ewodeba sruli orTomormirebu-

li sistema.

axla ganvixiloT zogierTi evkliduri sivrcisa da maTSi orTonor-

mirebuli sruli sistemebis (bazisebis) magaliTebi, amasTan vigulis-

xmebT, rom es sivrceebi wrfivi sivrceebia namdvili skalarebis mimarT.

1. namdvil ricxvTa R sivrceSi x da y ricxvebisaTvis skalaruli nam-

ravli ganimarteba ase: (x, y)=x⋅ y. cxadia, kmayofildeba skalaruli namrav-

lis ganmartebis oTxive aqsioma. amrigad, namdvil ricxvTa R sivrce evk-liduri sivrcea.

2. 2nR sivrcis x=(x1,x2,…,xn) da y=(y1,y2,…,yn) elementebisaTvis skalaruli

namravli SemoviRoT ase: (x, y)=1

.n

i ii

x y=∑ advili saCvenebelia, rom es operacia

akmayofilebs skalaruli namravlis yvela aqsiomas. maSasadame, 2nR sivr-

ce aris evkliduri sivrce. am sivrceSi ganvixiloT sistema {e1=(1,0,0, …,0), e2=(0,1,0,…,0),…, en=(0,0,0,…,n)} . es sistema qmnis baziss 2

nR sivrceSi. (ra-

tom?).

3. l2 sivrce evklidur sivrced gadaiqceva, Tu masSi ori x=(x1,x2,…,xn,…) da y=(y1,y2,…,yn,…) elementTa skalaruli namravlis operacias SemovitanT

Semdegnairad (x, y)=1

.i ii

x y∞

=∑ Tu gamoviyenebT helderis ganzogadebul uto-

lobas (ix. §1, lema 3), roca p=2, advilad davrwmundebiT, rom ukanaskneli

ricxviTi mwkrivi krebadia, amasTan, daculia skalaruli namravlis yve-

la aqsioma (ratom?). l2 sivrceSi umartives srul orTonormirebul siste-

mas qmnis veqtorebi e1=(1,0,0, …,0, …), e2=(0,1,0,…,0,…),…, en=(0,0,0,…,1,…),… . am sistemis orTonormirebuloba cxadia. vaCvenoT, rom es sistema srulia.

vTqvaT x=(x1,x2,…,xn,…)∈l2, e.i. 2

1

.ii

x∞

=

< ∞∑ maSin x(n)= (x1,x2,…,xn,0,0,…)=1

n

i ii

x e=∑ aris

e1, e2,…, en veqtorTa wrfivi kombinacia. meore mxriv,

|| x- x(n)||2=||(0,0, …, xn+1, xn+2,…)||2= 2

1

.ii n

x∞

= +∑

53

amitom limn→∞

|| x- x(n)||=0. maSasadame, ekuTvnis e1, e2,…, en,…sistemis sasrul qve-

sistemaTa wrfivi kombinaciebis Caketvas (ratom?).

4. L 2[a,b] sivrce aris evkliduri sivrce, Tu masSi ori uwyveti f da g funqciebis skalarul namravls ase ganvmartavT:

(f,g)= ( ) ( )b

a

f t g t dt∫ .

mkiTxvels vTxovT Seamowmos, rom ukanaskneli skalaruli namravlia. am

sivrceSi arsebul orTogonalur bazisTa Soris umniSvnelovanesia tri-

gonometriuli sistema, romelic Sedgeba funqciebisgan:

{ }2 /( ) , [ , ], 1, 2,...int b ae t a b nπ − ∈ = . (11.1) aseTia namdvil funqciaTa Semdegi sistemac:

1 2 2, cos , sin , [ , ], 1, 2,... .2

nt nt t a b nb a b aπ π⎧ ⎫∈ =⎨ ⎬− −⎩ ⎭

(11.1′)

am sistemis orTonormirebuloba martivad mtkicdeba (aCveneT). es sistema

srulicaa. am faqts me-12 paragrafSi davasabuTebT. 5. L2[E] sivrce aris evkliduri sivrce. rogorc cnobilia, L2[E] aris

lebegis azriT zomad E simravleze gansazRvruli zomad (lebegis az-

riT) funqciaTa simravle, romelTaTvisac 2( ) ;

E

f x dx < +∞∫

ukanaskneli integralic lebegis azriT gaigeba. es simravle wrfivi mra-

valsaxeobaa. marTlac, Tu f, g∈L2[E], maSin f+g∈L2[E] (es martivi Sedegia minkovskis integraluri utolobisa (ix. §1, lema 6)) da yoveli α ricxvi-saTvis α f ∈L2[E]. martivi saCvenebelia, rom L2[E] sivrce wrfivi sivrcis gansazRvris yvela aqsiomas akmayofilebs. helderis integraluri uto-

lobidan (ix. §1, lema 5) martivad gamomdinareobs, rom L2[E] sivrcis ori f da g funqciis namravlic kvlav L2[E] klasidanaa. ukanaskneli saSualebas

gvaZlevs am sivrceSi SemovitanoT skalaruli namravlis cneba. gansaz-

RvrebiT davuSvaT

(f,g)= ( ) ( )E

f x g x dt∫ .

martivi saCvenebelia, rom ukanaskneli skalaruli namravlis yvela Tvi-

sebas akmayofilebs. amrigad, L2[E] sivrce evkliduri sivrcea.

qvemoT Cven ganvixilavT separabelur evklidur sivrceebs. vaCvenoT,

rom aseT sivrceSi nebismieri {ϕα} orTonormirebuli sistema araumetes

Tvladia. marTlac, ganvixiloT am sistemis ori ϕα da ϕβ elementi. gvaqvs

||ϕα -ϕβ||2=(ϕα -ϕβ ,ϕα -ϕβ)=||ϕα||2-2(ϕα ,ϕβ)+||ϕβ||2=||ϕα||2+||ϕβ||2=2,

anu ||ϕα -ϕβ||=2. axla ganvixiloT B(ϕα,1/2) birTvTa erToblioba. isini ar Ta-

naikveTebian (ratom?). radgan sivrce separabeluria, amitom arsebobs

Tvladi {ψ n} simravle, romelic mkvrivia am sivrceSi. am simravlis erTi

elementi mainc moTavsdeba yovel birTvSi. es ki niSnavs, rom {ϕα} orTo-

normirebuli sistema araumetes Tvladia.

aRsaniSnavia, rom yovel separabelur evklidur sivrceSi arsebobs

sruli orTonormirebuli sistema. ukanasknelis dasamtkiceblad dagvWir-

deba Semdegi

54

Teorema 11.1 (Smidtis orTogonalizaciis meTodi). vTqvaT f1, f2,…, fn,…

evkliduri sivrcis wrfivad damoukidebeli sistemaa. maSin am sivrceSi

arsebobs iseTi orTonormirebuli sistema, rom

1. yoveli ϕn elementi aris f1, f2,…, fn elementTa wrfivi kombinacia:

ϕn=an1 f1+ an2 f2+…+ ann fn, amasTan ann≠0;

2. piriqiT, nebismieri fn elementi Semdegi saxiT warmoidgineba:

fn=bn1ϕ1+ bn2ϕ2+…+ bnnϕn, bnn≠0. (11.2) yoveli ϕn elementi 1 da 2 pirobiT calsaxad ganisazRvreba niSnis si-

zustiT (e.i. an1, an2,…, ann, bn1, bn2,…,bnn koeficientebi calsaxad ganisazRvre-

bian ±1-is sizustiT).

damtkiceba. ϕ1 veZioT Semdegi saxiT ϕ1=a11 f1, amasTan a11 ganisazRvreba

pirobiT

(ϕ1,ϕ1)=211a (f1,f1)=1,

saidanac

a11=1/b11= 1 11/ ( , ).f f± cxadia, ϕ1 calsaxad ganisazRvreba niSnis sizustiT. vTqvaT ϕk elementebi

(k<n), romlebic akmayofileben 1 da 2 pirobebs, ukve agebulia. maSin davu-

SvebT

fn=bn1ϕ1+bn2ϕ2+…+bn,n-1ϕn-1+hn, sadac (hn,ϕk)=0, roca k<n. marTlac, Sesabamisi bnk koeficientebi da, maSa-

sadame, hn elementic calsaxad ganisazRvreba pirobidan

(hn,ϕk)=( fn-bn1ϕ1- bn2ϕ2-…- bn,n-1ϕn-1,ϕk)=( fn,ϕk)-bnk(ϕk,ϕk)=0.

aRsaniSnavia, rom hn≠0 (winaaRmdeg SemTxvevaSi f1, f2,…, fn,… sistema iqneba wrfivad damokidebuli (ratom?)). maSasadame, (hn, hn)>0. davuSvaT, rom ϕn= = / ( , ).n n nh h h

induqciuri meTodiT agebidan naTelia, rom hn, da, maSasadame, ϕn gamoi-

saxeba f1, f2,…, fn sistemiT, anu ϕn=an1 f1+ an2 f2+…+ ann fn, sadac ann=1/ ( , ) 0.n nh h ≠ amis garda, (ϕn,ϕn)=1, (ϕn,ϕk)=0 (k<n) da

fn=bn1ϕ1+ bn2ϕ2+…+bnnϕn, (bnn= ( , ) 0n nh h ≠ ), anu ϕn akmayofilebs Teoremis pirobebs.

axla vaCvenoT, rom yovel separabelur evklidur sivrceSi arsebobs

sruli orTonormirebuli sistema (bazisi). marTlac, radgan sivrce

separabeluria, amitom am sivrceSi moiZebneba mkvrivi Tvladi simravle

{ψ1,ψ2,…, ψn,…}. {ψn} sistemidan gamovricxoT yvela iseTi ψk elementi, ro-

melic SeiZleba warmodgenil iqnas ψi (i<k) elementebis wrfivi kombinaci-

iT. miRebuli sistema (vTqvaT {ϕn} sistema) iqneba sruli (ratom?). Tu am

sistemis mimarT gamoviyenebT Smidtis orTogonalizaciis meTods, maSin

miRebuli sistema iqneba orTonormirebul sistema, amasTan (11.2) tolobis

ZaliT igi iqneba sruli sistema.

n-ganzomilebian evklidur sivrceSi e1,e2,…,en Tu orTonormirebuli

sistemaa (bazisia), maSin am sivrcis yoveli x elementi ase warmoidgineba:

1,

n

k kk

x c e=

= ∑ sadac ck=(x,ek).

55

axla gavarkvioT sakiTxi imis Sesaxeb, Tu usasrulo ganzomilebian

SemTxvevaSi ra saxe eqneba 1

n

k kk

x c e=

= ∑ warmodgenas.

vTqvaT ϕ1, ϕ2,…, ϕn,… evkliduri sivrcis orTonormirebuli sistemaa,

xolo f am sivrcis nebismieri elementia. f–s SevusabamoT ricxviTi mimdev-

roba: ck=(f,ϕk), k=1,2,…, romelTac f elementis furies koeficientebs vuwo-

debT {ϕk} sistemaSi ({ϕk} sistemis mimarT), xolo k kk

c ϕ∑ jams (SesaZlebe-

lia es mwkrivi arc iyos krebadi evklidur sivrceSi, amitom ukanaskneli

jami formaluri azriT gaigeba) ewodeba f elementis furies mwkrivi {ϕk}

sistemaSi.

bunebrivad ismis kiTxva: k kk

c ϕ∑ mwkrivi aris Tu ara krebadi (ra Tqma

unda, evkliduri sivrcis normis azriT)? Tu krebadia, maSin aris Tu ara

krebadi f elementisken? imisaTvis, rom am kiTxvebs vupasuxoT, winaswar

davsvaT aseTi sakiTxi: SesaZlebelia Tu ara mocemuli naturaluri n ricxvisTvis αk koeficientebi (k=1,2,…,n) ise SevarCioT, rom manZili f–sa

da Sn=1

n

k kk

α ϕ=

∑ Soris iyos minimaluri. gvaqvs

|| f-Sn||2=1 1

,n n

k k k kk k

f fα ϕ α ϕ= =

⎛ ⎞− − =⎜ ⎟⎝ ⎠

∑ ∑ ( f, f)-1

2( , )n

k kk

f α ϕ=

∑ +1 1

( , )n n

k k j jk j

α ϕ α ϕ= =

∑ ∑ =

=|| f||2-1

2n

k kk

cα=

∑ +2

1

n

kk

α=

∑ =|| f||2 2

1

n

kk

c=

−∑ + ( )2

1

n

k kk

cα=

−∑ .

cxadia, ukanaskneli gamosaxulebis minimumi ricxvTa α1,α2,…,αn sistemis

mimarT miiRweva maSin, roca k kcα = (k=1,2,…,n). am SemTxvevaSi

|| f-Sn||2=|| f||2 2

1

n

kk

c=

−∑ . (11.3)

amrigad, Cven vaCveneT, rom fiqsirebuli n-Tvis Sn=1

n

k kk

α ϕ=

∑ jamebs Soris

f elementidan umciresi gadaxra aqvs f elementis furies mwkrivis n-ur kerZo jams. radgan || f-Sn||2>0, amitom

2

1

n

kk

c=

∑ ≤|| f||2.

ukanaskneli utolobidan vaskvniT Sesabamisi mwkrivis krebadobas (ra-

tom?) da beselis Semdegi utolobis samarTlianobas

2

1k

kc

∞

=∑ ≤|| f||2. (11.4)

orTonormirebul sistemas ewodeba Caketili, Tu evkliduri sivrcis

nerbismieri f elementisaTvis adgili aqvs parsevalis tolobas:

2

1k

kc

∞

=∑ =|| f||2. (11.5)


Teorema 11.2. separabelur evklidur sivrceSi yoveli sruli orTo-

normirebuli sistema Caketilia da piriqiT.

56

damtkiceba. vTqvaT {ϕn} sistema Caketilia. maSin adgili aqvs parseva-

lis tolobas, rac imas niSnavs, rom (ix. (11.3)) lim || || 0.nnf S

→∞− = maSasadame,

evkliduri sivrcis nebismieri f elementi {ϕn} sistemis sasrul wrfiv

kombinaciaTa simravlis Sexebis wertilia (gaixseneT, rom Sn aris ϕ1, ϕ2, …,ϕn sistemis wrfivi kombinacia). amrigad, {ϕn} sistema srulia.

vTqvaT axla {ϕn} sistema aris sruli, e.i. yoveli f elementi nebismieri

sizustiT SeiZleba miaxlovebul iqnas 1

n

k kk

α ϕ=

∑ wrfivi kombinaciebiT.

rogorc naCvenebi iyo, 1

n

k kk

c ϕ=

∑ iZleva aranakleb zust miaxlovebas:

1

n

k kk

f c ϕ=

− ∑ ≤1

n

k kk

f α ϕ=

− ∑ ,

e.i.

1lim 0.

n

k kn kf c ϕ

→∞=

− =∑

aqedan (11.3) tolobis ZaliT vRebulobT:

2 2

1lim || || 0,

n

kn kf c

→∞=

⎛ ⎞− =⎜ ⎟⎝ ⎠

∑

anu adgili aqvs (11.5) tolobas.

Cven ukve vnaxeT (ix. (11.4)), rom samarTliania Semdegi winadadeba: imisa-

Tvis, rom ricxviTi (ck) mimdevroba iyos evkliduri sivrcis raime f ele-

mentis furies koeficientTa mimdevroba aucilebelia piroba 2

1k

kc

∞

=∑ <+∞.

aRsaniSnavia, rom sruli evkliduri sivrceebisaTvis ukanaskneli sakmari-

sicaa. kerZod, samarTliania Semdegi

Teorema 11.3 (risi da fiSeri). vTqvaT E sruli evkliduri sivrcea da

{ϕn} am sivrceSi nebismieri orTonormirebuli sistemaa. vigulisxmoT ag-

reTve, rom (cn) mimdevroba akmayofilebs pirobas 2

1.k

kc

∞

=

< +∞∑ maSin E sivr-

ceSi arsebobs iseTi f elementi, rom ck=( f,ϕn) da 2

1k

kc

∞

=∑ =( f, f)=|| f||2. (11.6)

damtkiceba. vTqvaT fn=1

n

k kk

c ϕ=

∑ . maSin {ϕk} sistemis orTonormirebulobis

gamo

2

n p nf f+ − =2

1 1 ...n n n p n pc cϕ ϕ+ + + ++ + =2

1.

n p

kk n

c+

= +∑

radgan 2

1,k

kc

∞

=

< +∞∑ amitom (fn) aris fundamenturi mimdevroba da, maSasadame,

E sivrcis sisrulis gamo, (fn) mimdevroba krebadia E sivrcis raime f ele-

mentisken, e.i. lim 0.nnf f

→∞− = vigulisxmoT, rom n≥i. gvaqvs

( f,ϕi)=( fn,ϕi)+(f- fn,ϕi). (11.7) amis garda,

( fn,ϕi)=(c1ϕ1+c2ϕ2+…+cnϕn, ϕi)=ci(ϕi,ϕi)=ci.

57

radgan

|(f- fn,ϕi)|≤|| f- fn||⋅||ϕi||→0, n→∞,

amitom, Tu (11.7) tolobaSi gadavalT zRvarze, roca n→∞, miviRebT

lim( , ) ,i inf cϕ

→∞= anu ( f,ϕi)= ci.

meore mxriv,

2nf f− =

1 1( , )

n n

k k k kk k

f c f cϕ ϕ= =

− −∑ ∑ = ( , )f f - 2

1

n

kk

c=

∑ →0, n→∞.

amrigad, sruldeba (11.6).

risisa da fiSeris Teoremas emyareba Semdegi debulebis mtkiceba.

Teorema 11.4. imisaTvis, rom srul, separabelur evklidur E sivrceSi orTonormirebuli sistema {ϕn} iyos sruli (iyos bazisi), aucilebelia

da sakmarisi, rom E-Si ar arsebobdes aranulovani elementi, romelic

orTogonaluri iqneba {ϕn} sistemis nebismieri elementis.

damtkiceba. vTqvaT {ϕn} sistema srulia. maSin igi Caketilicaa. Tu

davuSvebT, E sivrcis iseTi f elementis arsebobas, romelic orTogona-

luria yoveli ϕn-is, maSin miviRebT, rom f elementis nebismieri furies

cn=( f,ϕn) koeficienti nulis toli iqneba da parsevalis (11.5) tolobis Za-

liT || f||=0, anu f=0. piriqiT, davuSvaT ar arsebobs E sivrcis iseTi aranulovani elementi,

romlis yvela furies koeficienti nulia. vaCvenoT, rom {ϕn} sruli sist-

emaa. davuSvaT sawinaaRmdego – vTqvaT {ϕn} sistema sruli ar aris. maSin

is arc Caketili iqneba, e.i. iarsebebs iseTi elementi g≠0, rom

||g||2=( g, g)> 2

1k

kc

∞

=∑ (ck=(g,ϕk)).

meore mxriv, risisa da fiSeris Teoremis ZaliT moiZebneba iseTi f∈E el-

ementi, romlisTvisac (f,ϕk)=ck da || f||2=( f, f)= 2

1k

kc

∞

=∑ . gvaqvs

( f-g,ϕk)=( f,ϕk)-(g,ϕk)=ck -ck=0, k=1,2,…. amitom daSvebis ZaliT f-g=0. magram

( f, f)= 2

1k

kc

∞

=∑ <( g, g), e.i. || f||<|| g||.

es ki niSnavs, rom f≠g, anu f-g≠0. miviReT winaaRmdegoba.

§12. rimanis lokalizaciis principi. furies

mwkrivis krebadobis pirobebi. trigonometriuli

sistemis sisrule

furies mwkrivebis krebadobis sakiTxs bevri Zneli da metad faqizi

problema ukavSirdeba. am problemebma ganapirobes funqciaTa Teoriis

safuZvliani gadamuSaveba. bevri didi maTematikosis, maT Soris rimanis,

kantoris, lebegis, kolmogorovis saxelebi, mWidrod aris dakavSirebu-

li am dargTan, romlis Sesaxebac SeiZleba iTqvas, rom Cvens droSi igi

58

Tavis mravalricxovan ganzogadoebebTan da ganStoebebTan erTad centra-

lur adgils iWers analizSi.

Cven, ra Tqma unda, ver SevudgebiT am Teoriis farTo da Rrma ana-

lizs. davkmayofildebiT mxolod ramdenime ZiriTadi sakiTxis ganxil-

viT.

sasrul jams

01

( ) ( cos sin )N

n nn

f x a a nx b nx=

≡ + +∑ (12.1)

sadac x∈R, a0, a1, a2, …, aN, b1, b2, …, bN, sazogadod, kompleqsuri ricxvebia,

ewodeba trigonometriuli mravalwevri.

Tu gamoviyenebT eileris formulas

cos ,2

ix ixe ex−+

= sin ,2

ix ixe exi

−−=

maSin advilad davrwmundebiT, rom (12.1) tolobiT gansazRvruli f funq-cia asec SeiZleba Caiweros:

( ) , .N

inxn

n Nf x c e x R

=−

= ∈∑ (12.2)

cxadia, yoveli trigonometriuli mravalwevri 2π-perioduli funqciaa,

amasTan (12.2) mravalwevri namdvili funqciaa maSin da mxolod maSin, ro-

ca n nc c− = (ratom?).

Tu n nulisagan gansxvavebuli mTeli ricxvia, maSin /inxx e in→ funqcia

(igi warmoadgens inxx e→ funqciis erT-erT pirvelyofils) aris kvlav

2π-perioduli funqcia, amitom

1, 0,10, 1, 2,....2

inx ne dx

n

π

ππ −

=⎧= ⎨ = ± ±⎩

∫Tu

Tu (12.3)

es niSnavs, rom / 2inxx e π→ sistema [-π,π] segmentze qmnis orTonormire-

bul sistemas (ix, (11.1)); aseTia namdvil funqciaTa Semdegi sistemac (ix,

(11.1′)): 1 cos sin, , , [ , ), 1, 2,... .2

nt nt t nπ ππ π π

⎧ ⎫∈ − =⎨ ⎬⎩ ⎭

gavamravloT (12.2) tolobis orive mxare eimx–ze, sadac m mTeli ricxvia

da namravli vaintegroT [-π,π] Sualedze. maSin (12.3)-is ZaliT miviRebT:

1 ( ) ,2

imxmc f x e dx

π

ππ−

−

= ∫ (12.4)

roca |m|≤N (ratom? SeamowmeT). Tu |m|>N, maSin integrali (12.4) warmodgena-Si 0-is tolia.

(12.2) tolobis Sesabamisad Cven ganvmartavT trigonometriul mwkrivs

Semdegi saxiT:

,inxn

nc e

+∞

=−∞∑ x∈R, (12.5)

am mwkrivs aRvniSnavT simboloTi S(x), xolo mis N–ur kerZo jams, rog-

orc wesi, - simboloTi SN(x) (SN(x)=N

inxn

n N

c e=−∑ ). S(x)-s sazogadod simboluri

azri aqvs, radgan CvenTvis ar aris cnobili (12.5) mwkrivi aris Tu ara

krebadi. aRsaniSnavia, rom (12.5) mwkrivSi cn ricxvebi nebismieria.

59

Tu davuSvebT, rom (12.5) mwkrivi TiTqmis yvelgan krebadia [-π,π) Sua-ledze raRac funqciisaken, amasTan vigulisxmoT, rom am mwkrivis kerZo

jamTa mimdevrobisaTvis arsebobs [-π,π) periodze lebegis azriT jamebadi

maJoranti, maSin lebegis kargad cnobili debulebis Tanaxmad (igulis-

xmeba integralis niSnis qveS zRvarze gadasvlis erTi Teorema) davas-

kvniT, rom f iqneba 2π–perioduli, periodze jamebadi funqcia da koefi-

cientebi ganisazRvreba (12.4) tolobebiT. marTlac, davafiqsiroT mTeli

ricxvi m. maSin radgan

( ) ( ) ,imx imxNS x e f x e− −⎯⎯⎯→T.Yy.

roca N→∞, amitom 1 1 1lim ( ) lim ( ) ( ) .

2 2 2imx imx imx

N NN NS x e dx S x e dx f x e dx

π π π

π π ππ π π− − −

→∞ →∞− − −

= =∫ ∫ ∫ (12.6)

Tu gaviTvaliswinebT 12

inxeπ

⎧ ⎫⎨ ⎬⎩ ⎭

sistemis orTonormirebulobas, maSin

|m|≤N SemTxvevaSi gveqneba

1 ( )2

imxNS x e dx

π

ππ−

−

=∫1

2

Ninx imx

nn N

c e e dxπ

ππ−

=−−

=∑∫

= 1 .2

Ninx imx

n mn N

c e e dx cπ

ππ−

=− −

=∑ ∫

amitom (12.6)-dan miviRebT, rom dasaxelebul pirobebSi (12.5) mwkrivis cn koeficientebi ganisazRvrebian (12.4) tolobiT.

Catarebuli msjeloba gvibiZgebs ganvixiloT 2π–perioduli funqciebi,

romlebic jamebadia periodze. maTTvis azri eqneba

1 ( )2

inxf x e dxπ

ππ−

−∫

integrals, romelsac (12.4) tolobis analogiiT cn-iT aRvniSnavT. maT vu-

wodebT f funqciis furies koeficientebs trigonometriuli sistemis mi-

marT, xolo amgvarad gansazRvruli cn koeficientebisaTvis (12.5) mwkrivs ewodeba f funqciis furies mwkrivi trigonometriuli sistemis mimarT da

am faqts ase weren:

( ) .inxn

nf x c e

+∞

=−∞∑∼

amjerad, ismis kiTxva: f funqciis furies mwkrivi aris Tu ara krebadi

(vTqvaT wertilovnad, Tanabrad x–is mimarT da sxva)? Tu krebadia, ras

udris misi jami? ganisazrvreba Tu ara f funqcia Tavisi furies koefici-

entebiT? an, sxvagvarad rom vTqvaT, Tu CvenTvis cnobilia furies koefi-

cientebi, maSin SegviZlia Tu ara vipovoT es funqcia? Tu am funqciis

povna SesaZlebelia, maSin ra gziT SeiZleba misi povna?

furies trigonometriuli mwkrivebis Seswavlisas Cven SevexebiT or-

gvar trigonometriul mravalwevrs:

0

1( ) , ( ) ( ).1

n nimx

n m n mm n m

D x c e K x D xn=− =

= =+∑ ∑ (12.7)

pirvel maTgans ewodeba dirixles guli, xolo meores – Cezaros guli.

SeviswavloT maTi zogierTi Tviseba.

Teorema 12.1. yoveli n=0,1,2,…–Tvis adgili aqvs tolobebs:

60

sin( 1/ 2)( ) , 0,sin( / 2)n

n xD x xx+

= ≠ (12.8)

1 1 cos( 1)( ) , 0,1 1n

n xK x xn cosx

− += ⋅ ≠

+ − (12.9)

1 1( ) ( ) 1,2 2n nD x dx K x dx

π π

π ππ π− −

= =∫ ∫ (12.10)

20 ( ) (0 | | ).( 1)(1 cos )nK x xn

δ πδ

≤ ≤ < ≤ ≤+ −

(12.11)

damtkiceba. (12.7)–is ZaliT gvaqvs:

( ) ( 1)1 ( ) ( ).ix i n x inxne D x e e+ −− = − ratom? (12.12)

(12.8) tolobis misaRebad (12.12) tolobis orive mxare gavamravloT e-ix/2-ze da gamoviyenoT eileris formula. Tu ( )nK x gulis ganmartebaSi (ix. (12.7)) gaviTvaliswinebT (12.12)-s, maSin

( 1) ( 1) ( 1)

0( 1) ( )( 1)( 1) ( 1) ( ) 2 ,

nix ix ix i m x imx i n x i n x

nm

n K x e e e e e e e− − + − + − +

=

+ − − = − − = − −∑

saidanac miiReba (12.9) toloba (ratom?). maSasadame, ( )nK x ≥0 da sruldeba

(12.11). (12.10)-is samarTlianoba uSualod gamomdinareobs (12.7)–dan. am paragrafSi Cven saqme gveqneba mxolod trigonometriul sistemas-

Tan. vigulisxmebT, rom f funqcia Tavdapirvelad gansazRvrulia (-π,π] Sualedze da Semdeg 2π-periodulad gagrZelebulia mTel namdvil ricx-

vTa simravleze. aRsaniSnavia, rom amgvari gagrZeleba SeuZlebelia, Tu

f(-π)≠f(π). magram, Tu f∗ funqcia iseTia, rom f∗(x)= f(x), roca x∈(-π,π) da f∗(-π)=

=f(π), maSin SeiZleba gavagrZeloT periodulad mTel RerZze, xolo f∗–is

furies koeficientebi, cxadia, f funqciis Sesabamisi koeficientebis to-lia.

Tu f funqciis furies koeficientebi moicema (12.4) tolobiT, maSin misi

furies mwkrivis n–ur kerZo jams eqneba saxe:

1( ) ( ; ) ( )2

n nimx imt imx

n n mm n m n

S x S f x c e f t e dteπ

ππ−

=− =− −

≡ ≡ =∑ ∑ ∫ =

( )1 ( ) ,2

nim x t

m n

f t e dtπ

ππ−

=−−

⎡ ⎤= ⎢ ⎥⎣ ⎦

∑∫

anu (ix. (12.8)) gaviTvaliswinebT, ra im faqts, rom integrali [a,a+2π] Sua-ledze lokalurad jamebadi, 2π-perioduli funqciidan ar aris damoki-

debuli a ricxvze, miviRebT

1 1( ; ) ( ) ( ) ( ) ( )2 2n n nS f x f t D x t dt f x t D t dt

π π

π ππ π− −

= − = −∫ ∫ (ratom?). (12.13)

Teorema 12.2. (rimanis lokalizaciis principi). vTqvaT f lebegis az-

riT jamebadi funqciaa [-π,π] Sualedze (f∈L[-π,π]) da , 0<δ<π, maSin

lim ( ) ( ) 0nnf x t D t dt

δ π

π δ

−

→∞−

⎛ ⎞+ − =⎜ ⎟

⎝ ⎠∫ ∫ . (12.14)

damtkiceba. davafiqsiroT x wertili da g funqcia segmentze ganvmar-toT Semdegnairad:

61

0, | | ,( ) ( ) , | | ,

sin( / 2)

tg t f x t t

t

δ

δ π

<⎧⎪= −⎨ < ≤⎪⎩

roca

roca

Dda Semdeg gavagrZeloT 2π–periodulad. ( )nD x –is warmodgenis Tanaxmad (ix. (12.8)) gveqneba:

( ) ( )nf x t D t dtδ π

π δ

−

−

⎛ ⎞+ − =⎜ ⎟

⎝ ⎠∫ ∫ ( )sin( 1/ 2)g t n tdt

π

π−

+ =∫

= 1 2( ) cos sin ( )sin cos ( ) ( ).2 2t tg t ntdt g t ntdt A n A n

π π

π π− −

⎛ ⎞ ⎛ ⎞+ ≡ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫ ∫ (12.15)

vaCvenoT, rom A1(n)→0 da A2(n) →0, roca n→∞.

SemoviRoT aRniSvna ϕ(x)=g(t)cos (t/2). es funqcia jamebadia [-π,π] Sualed-

ze. CvenTvis cnobilia, rom yoveli ε>0 ricxvisaTvis arsebobs iseTi uw-

yveti ϕe funqcia, rom

( ) ( ) .x xπ

επ

ϕ ϕ ε−

− <∫ (12.16)

cxadia, ϕe∈L2[-π,π]. radgan L2[-π,π] aris evkliduri sivrce (ix. §11, punqti 5), amitom beselis utolobis Tanaxmad (ix. (11.4)) ϕe funqciis furies cn(ϕe) koeficientTa mimdevroba usasrulod mcirea ( lim ( ) 0nn

c εϕ→∞

= ). koSisa da

Svarcis utolobis ZaliT ϕ funqciis furies cn(ϕ) koeficientTaTvis gveq-

neba

|cn(ϕ)|= [ ]1 1( ) ( ) ( )2 2

inx inxx e dx x x e dxπ π

επ π

ϕ ϕ ϕπ π

− −

− −

≤ − +∫ ∫

+ 1 1( ) ( ) ( )2 2

inxx e dx x x dxπ π

ε επ π

ϕ ϕ ϕπ π

−

− −

≤ − +∫ ∫

+1(1) (1), .2

o o nεπ

≤ + → ∞ (12.17)

aqedan davaskvniT, rom

1 ( ) (1), .2

inxx e dx o nπ

π

ϕπ

−

−

= → ∞∫

maSasadame, ukanaskneli gamosaxulebis namdvili da warmosaxviTi nawile-

bi 0-ken krebadia. kerZod, (ix. (12,15)) A1(n)→0, roca n→∞.

analogiurad vaCvenebT, rom A2(n)→0, n→∞.

Tu gaviTvaliswinebT funqciis furies mwkrivis kerZo jamebis warmod-

genas (ix. (12.13)), maSin ukanaskneli Teoremis ZaliT, SegviZlia davaskvnaT

Semdegi: ( ( ; )nS f x ) mimdevrobis krebadobis sakiTxi daiyvaneba

1 1( ) ( ) ( ) ( )2 2

x

n nx

f x t D t dt f u D x u duδ δ

δ δπ π

+

− −

− = −∫ ∫

integralis Seswavlaze (nebismieri dadebiTi δ>0 ricxvisaTvis) (ratom?).

aqedan ki cxadia, rom ukanasknel integralSi gaiTvaliswineba f funqciis mniSvnelobebi x–is δ-midamoSi da integralebis yofaqcevisaTvis (krebado-

bis TvalsazrisiT) ar aqvs mniSvneloba, Tu rogoria f funqcia [-π,-δ]∪[δ,π] simravleze. amrigad, ( ; )nS f x -is krebadoba-ganSladobis sakiTxi lokalu-

62

ri xasiaTisaa, anu x( ; )nS f -is krebadoba-ganSladobis sakiTxi mTlianad

damokidebulia f funqciis yofaqcevaze x wertilis raime midamoSi. swo-

red, am faqtis gamo ewodeba ukanasknel Teoremas lokalizaciis princi-

pi.

Catarebuli msjelobidan gamomdinareobs, rom ori funqciis furies

mwkrivi SesaZlebelia erTi da igive yofaqcevisa iyos erT intervalze da

sxvadasxvagvari yofaqcevisa sxva intervalebze. aRsaniSnavia, rom furies

mwkrivTa es Tviseba kontrastulia xarisxovan mwkrivTa erTaderTobis

TvisebasTan. kerZod, rogorc cnobilia, Tu ori xarisxovani mwkrivis

mniSvnelobebi emTxveva erTmaneTs raime wertilis midamoSi, maSin es xar-

isxovani mwkrivebi erTmaneTs emTxveva maTi krebadobis areze.

ganvixiloT dirixles gulis erTi Tviseba. rogorc cnobilia samarT-

liania utoloba |sint|≥2|t|/π, roca t∈[-π/2,0)∪(0, π/2]. amitom gvaqvs (ix. (12.8)): sin( 1/ 2) 1( ) .

sin( / 2) sin( / 2) | |nn xD x

x x xπ+

= ≤ ≤ (12.18)

Teorema 12.3. (dini). vTqvaT f funqcia 2π-perioduli, lokalurad ja-

mebadi funqciaa. vigulisxmoT, rom raime x0 wertilisaTvis

0

0

( ),x t

dtt

δ ϕ< +∞∫ (12.19)

sadac

[ ]0 0 0 0

1( ) ( ) ( ) 2 ( )2x t f x t f x t f xϕ = + + − −

da 0<δ<π. maSin f funqciis furies mwkrivi krebadia f funqciisken x0 wer-

tilSi:

Sn(x0;f)-f(x0)→0, n→∞. damtkiceba. Tu gaviTvaliswinebT, rom dirixles guli luwi funqciaa,

maSin (12.13)-dan (12.10)-is gaTvaliswinebiT gveqneba

[ ]0 0 0 01( ; ) ( ) ( ) ( ) ( )

2n nS f x f x f x t f x D t dtπ

ππ −

− = − − =∫

= [ ] [ ]0

0 0 0 00

1 1( ) ( ) ( ) ( ) ( ) ( )2 2n nf x t f x D t dt f x t f x D t dt

π

ππ π−

− − + − − =∫ ∫

= [ ]0 0 00

1 ( ) ( ) 2 ( ) ( )2 nf x t f x t f x D t dt

π

π+ + − − =∫

= 0

00 0

( )1 1( ) ( ) sin( 1/ 2)sin( / 2)

xx n

tt D t dt n tdt

t

π π ϕϕ

π π= + =∫ ∫

=( ) ( )

0 0

0 0

( ) cos / 2 ( )sin / 21 1sin cos .sin( / 2) sin( / 2)

x xt t t tntdt ntdt

t t

π πϕ ϕπ π

⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫ ∫ (12.20)

vTqvaT h1 aris 2π-perioduli funqcia, romelic (-π,π] Sualedze asea

gansazRvruli:

0

1

( ) cos / sin( / 2), (0, ],( )

0, ( ,0].x t t t t

h tt

ϕ π

π

∈⎧⎪= ⎨∈ −⎪⎩

Tu

Tu

analogiurad, vigulisxmoT, rom h2–ic 2π-perioduli funqciaa da

63

0

2

( )sin / sin( / 2), (0, ],( )

0, ( ,0].x t t t t

h tt

ϕ π

π

∈⎧⎪= ⎨∈ −⎪⎩

Tu

Tu

(12.19) pirobidan gamomdinareobs, rom h1 da h2 funqcia 2π-perioduli, lo-

kalurad jamebadi funqciaa. Amis garda, am funqciebis gansazRvris safu-

Zvelze, (12.20)–dan miviRebT

0 0 1 20 0

1 1( ; ) ( ) ( )sin ( )cos .nS f x f x h t ntdt h t ntdtπ π

π π− = +∫ ∫

maSasadame, 0 0( ; ) ( )nS f x f x− SeiZleba warmovadginoT ori lokalurad jame-

badi funqciis furies koeficientTa jams. zustad iseve, rogorc es gava-

keTeT (12.17) Sefasebisas, davaskvniT, rom lokalurad jamebadi funqciis

koeficientTa mimdevroba 0-ken krebadia, anu

0 0( ; ) ( ) (1), .nS f x f x o n− = → ∞ SemoviRoT gansazRvreba. vityviT, rom f funqcia x wertilSi akmayofi-

lebs lipSicis pirobas, Tu arsebobs iseTi M da δ0 dadebiTi ricxvebi,

rom nebismieri y wertilisaTvis, romlisTvisac |x-y|<δ0, gvaqvs: |f(x)-f(y)|≤M⋅|x-y|. (12.21)

Sedegi. Tu 2π-perioduli funqcia f∈L[-π,π], amasTan x wertilSi akmayo-

filebs lipSicis pirobas, maSin f funqciis furies mwkrivi krebadia x wertilSi f(x)-ken (ratom?).

Teorema 12.4 (dini da lipSici). vTqvaT 2π-perioduli f funqcia akma-yofilebs lipSicis pirobas (anu (12.21) pirobas) Tanabrad x–isa da y –is mimarT. maSin f funqciis furies mwkrivi Tanabrad krebadia f funqciisken.

damtkiceba. (12.10) tolobisa da (12.13) warmodgenis ZaliT

[ ]1( ; ) ( ) ( ) ( ) ( ) .2n nS f x f x f x t f x D t dt

π

ππ −

− = − −∫

davasaxeloT ε>0 ricxvi da SevarCioT δ ricxvi ise, rom 0< δ<δ0 da M⋅δ<ε/2. gvaqvs warmodgena:

( )1 1 1( ; ) ( ) ( ) ( ) ( )2 2 2n nS f x f x f x t f x D t dt

δ δ π

δ π δπ π π

−

− −

⎧ ⎫− = + + − − ≡⎨ ⎬

⎩ ⎭∫ ∫ ∫

1 2 3( ) ( ) ( ).B n B n B n≡ + +

SevafasoT B1(n). radgan f funqcia akmayofilebs lipSicis (12.21) piro-bas Tanabrad x–isa da y–is mimarT da adgili aqvs (12.18) Sefasebas, ami-tom

| B1(n)|≤ 1 ( ) ( ) ( )2 nf x t f x D t dt

δ

δπ −

− −∫ ≤ 1 ( )2 nM t D t dt

δ

δπ −

≤∫

≤1 2 .

2 | | 2 2M t dt M M

t

δ

δ

π δ επ δπ π−

= = <∫

arsebobs iseTi N(ε) ricxvi, rom roca n> N(ε), maSin

2 3| ( ) ( ) | .2

B n B n ε+ <

marTlac, ganvixiloT

( )( ) ( ) sin( 1/ 2)sin( / 2)nf x tf x t D t dt n tdt

t

π π

δ δ

−− = + =∫ ∫

64

=( ) ( )cos( / 2)sin sin( / 2)cos

sin( / 2) sin( / 2)f x t f x tt ntdt t ntdt

t t

π π

δ δ

− −+ ≡∫ ∫

≡C1(n)+ C1(n). SevafasoT C1(n) (C2(n) Sefasdeba analogiurad). SemoviRoT aRniSvna:

( )( ) cos( / 2).sin( / 2)xf x tt t

tχ −

≡

cxadia, es funqcia uwyvetia [δ,π] Sualedze. gveqneba

C1(n)= ( )sin .x t ntdtπ

δ

χ∫

movaxdinoT cvladTa gardaqmna t=τ +π/n. miviRebT

C1(n)=/ /

/ /

( / )sin ( / ) ( / )sinn n

x xn n

n n n d n n dπ π π π

δ π δ π

χ τ π τ π τ χ τ π τ τ− −

− −

+ + = − +∫ ∫ ,

anu

2 C1(n)= [ ]/

( ) ( / ) sinn

x xt t n ntdtπ π

δ

χ χ π−

− + +∫

/ /

( ) sin ( / )sin .x xn n

t ntdt t n ntdtπ δ

π π δ π

χ χ π− −

+ − +∫ ∫

amrigad,

|C1(n)|≤/

,/

2( ) ( / ) max ( ) ;n

x x xx tn

t t n dt tn

π π

δ π

πχ χ π χ−

−

− + +∫

radgan χx funqcia SemosazRvrulia da Tanabrad uwyvetia [δ/2,π] Sualedze

(ratom?), amitom

C1(n)=o(1) Tanabrad x–is mimarT, roca n→∞.

maSasadame, f funqciis furies mwkrivi Tanabrad krebadia f funqciisken.

SesaZlebelia am Teoremis pirobebis ise Sesusteba, rom miRebuli de-

bulebis Sedegi ZalaSi darCes. Cven am sakiTxs ar SevexebiT.

meore mxriv, yoveli lokalurad jamebadi, 2π-perioduli funqciis fu-

ries mwkrivi ar aris krebadi dasaxelebul wertilSi, ufro metic – Se-

saZlebelia ganSladi iyos yvelgan. arsebobs iseTi uwyveti funqcia, rom-

lis furies mwkrivi ganSladia raRac araTvlad simravleze. aRsaniSna-

via, rom amgvari sakiTxebi Zalze rTul problemebs ukavSirdeba. amitom

Cven maT ar ganvixilavT.

mdgomareoba mniSvnelovnad umjobesdeba, Tu Sn(x,f)–ebis magivrad ganvi-xilavT maT saSualo ariTmetikulebs (e.w. Cezaros saSualoebs)

[ ]0 11( , ) ( , ) ( , ) ... ( , ) .

1 nx f S x f S x f S x fn

σ = + + ++

(12.22)

amis dasturia Semdegi

Teorema 12.5 (feieri). Tu f uwyveti, 2π-perioduli funqciaa, maSin misi

Cezaros saSualoebi Tanabrad x∈R–is mimarT aris krebadi f funqciisken, roca n→∞.

damtkiceba. (12.22)-isa da (2.13)–is ZaliT gveqneba:

( , )x fσ =0

1 1( ) ( ) .2 1

n

mm

f x t D t dtn

π

ππ =−

⎛ ⎞− ⎜ ⎟+⎝ ⎠∑∫

65

Tu gaviTvaliswinebT Cezaros gulis gansazRvras (ix. (12.7)), miviRebT:

( , )x fσ =1 ( ) ( ) .

2 nf x t K t dtπ

ππ −

−∫

meore mxriv (12.10)-dan gamomdinareobs, rom 1( ) ( ) ( ) .

2 nf x f x K t dtπ

ππ −

= ∫

amitom

[ ]1( ; ) ( ) ( ) ( ) ( ) .2n nf x f x f x t f x K t dt

π

π

σπ −

− = − −∫

davasaxeloT ε>0 ricxvi. radgan f funqcia uwyvetia, amitom igi Semo-sazRvrulia [-π,π] segmentze da, maSasadame, misi periodulobis gamo uw-

yvetia mTel RerZze. vTqvaT |f(x)|≤M, x∈R. imis gamo, rom f funqcia Tanabrad

uwyvetia, davaskvniT: arsebobs iseTi δ>0 ricxvi, rom Tu |x-y|<δ, maSin |f(x)-f(y)|<ε/2. (12.23)

(12.11) Sefasebis ZaliT moiZebneba iseTi N, rom roca n≥N da δ≤|t|≤π, gveqne-ba:

0≤Kn(t)≤ .4Mε (12.24)

(12.23) Sefaseba gvaZlevs:

( ) ( ) ( ) ( ) .2n nf x t f x K t dt K t dt

δ π

δ π

ε π ε− −

− − ≤ = ⋅∫ ∫ (12.25)

Tu axla gamoviyenebT (12.24)-s, miviRebT:

( ) ( ) ( ) 2 .4nf x t f x K t dt Mdt

M

δ π π

π δ π

ε π ε−

− −

⎛ ⎞+ − − ≤ = ⋅⎜ ⎟

⎝ ⎠∫ ∫ ∫ (12.26)

bolos, gaviTvaliswinebT ra (12.25)-s da (12.26) Sefasebas, davwerT:

( ; ) ( )n f x f xσ ε− < yoveli x–Tvis da naturaluri n≥N ricxvisTvis.

Teorema 12.6. Tu 2π-perioduli, lokalurad jamebadi funqciis yvela

furies koeficienti 0-ia, maSin f funqcia TiTqmis yvelgan 0-is tolia.

damtkiceba. rogorc cnobilia (ix. (12.16)), yoveli naturaluri n ricxvisaTvis arsebobs iseTi uwyveti, 2π-perioduli fn funqcia, romelic

akmayofilebs utolobas

1( ) ( ) .nf x f x dxn

π

π−

− <∫ (12.27)

fubinis Teoremis gamoyenebiT

| ( , ) ( , ) |m n mx f x f dxπ

π

σ σ−

−∫ = | ( , ) |m nx f f dxπ

π

σ−

−∫ ≤

≤ ( )( )1 | | ( )2 n nf f x t K t dtdx

π π

π ππ − −

− −∫ ∫ = ( )( )1 ( ) | |2 n nK t dt f f x t dx

π π

π ππ − −

− −∫ ∫

=1| ( ) ( ) |.nf u f u du

n

π

π−

− <∫

meore mxriv, feieris saSualoebis ganmartebis Tanaxmad (ix. (12.22)) σn(x,f)≡ ≡0, amitom

66

1| ( , ) | .m nx f dxn

π

π

σ−

<∫

feieris 12.5 Teoremis Tanaxmad σm(·, fn) Tanabrad krebadia funqciisaken, ro-ca m→∞. amitom ukanaskneli utolobidan miviRebT

1( ) .nf x dxn

π

π−

≤∫

(12.27)-dan gveqneba 2( ) .f x dxn

π

π−

<∫

amrigad,

( ) 0,f x dxπ

π−

=∫

anu f(x)=0 TiTqmis yvelgan.

am debulebidan gamomdinareobs

Sedegi. Tu ori f da g jamebadi funqciis furies koeficientebi erT-

maneTs emTxveva, maSin es funqciebi TiTqmis yvelgan emTxveva erTmaneTs

(ratom?).

savarjiSo. feieris Teoremisa da (12.27) Sefasebis gamoyenebiT aCveneT,

rom trigonometriuli sistema srulia L[-π,π] sivrceSi, anu nebismieri

ε–Tvis arsebobs trigonometriuli sistemis raRac sasruli qvesistemis

wrfivi kombinacia Tε, rom

( ) ( ) .f x T x dxπ

επ

ε−

− <∫

Teorema 12.7 (stouni da vaierStrasi). vTqvaT f funqcia uwyvetia [a,b] segmentze, maSin moiZebneba mravalwevrTa iseTi Pn mimdevroba, romelic

Tanabrad krebadia amave Sualedze.

damtkiceba. ganvixiloT [0,π] segmentis wrfivi asaxva [a,b] segmentze:

, [0, ].b ax a t t ππ−

= + ∈ (12.28)

SemoviRoT aRniSvna: g(t)=f(a+(b-a)t/π). cxadia, funqcia, rogorc uwyvet fun-qciaTa kompozicia, uwyvetia [0,π] segmentze. es funqcia uwyvetad gavagr-ZeloT [-π,0] Sualedze, ise rom amgvarad miRebuli ϕ funqciisaTvis ϕ(-π)= =ϕ(π)=g(π). ra Tqma unda, g funqciis [-π,0] segmentze gagrZelebis mravali

SesaZlebloba arsebobs. magaliTad, erT-erT aseT gagrZelebas miviRebT,

Tu g funqcias gavagrZelebT luwad, anu vigulisxmebT: ϕ(t)=g(-t), t∈[-π,0]. Semdeg, ϕ funqcia kvlav gavagrZeloT, amjerad, - 2π-periodulad mTel

RerZze da miRebuli funqcia simartivisaTvis kvlav aRvniSnoT ϕ–iT.

feieris Teoremis Tanaxmad (ix. Teorema 12.5) moiZebneba trigonometri-ul mravalwevrTa iseTi Pn mimdevroba, romelic [-π,π] segmentze Tanabrad

iqneba krebadi ϕ funqciisken, anu yoveli ε>0 ricxvisTvis arsebobs iseTi

N, rom n>N naturaluri ricxvisTvis

( ) ( ) , [ , ].2nt P t tεϕ π π− < ∈ − (12.29)

aviRoT nebismieri n>N naturaluri ricxvi. ra Tqma unda Pn warmoadgens

sinkt da coskt funqciebis wrfiv kombinacias (gaixseneT feieris saSualo-

67

ebis struqrura (ix. (12.22))), amasTan, es funqciebi analizuria mTel Rer-

Zze; e.i. TiToeuli maTgani da, maSasadame, Pn-ic nebismier sasrul Sua-

ledze (kerZod, [-π,π] segmentze) SeiZleba warmodgenil iqnas Tanabrad

krebadi xarisxovani mwkrivis jamis saxiT. es imas niSnavs, rom dasaxele-

buli ε>0 ricxvisTvis moiZebneba aRniSnuli xarisxovani mwkrivis iseTi

kerZo jami

Sm(t)=( )

( )

0, [ , ],

m nn k

kk

t tλ π π=

∈ −∑

rom

( ) ( ) , [ , ].2n mP t S t tε π π− < ∈ −

(12.29) da ukanaskneli Sefasebebidan davaskvniT

( ) ( ) ( ) ( ) ( ) ( ) , [ , ].2 2m n n mt S t t P t P t S t tε εϕ ϕ ε π π− ≤ − + − < + = ∈ − (12.30)

axla (12.28)-dan gveqneba ( [0, ], [ , ]).x at t x a bb a

π π−= ∈ ∈

−maSasadame, (12.30)-is

Tanaxmad davaskvniT

( ) , [ , ]m mx a x a x af x S S x a bb a b a b a

π ϕ π π ε− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − < ∈⎜ ⎟ ⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠(ratom?).

bolos, Zneli ar aris imis warmodgena, rom mx aSb a

π −⎛ ⎞⎜ ⎟−⎝ ⎠

aris mravalwevri

x–is mimarT.

§13. hilbertis sivrce

hilbertis sivrce ewodeba H simravles, romelic akmayofilebs piro-

bebs:

1. H simravle evklides sivrcea;

2. H aris sruli sivrce am sivrcis (evkliduri sivrcis) normis az-

riT;

3. H sivrce aris usasruloganzomilebiani.

umravles SemTxvevaSi ganixilaven separabelur hilbertis sivrceebs.

separabeluri hilbertis sivrcis magaliTia l2 sivrce (ratom?). or Ex da Ey evklidur sivrces ewodebaT izomorfuli, Tu am sivrceebs

Soris, rogorc wrfiv sivrceTa Soris arsebobs iseTi izomorfizmi

ϕ : Ex → Ey (ix. §9), rom Ex sivrcis yoveli x(1) da x(2)

elementisTvis (x(1), x(2)

)=

=(ϕ(x(1)),ϕ( x(2)

)). sxvagvarad rom vTqvaT, or evklidur sivrces Soris izo-

morfuli Tanadoba am sivrceTa Soris iseTi urTierTcalsaxa Tanadobaa,

romelic inarCunebs rogorc wrfiv operaciebs (romlebic am sivrceSia

ganmartebuli), aseve skalarul namravlsac. Tu gaviTvaliswinebT normis

ganmartebas evklidur sivrceSi (ix. §11), davaskvniT

||x(1)||2=(x(1)

, x(1))=(ϕ(x(1)

),ϕ( x(2)))=||ϕ(x(1)

)||2,

anu

||x(1)||=||ϕ(x(1)

)||.

68

aqedan gamomdinare, ori izomorfuli evkliduri sivrce erTdroulad

srulia, an ar aris sruli (ratom?).

n-ganzomilebiani En sivrce SeiZleba ganvixiloT rogorc evkliduri

sivrce. marTlac, vTqvaT e1, e2,…, en aris am sivrcis raime bazisi. maSin am

sivrcis nebismieri x da y elementi SeiZleba warmodgenil iqnas

tolobebiT:

1 1 2 2 ... n nx x e x e x e= + + + da 1 1 2 2 ... n ny y e y e y e= + + + .

am ori elementis skalaruli namravli ase ganvmartoT: 1

( , )n

i ik

x y x y=

= ∑ . ase

amrigad, En sivrce evklidur sivrced gadaiqceva. vTqvaT x=(x1, x2,..., xn) da y=(y1, y2,..., yn)∈ 2

nR . ganvixiloT asaxva ϕ, romlisTvisac ϕ( x )= x. advilia imis

Cveneba, rom ϕ aris izomorfizmi wrfiv En da 2nR sivrceTa Soris, amasTan

( x , y )=(ϕ( x ),ϕ( y ))=(x, y). n-ganzomilebiani sivrceebi SeiZleba ganvixiloT rogorc evkliduri

sivrceebi, ufro metic, - izomorfuli evkliduri sivrceebi. magram

usasruloganzomilebiani evkliduri sivrceebi sazogadod ar arian izo-

morfulni. magaliTisaTvis SeiZleba davasaxeloT l2 da L 2[a,b] evkliduri

sivrce. l2 sruli sivrcea, xolo L 2[a,b] ar aris sruli (ix, §4, punqti 4 da 9), amitom isini ar iqnebian izomorfulni. miuxedavad ganxiluli magali-

Tisa, samarTliania

Teorema 13.1 (hilbertis sivrceTa izomorfizmis Sesaxeb). Yyoveli

ori separabeluri hilbertis sivrce erTmaneTis izomorfulia.

damtkiceba. sakmarisia davamtkicoT, rom yoveli separabeluri hilber-

tis sivrce l2 sivrcis izomorfulia. separabelur hilbertis H sivrceSi aviRoT sruli orTonormirebuli {ϕk} sistema (aseTi sistema arsebobs

(ix. §11)). ganvixiloT nebismieri elementi f∈H da mas SevuTanadoT amave

elementis furies ck=( f,ϕk), k=1,2,… , koeficientTa mimdevroba c =( c1, c2,…, cn,…). Aes Sesa-bamisoba (asaxva) aRvniSnoT simboloTi ℑ. beselis

utolobis ZaliT 2

1.k

kc

∞

=

< +∞∑ amrigad, ℑ: H→l2. 11.4 Teoremis Tanaxmad ℑ

ineqciuri asaxvaa (ratom?). meore mxriv, risisa da fiSeris Teoremis

ZaliT es asaxva sureq--ciulicaa (ratom?) da, maSasadame, aris bieqcia.

skalaruli namravlis Tvisebebidan gamomdinare martivad davaskvniT,

rom ℑ asaxva aris izo-morfizmi wrfivi H sivrcisa wrfiv l2 sivrceze.

amis garda, radgan {ϕk} sistema srulia, amitom Caketilicaa. amrigad, Tu

f∈H elementis furies koeficientTa mimdevrobaa d =( d1, d2,…, dn,…), maSin

(f, f)= 2

1

,nn

c∞

=∑ (g, g)= 2

1

,nn

d∞

=∑

da

(f+g, f+g)=(f, f)+2(f,g)+(g, g). meore mxriv,

( )2 2 2

1 1 1 1

2 .n n n n n nn n n n

c d c c d d∞ ∞ ∞ ∞

= = = =

+ = + +∑ ∑ ∑ ∑

ukanaskneli tolobebidan gamomdinareobs, rom

(f,g)= 1

n nn

c d∞

=∑ .

69

Tu gavixsenebT l2 sivrceSi skalaruli namravlis ganmartebas (ix. §11, pun-

qti 3), maSin advilad davrwmundebiT, rom 1

n nn

c d∞

=∑ aris c, d∈l2 elementebis

skalaruli namravli.

hilbertis sivrceTa qvesivrceebs axasiaTebT mTeli rigi specifikuri

(da, amave dros, Zalze mniSvnelovani) Tvisebebi. gavecnoT hilbertis siv-

rcis zogierT qvesivrces.

1. vTqvaT h aris H sivrcis nebismieri elementi. yvela im f elementTa

simravle, romlebis orTogonaluria h elementis ((f,h)=0), qmnis H sivrcis qvesivrces (ratom?).

2. l2 sivrcis yvela im x =( x1, x2,…, xn,…) elementTa simravle, romelTaTvi-

sac xn=0, roca n=2,4,6…, qmnis qvesivrces.

aRsaniSnavia, rom bolo or magaliTSi ganxiluli hilbertis sivrce-

Ta qvesivrceebi Caketili qvesivrceebia (ratom?).

vTqvaT M aris H sivrcis raime Caketili qvesivrce (qvemoT hilbertis

sivrcis Caketili qvesivrceebis rolSi vigulisxmebT Caketil qvesivrce-

ebs). simboloTi M⊥ aRvniSnavT yvela im g∈H elementTa simravles, rom-

lebic orTogonaluria M qvesivrcis nebismieri elementis. M⊥ simravles

ewodeba M qvesivrcis orTogonaluri damateba. radgan tolobidan (g1, f)= =(g2, f)=0 gamomdinareobs rom (α1g1+α2g2, f)=0, amitom M⊥

simravlis wrfivoba

cxadia. axla davamtkicoT M⊥ simravlis Caketiloba. vTqvaT g aris M⊥

-is

Sexebis wertili. maSin am simravleSi moiZebneba (gn) mimdevroba (gn∈M⊥),

romelic krebadi iqneba g elementisken, anu lim 0.nng g

→∞− = vTqvaT f aris M

qvesivrcis nebismieri elementi, maSin

|(g, f)-(gn, f)|=|(g-gn, f)|≤|| g-gn||⋅|| f||→0, n→∞, e.i. lim( , ) ( , ).nn

g f g f→∞

= radgan gn∈M⊥ da f∈M, amitom (gn, f)=0. maSasadame, (g, f)=0.

aqedan gamomdinareobs, rom g∈M⊥.

samarTliania Semdegi mniSvnelovani

Teorema 13.2. Tu M hilbertis H sivrcis Caketili, wrfivi qvesivrcea,

maSin H sivrcis nebismieri f elementi erTaderTi saxiT ase warmoidgine-

ba: f=h+h′, sadac h∈M, xolo h′∈M⊥. damtkiceba. jer vaCvenoT aseTi warmodgenis samarTlianoba. upirveles

yovlisa SevniSnoT, rom M separabeluri hilbertis sivrcea Semdeg gare-

moebaTa gamo: 1. rogorc cnobilia, separabeluri metrikuli sivrcis qve-

sivrce separabeluria (ix. Teorema 3.7 ), 2. sruli sivrcis Caketili qvesiv-

rce srulia (ix. Teorema 4.1 ), 3. evkliduri sivrcis qvesivrce evkliduria.

separabelur evklidur sivrceSi yovelTvis arsebobs sruli orTonor-

mirebuli sistema. vigulisxmoT, rom aseTi sistemaa {ϕn}. vTqvaT cn=( f,ϕn),

n=1,2,… . amitom 2

1.n

nc

∞

=

< +∞∑ davuSvaT m>n naturaluri ricxvia da ganvixi-

loT 2

1 1

m n

k k k kk k

c cϕ ϕ= =

− =∑ ∑2

1

m

k kk n

c ϕ= +

=∑ 2

1

0k

m

k nc

= +

→∑ , .n → ∞

maSasadame, Sn=1

n

k kk

c ϕ=

∑ , n=1,2, …, mimdevroba fundamenturia M qvesivrceSi.

amitom M-is sisrulis gamo (Sn) krebadia hilbertis H sivrcis normis az-

70

riT. vTqvaT is krebadia h∈H elementisken ( )lim 0 .nnS h

→∞− = am faqts ase Cav-

werT 1

.k kk

h c ϕ∞

=

= ∑ vTqvaT h′=f-h. maSin

(h′,ϕk)=(f,ϕk)-(h,ϕk)=ck-(h,ϕk).

vaCvenoT, rom (h′,ϕk)=0, e.i. davamtkicoT, rom ck=(h,ϕk). marTlac, roca m>n, gvaqvs

(h,ϕk)=(h-Sn,ϕk)+(Sn,ϕk)=(h-Sn,ϕk)+ ck.

pirveli Sesakrebi ase fasdeba

|(h-Sn,ϕk)|≤|| h-Sn||⋅||ϕk||→0, n→∞.

amrigad, (h,ϕk)= lim( , ) .k knh cϕ

→∞=

unda vaCvenoT, rom h′ aris orTogonaluri M qvesivrcis nebismieri ζ elementis. marTlac, radgan {ϕn} sistema aris sruli M-Si, amitom moiZeb-

neba iseTi ak koeficientebi, rom 1

lim 0n

k kn k

aζ ϕ→∞

=

− =∑ (ratom?). gvaqvs

|(h′,ζ)|=|(h′,1

n

k kk

a ϕ=

∑ )+(h′,ζ-1

n

k kk

a ϕ=

∑ )|=|(h′,ζ-1

n

k kk

a ϕ=

∑ )|≤

≤|| h′||⋅||ζ-1

n

k kk

a ϕ=

∑ ||→0, n→∞.

ukanaskneli utolobis marjvena mxare ar aris damokidebuli n–ze, ami-tom (h′,ζ)=0. maSasadame, h′∈M⊥

.

axla f=h+ h′ warmodgenis erTaderToba vaCvenoT. vTqvaT meore warmod-

genac arsebobs: f=h1+ '1h , sadac h1∈M, '

1h ∈M⊥. maSin cn=( f,ϕn)=(h1+ '

1h ,ϕn)=(h1,ϕn)+( '1h ,ϕn)=(h1,ϕn).

meore mxriv, rogorc zemoT iyo naCvenebi, (h,ϕn)=cn. amrigad, M sivrcis h da h1 elementebs M sivrceSi sruli orTonormirebuli sistemis mimarT

gaaCnia erTidaigive furies koeficientebi, e.i. h-h1 elementis yvela furi-

es koeficienti 0-ia. amitom 11.4 Teoremis ZaliT h=h1. es ki imas niSnavs,

rom h′= '1h .

§14. wrfivi operatoris uwyvetoba da SemosazRvruloba. SeuRlebuli sivrce

vTqvaT Ex da Ey warmoadgenen normirebul sivrceebs ricxvTa erTi da

igive velis mimarT. vigulisxmoT agreTve, rom A aris wrfivi operatori, romelic asaxavs D(A)⊂Ex mravalsaxeobas Ey sivrceSi (D(A) simravles ewo-

deba A operatoris gansazRvris simravle).

rogorc cnobilia, raime E simravleze gansazRvrul namdvil an komp-

leqsur funqcias ewodeba SemosazRvruli, Tu arsebobs iseTi arauaryo-

fiTi M ricxvi, rom yoveli x∈E-Tvis |f(x)|≤M. am gziT wrfivi A operatoris SemosazRvrulobis ganmarteba araefeqturia. marTlac, vTqvaT aranulova-

ni A operatori gansazRrulia wrfiv D(A) mravalsaxeobaze. aviRoT iseTi

x0∈D(A) wertili, rom Ax0≡y0≠0. A operatoris wrfivobis gamo yoveli α ska-

71

larisaTvis A(αx0)=α Ax0=α y0. aqedan gamomdinare {α y0} simravle da, miTume-

tes, A(D(A)) ar iqneba SemosazRvruli.

SemoviRoT normirebul sivrceze gansazRvruli wrfivi operatoris

SemosazRvrulobis cneba.

vTqvaT Ex da Ey normirebuli sivrceebia erTi da igive skalaruli ve-

lis mimarT, amasTan A wrfivi operatoria, romelic asaxavs Ex-s Ey–Si.

gansazRvreba a). vityviT, rom A operatori SemosazRvruli operato-

ria, Tu is Ex normirebuli sivrcis yovel SemosazRvrul simravles asa-

xavs Ey normirebuli sivrcis SemosazRvrul simravleSi.

gansazRvreba b). vityviT, rom A: Ex→Ey operatori aris SemosazRvruli,

Tu yoveli arauaryofiTi m ricxvisaTvis arsebobs iseTi M (M≥0) ricxvi, rom pirobidan x∈Ex,

xEx ≤m, gamomdinareobs

EyAx ≤M.

momavalSi Caweris simartivis mizniT Ex da Ey sivrceebSi normas erTi

da igive ||⋅|| simboloTi aRvniSnavT.

gansazRvreba g). vityviT, rom wrfivi A: Ex→Ey operatori aris Semosaz-Rvruli, Tu es operatori aris SemosazRvruli Ex sivrcis erTeulovan

birTvze, anu

[0,1]sup

x BAx

∈=

1sup

xAx

≤<+∞.

gansazRvreba d). vityviT, rom wrfivi A: Ex→Ey operatori aris Semosaz-Rvruli, Tu es operatori aris SemosazRvruli Ex sivrcis erTeulovan

sferoze, anu

[0,1]sup

x SAx

∈=

1sup

xAx

=<+∞.

gansazRvreba e). wrfiv A: Ex→Ey operators ewodeba SemosazRvruli, Tu

arsebobs iseTi arauaryofiTi M ricxvi, rom nebismieri x∈Ex wertilisa-

Tvis

Ax ≤M x . (14.1) Camoyalibebuli ganmartebebi tolfasia. amisTvis sakmarisia vaCvenoT,

rom a) ⇒ b) ⇒ g) ⇒ d) ⇒ e) ⇒ a).

vTqvaT adgili aqvs a) ganmartebas. radgan fiqsirebuli m ricxvisaTvis

birTvi B[0,m] SemosazRvruli simravlea Ex-Si, amitom {Ax: x∈B[0,m]} simrav-le iqneba SemosazRvruli Ey-Si, anu iarsebebs M (M≥0) ricxvi, rom

Ax ≤M, Tu x ≤1. b) ⇒ g) ⇒ d) winadadebebis samarTlianoba cxadia.

d) ⇒ e). marTlac, x=0∈Ex wertilisaTvis (14.1) Sefaseba sruldeba. da-

vuSvaT x≠0 da ganvixiloT x =x/||x||∈Ex. radgan ||| x ||=1, amitom simravle { Ax }

SemosazRvrulia, anu arsebobs iseTi M (M≥0) ricxvi, rom Ax = ( )/|| ||A x x = Ax / x ≤M,

anu adgili aqvs (14.1) Sefasebas. winadadebis e) ⇒ a) samarTlianoba cxadia.

SemosazRvruli wrfivi operatorisaTvis ganimarteba norma.

ganvixiloT ramdenime gansazRvreba.

A operatoris norma aris arauaryofiTi ricxvi, romelic sxvadasxva

saxiT SeiZleba iqnas warmodgenili:

1. 1

A =1

supx

Ax≤

;

72

2. 2

A =1

supx

Ax=

;

3. 3

A =inf{M}, sadac M–is rolSi ganixileba yvela is ricxvi, rome-

lic akmayofilebs (14.1) utolobas. Teorema 14.1. samarTliania tolobebi

1A =

2A =

3A . (14.2)

damtkiceba. (14.1)–s Tanaxmad 3

A =inf{M}, sadac yoveli M∈{M}ricxvi

aris { Ax / x , x≠0} simravlis maJoranti. namdvil ricxvTa simravlis qve-

simravlis supremumis gansazRvris Tanaxmad

0supx≠

{ }/Ax x =0

supx≠

( ){ }/A x x =inf{M}.

amrigad,

3A =

0supx≠

( ){ }/A x x .

magram /x x =1, amitom

3A ≤

2A ≤

1A .

meore mxriv, Tu (14.1) utolobaSi vigulisxmebT x ≤1, miviRebT

Ax ≤M. maSasadame,

1A =

1sup

xAx

≤≤M,

e.i.

1A ≤inf{M}=

3A .

(14.2) toloba saSualebas gvaZlevs normirebul sivrceebSi Semosaz-

Rvruli operatoris norma CavweroT CvenTvis sasurveli 1), 2) an 3) for-

miT. amitom, bunebrivia, A operatoris normis aRsaniSnavad visargebloT

||A|| simboloTi da mis rolSi vigulisxmoT erT-erTi 1

A , 2

A , 3

A ricx-vTagan.

Zneli ar aris imis mixvedra, rom SemosazRvruli A operatorisTvis

adgili aqvs utolobas

Ax ≤ A x .M (14.3) vTqvaT A aris wrfivi operatori: A: Ex→Ey (Ex da Ey kvlav normirebuli

sivrceebia ricxvTa erTi da igive velis mimarT) A operatoris uwyvetoba x0∈Ex wertilSi funqciis uwyvetobis kargad cnobili ganmartebis analo-

giuria.

A operators ewodeba uwyveti x0∈Ex wertilSi, Tu nebismieri ε>0 ric-xvisaTvis arsebobs iseTi δ>0, rom Tu ||x-x0||<δ, maSin ||Ax-Ax0||<ε.

Teorema 14.2. Tu wrfivi A operatori uwyvetia wrfivi Ex sivrcis nu-

lovan wertilSi, maSin igi uwyveti iqneba am sivrcis nebismier wertilSi. damtkiceba. A operatoris nulovan wertilSi uwyvetobidan gamomdina-

reobs, rom nebismieri ε>0-Tvis arsebobs iseTi δ>0, rom Tu x∈D(A) da

||x||<δ, maSin ||Ax||<ε. vTqvaT x0 D(A)-s nebismieri fiqsirebuli wertilia da

x∈D(A). vigulisxmoT, rom z=x-x0. maSin Tu ||z||<δ, gveqneba ||Az||<ε, anu pirobi-dan ||x-x0||<δ unda gamomdinareobdes, rom ||Ax-Ax0||=||A(x-x0)||<ε.

73

Teorema 14.3. imisaTvis, rom wrfivi A (A: Ex→Ey) operatori iyos Semo-

sazRvruli aucilebeli da sakmarisia, rom is iyos uwyveti.

damtkiceba. (sakmarisoba) vTqvaT A operatori uwyvetia. maSin A-s 0∈Ex

wertilSi uwyvetobidan, kerZod, gamomdinareobs, rom ε=1-s Seesabameba is-eTi δ>0 ricxvi, rom Tu x∈D(A) da ||x||<δ, gveqneba ||Ax||<1. vTqvaT x∈D(A) da

x≠0. radgan 2 2

xx

δ δ δ= < , amitom 12

xAx

δ⎛ ⎞<⎜ ⎟⎜ ⎟

⎝ ⎠. A

aqedan miviRebT

( ) 2 ,A x xδ

<

anu A SemosazRvruli operatoria.

(aucilebloba) vTqvaT A aris aranulovani SemosazRvruli operatori,

e.i. arsebobs iseTi x∈D(A) wertili, rom ||Ax||>0, anu ||A||>0 (ratom? ix. (14.1)). ganvixiloT nebismieri ε>0 ricxvi da davuSvaT δ=ε/||A||. maSin Tu x∈D(A) da ||x||<ε/||A||, gveqneba (ix. (14.3))

Ax ≤ A ⋅ x < AAε =ε,

maSasadame, A operatori aris uwyveti 0∈Ex wertilSi, anu uwyvetia Ex siv-

rcis nebismier wertilSi.

wrfivi funqcionalis cneba ganxiluli iyo me-9 paragrafSi. radgan

funqcionalis mniSvnelobebi skalarebia, amitom ||f(x)||=|f(x)|, sadac f raRac funqcionalia. ase rom, wrfivi f funqcionalis SemosazRvruloba gani-

marteba Semdegnairad. E normirebul sivrceze gansazRvrul wrfiv f funq-cionals ewodeba SemosazRvruli, Tu am sivrcis nebismieri x∈E werti-lisTvis sruldeba utoloba

|f(x)|≤M x , sadac M raRac arauaryofiTi ricxvia. M ricxvTa simravlis infimums

ewodeba f funqcionalis norma da igi ase aRiniSneba: f . wrfivi opera-torebisTvis damtkicebuli 14.1 Teoremidan gamomdinareobs, rom

f =1 1

sup ( ) sup ( ) .x x

f x f x≤ =

=

amis garda, 14.2 Teoremidan miiReba, rom f funqcionalis SemosazRvrulo-

bisaTvis aucilebeli da sakmarisia, rom is iyos uwyveti E sivrceze (an rac igivea, iyos uwyveti 0∈E wertilSi).

vTqvaT Ex da Ey wrfivi normirebuli sivrceebia ricxvTa erTi da igive

velis mimarT. vigulisxmoT agreTve, rom A da B wrfivi asaxvebia Ex-isa Ey sivrceSi. αA-Ti aRvniSnoT iseTi operatori, romlisTvisac (αA)x=αAx yoveli x∈Ex-Tvis. C–s ewodeba A da B operatorebis jami (C= A+B), Tu Ex-

sivrcis nebismieri x wertilisaTvis Cx=Ax+Bx. cxadia, C(αx)= A(αx)+B(αx)= = αAx+αBx da C(x+y)=A(x+y)+B(x+y)= Ax+Bx+Ay+By. amrigad, C wrfivi operato-ria.

aRvniSnoT L(Ex,Ey)–iT simravle yvela uwyveti wrfivi operatorebisa,

romlebis asaxaven Ex-s Ey–Si da sadac Sekrebisa da skalarze gamravle-

bis operacia ukve ganxiluli gziT aris SemoRebuli. advili saCvenebe-

lia, rom L(Ex,Ey) wrfivi sivrcea (aCveneT). L(Ex,Ey) sivrce normirebuli

sivrcecaa. amisaTvis sakmarisia, aviRoT am sivrcis nebismieri elementi

74

(SemosazRvruli A operatori) da vaCvenoT rom operatoris norma ||⋅|| normis ganmartebis aqsiomebs akmayofilebs. marTlac,

1. ||A||≥0 da ||A||=0 ⇔ A=0.; 2. ||αA||=

1sup

xAxα

== ( )

1sup

xAxα

== ( )

1sup

xAxα

== ;Axα

3. Tu A,B∈L(Ex,Ey) da ||x||=1, maSin Ax Bx Ax Bx+ ≤ + ≤ ( )

1sup

xAx

=+ ( )

1sup

xBx

== ,A B+

maSasadame,

A B+ =1

supx =

Ax Bx+ ≤ .A B+

Teorema 14.4. vTqvaT Ex da Ey wrfivi normirebuli sivrceebia erTi da

igive skalaruli velis mimarT. Tu Ey srulia, maSin L(Ex,Ey) sruli sivr-

cea.

damtkiceba. vTqvaT (An) wrfiv, SemosazRvrul operatorTa fundamentu-

ri mimdevrobaa (An∈L(Ex,Ey)), e.i. nebismieri ε>0 ricxvisaTvis arsebobs N ricxvi, rom roca naturaluri m,n>N, maSin m nA A ε− < . aqedan aRniSnuli

naturaluri m da n ricxvebisaTvis

m nA x A x− ≤ .m nA A x xε− ⋅ < ukanasknelidan gamomdinareobs, rom fiqsirebuli x–Tvis (Anx) aris fun-damenturi mimdevroba Ey normirebul sivrceSi. Ey-is sisrulis gamo arse-

bobs am sivrcis iseTi elementi, vTqvaT Ax, rom nA x A x− →0, roca x→∞. Tu yoveli x–Tvis aseT msjelobas CavatarebT, miviRebT asaxvas x→ Ax. vaCvenoT am operatoris erTgvarovneba. An operatoris erTgvarovnebis ga-

mo gvaqvs

||αAx-A(αx)||= ||αAx-αAnx+An(αx)-A(αx)||≤ ||αAx-αAnx||+ ||An(αx)-A(αx)||= =|α|⋅||Ax-Anx||+|| An(αx)-A(αx)||→0, roca n→0.

radgan ||αAx-A(αx)|| ar aris damokidebuli n-ze, amitom davaskvniT ||αAx- -A(αx)||=0, e.i. αAx-A(αx)=0.

A operatoris aditiuroba analogiurad mtkicdeba (aCveneT).

normis Tvisebebidan martivad gamomdinareobs, rom

m n m nA A A A− ≤ − (aCveneT). (14.4)

Cvens SemTxvevaSi m nA A ε− < , amitom (14.4)–is Tanaxmad

.m nA A ε− <

amrigad, ( )nA ricxviTi mimdevroba fundamenturia, amitom SemosazRvru-

licaa. vTqvaT nA ≤M. A operatoris SemosazRvrulobis gamo

nA x ≤ nA ⋅ x ≤M x ; magram

Ax ≤M x . maSasadame, A SemosazRvruli operatoria.

yoveli naturaluri m,n>N ricxvebisaTvis da ||x||≤1 –Tvis gvaqvs

.m n m n m nA x A x A A x A A ε− ≤ − ≤ − <

Tu ukanasknel utolobaSi m–is mimarT gadavalT zRvarze, miviRebT

nAx A x ε− ≤ .

75

amrigad,

nA A− =1

supx =

nAx A x ε− ≤ .

maSasadame, (An) operatorTa mimdevroba krebadia A operatorisken L(Ex,Ey) sivrceSi (L(Ex,Ey) sivrcis normis azriT).

vTqvaT E normirebuli sivrcea, sazogadod, kompleqsur ricxvTa vel-

is mimarT. L(Ex,C) sivrces, sadac C kompleqsur ricxvTa simravlea, ewode-

ba E sivrcis SeuRlebuli sivrce da aRiniSneba simboloTi E∗ (E∗=L(Ex,C)). radgan C sivrce srulia, amitom 14.4 Teoremis ZaliT E∗

sruli sivrcea

(maSin roca E sivrce SeiZleba ar iyos sruli).

igive SeiZleba sityva-sityviT gavimeoroT im SemTxvevaSi, roca E aris normirebuli sivrce namdvil ricxvTa velis mimarT.

Teorema 14.5. vTqvaT A aris wrfivi operatori, romelic n-ganzomile-

bian En sivrces asaxavs normirebul Ey sivrceSi. maSin A aris SemosazRvru-li operatori.

damtkiceba. vTqvaT e1, e2,…, en bazisia En sivrceSi, nE

⋅ aris norma En-Si, da

x =1

.n

i i ni

x e E=

∈∑ maSin A operatoris wrfivobisa da koSisa da Svarcis uto-

lobis ZaliT

|| A x ||=1

n

i ii

x Ae=∑ ≤

1

| |n

i ii

x Ae=∑ ≤

1/ 22

1

n

ii

Ae=

⎛ ⎞ ⋅⎜ ⎟⎝ ⎠∑

1/ 22

1

n

ii

x=

⎛ ⎞⎜ ⎟⎝ ⎠∑ ,

anu

Ax ≤Mn2nR

x ,

sadac Mn≡1/ 2

2

1

n

ii

Ae=

⎛ ⎞ ⋅⎜ ⎟⎝ ⎠∑ Teoremis dasamtkiceblad sakmarisia gaviTvaliswi-

noT (10.10) Sefaseba.

am Teoremis safuZvelze davaskvniT, rom sasrulganzomilebian normi-

rebul sivrceze gansazRvruli yoveli wrfivi funqcionali aris uwyveti

funqcionali.

§15. hanisa da banaxis Teorema normirebuli

sivrceebisaTvis da misi Sedegebi

hanisa da banaxis Teorema normirebuli sivrceebisaTvis ase SeiZleba

Camoyalibdes.

Teorema 15.1 (hani da banaxi (normirebuli sivrceebisaTvis)). vTqvaT E namdvili normirebuli sivrcea (anu normirebuli sivrcea namdvil ricxv-

Ta velis mimarT), L misi qvesivrcea (ara aucileblad Caketili qvesivrce),

xolo f0 wrfivi, SemosazRvruli funqcionalia am qvesivrceze. maSin arse-

bobs E sivrceze gansazRvruli iseTi uwyveti f funqcionali, romelic

emTxveva f0 funqcionals L qvesivrceze (f(x)= f0(x), x∈L), amasTan

0 .L E

f f= (15.1)

76

damtkiceba. SemoviRoT aRniSvna 0 .L

f k≡ vTqvaT P(x)=k||x||. cxadia, P aris dadebiTad erTgvarovani, amozneqili funqcionali (SeamowmeT), amasTan

yoveli x∈L-Tvis | f0(x)|≤ k||x||. (ratom?)

kerZod, f0(x)≤ k||x||. wrfivi sivrceebisaTvis hanisa da banaxis 10.1 Teoremis

Tanaxmad f0 SeiZleba gavagrZeloT E sivrceze wrfiv f funqcionalamde ise,

rom

f(x)≤k||x||, x∈E. ukanasknel utolobaSi, Tu x–is nacvlad ganvixilavT - x-s, miviRebT

-f(x)=f(-x)≤ k||-x||=k||x||. amrigad,

| f(x)| ≤k||x||, e.i.

0 .E L

f f≤ (15.2) meore mxriv,

0 01 1 1

sup ( ) sup ( ) sup ( ) ,x E x L x L

E Lx x x

f f x f x f x f∈ ∈ ∈= = =

= ≥ = =

anu

Ef ≥ 0 .

Lf (15.3)

(15.2) da (15.2) Tanafardobebidan miiReba (15.1).

am SemTxvevaSi amboben, rom f funqcionali warmoadgens f0 funqciona-

lis gagrZelebas normis Seucvlelad.

Sedegi 1. Tu x0 aris normirebuli E sivrcis aranulovani elementi,

maSin E sivrceze arsebobs iseTi wrfivi, uwyveti f funqcionali, rom

f =1 da f(x0)= ||x0||. (15.4) damtkiceba. erTganzomilebian L≡{αx0} qvesivrceze (α skalaria; misi

cvlilebiT miiReba simravle {αx0}) ganvsazRvroT f0 funqcionali Semdeg-

nairad

f0(αx0)=α ||x0||. cxadia,

|| f0||00

0 01

sup ( )x Lx

f xα

αα

∈=

= ( )00

01

supx Lx

xα

αα

∈=

=00

01

supx Lx

xα

αα

∈=

= =1.

15.1 Teoremis ZaliT es funqcionali SeiZleba gavagrZeloT mTels E sivrceze. miviRebT E sivrceze gansazRvrul funqcionals, romelic akma-

yofilebs (15.4) pirobas. Sedegi 1-dan gamomdinareobs Sedegi 2. yoveli x0≠0-Tvis arsebobs iseTi wrfivi, uwyveti f funqcio-

nali, rom f(x0)≠0. Sedegi 2 SeiZleba CamovayaliboT Semdegi eqvivalenturi formiT.

Sedegi 20. nebismieri gansxvavebuli x′ da x′′ wertilebisaTvis (x′,x′′∈E) arsebobs iseTi wrfivi, uwyveti funqcionali f, rom f(x′)≠f(x′′) (daamtkiceT).

Sedegi 3. vTqvaT M aris normirebuli E sivrcis raime Caketili mra-

valsaxeoba (qvesivrce). vigulisxmoT, rom x0∈X\ M. maSin arsebobs mTel E sivrceze gansazRruli iseTi wrfivi, uwyveti funqcionali, rom

1. f(x)=0, roca x∈M; 2. f(x0)=1;

77

3. f =1/d, sadac d= 0inf ( , )M

xτ

ρ τ∈

= 0inf )M

xτ

τ∈

− .

damtkiceba. aviRoT M′={tx0+τ}, sadac τ∈M, xolo t nebismieri namdvili

ricxvia. M′ simravlis yoveli elementi calsaxad warmoidgineba y= tx0+τ saxiT. davuSvaT y=t1x0+τ1, sadac τ∈M, t∈R. aqedan miviRebT

tx0+τ = t1x0+τ1 ⇔ (t-t1)x0=τ1-τ. (15.4) cxadia, τ1-τ∈M, amitom (t-t1)x0∈M. Tu t-t1≠0, maSin (15.4) tolobidan gveqneba:

x0=(τ1-τ)/(t-t1)∈M, es ki debulebis pirobas ewinaaRmdegeba. Tu t=t1, maSin

(15.4)-dan davaskvniT, rom τ1=τ, es ki aRniSnuli warmodgenis erTaderTo-

bas niSnavs. ganvsazRvroT M′ mravalsaxeobaze funqcionali f0 ase: f0(y)=t. Tu y∈M, maSin t=0 e.i. sruldeba piroba 1. Tu y=x0, maSin t=0, e.i. sruldeba

piroba 2. axla vTqvaT t≠0 y≠0∈E da ganvixiloT

|f0(y)|=|t|= 0/ / /t y y y x tτ= + . (15.5)

Tu gaviTvaliswinebT, rom 0 /x tτ+ = ( )0 /x tτ− − da / tτ− ∈M, miviRebT

( )0 /x tτ− − ≥d, saidanac (15.5)–is safuZvelze gveqneba

|f0(y)|= 0/ /y x tτ+ ≤ / .y d

aqedan gamomdinareobs utoloba 0 1/ .f d≤

axla vaCvenoT, rom 0 1/ .f d≥ Tu gaviTvaliswinebT d manZilisa da in-

fimumis ganmartebas, maSin movZebniT M qvesivrcis iseT (τn) mimdevrobas,

rom d= 0lim .nnx τ

→∞− koSisa da Svarcis utolobis gamoyenebiT davaskvniT

0 0 0 0 0 01 ( ) ( ) .n nf x f x x fτ τ= = − ≤ − ⋅

Tu ukanasknel SefasebaSi gadavalT zRvarze, roca n→∞, miviRebT 1≤d⋅||f0||, e.i ||f0||=d. hanisa da banaxis 15.1 Teoremis gamoyenebiT f0 funqcionali SeiZ-

leba gavagrZeloT mTel E sivrceze normis Seucvlelad.

§16. SeuRlebul sivrceTa magaliTebi.

meore SeuRlebuli sivrce

me-14 paragrafSi SemoRebuli iyo SeuRlebuli sivrcis cneba. axla

ganvixiloT aseTi sivrcis ramdenime mniSvnelovani magaliTi.

1. vTqvaT En aris n–ganzomilebiani wrfivi sivrce namdvili an kompleq-

sur ricxvTa velis mimarT. masSi avirCioT raime bazisi e1,e2, …,en. maSin En–is nebismieri elementi erTaderTi saxiT warmoidgineba aseTi saxiT x=

=1

.n

i ii

x e=∑ Tu f aris En sivrceze gansazRvruli wrfivi funqcionali, maSin,

cxadia,

f(x)=1

( ).n

i ii

x f e=∑ (16.1)

amrigad, wrfivi funqcionali calsaxad ganisazRvreba misi mniSvnelobe-

biT e1, e2, …, en bazisis veqtorebze, amasTan es mniSvnelobebi SeiZleba mo-

ices nebismierad. yoveli i-Tvis (i=1,2,…,n) ganvixiloT e1, e2, …, ei-1, ei+1,…, en

78

veqtorebze “moWimuli” wrfivi mravalsaxeoba (anu E sivrcis minimaluri

qvesivrce, romelic e1, e2, …, ei-1, ei+1,…, en veqtorebs Seicavs). aRvniSnoT igi

simboloTi Mi. cxadia, Mi iqneba E sivrcis n–ganzomilebiani wrfivi qve-

sivrce. hanisa da banaxis Teoremis Sedegi 3-is ZaliT (ix. §15) iarsebebs En sivrceze gansazRvruli wrfivi, uwyveti funqcionali gj (j=1,2,…,n), rom

gj(ei)=1, ,0, .

i ji j=⎧

⎨ ≠⎩

Tu

Tu

es funqcionalebi wrfivad damoukidebelia. marTlac, vTqvaT

C1g1 + C2g2+…+ Cngn=0, anu E sivrcis yoveli x elementisaTvis

C1g1(x) + C2g2(x) +…+ Cngn(x)=0, sadac C1,C2,…, Cn raRac mudmivebia. kerZod, nebismieri j-Tvis (j=1,2,…,n) mi-viRebT

C1g1(ej) + C2g2(ej) +…+ Cngn(ej)=0, anu Cjgj(ej)=0, saidanac davaskvniT Cj=0, j=1, 2,…, n. meore mxriv, cxadia, rom

gj(x)=1

( ) .n

i j i ji

x g e x=

=∑

amitom (16.1) toloba SeiZleba asec Caiweros

f(x)=1

( ) ( ),n

i ii

f e g x=∑ x∈En,

anu, rac igivea,

f=1

( ) .n

i ii

f e g=∑

amrigad, gj (j=1,2,…,n) funqcionalebi warmoadgenen baziss En sivrcis SeuR-

lebul (En)∗≡ nE∗

sivrceSi; e.i. n-ganzomilebiani sivrcis SeuRlebuli sivrce n-ganzomilebiania.

vTqvaT En sivrceSi moqmedebs norma 1/

1,

pnp

ii

x x=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ 1<p<∞. advili Sesa-

mowmebelia, rom ukanaskneli normis ganmartebis yvela pirobas akmayofi-

lebs. SemoviRoT aRniSvna f(ei)≡fi da vaCvenoT, rom f funqcionalis (f∈ nE∗)

norma moicema tolobiT

1/

1

,qn

qi

if f

=

⎛ ⎞= ⎜ ⎟⎝ ⎠∑ sadac

1 1 1p q

+ = . (16.2)

marTlac, (16.1) tolobisaTvis gamoviyenebT ra helderis utolobas (ix.

(1,1)), miviRebT

1( )

n

i ii

f x f x=

≤ ∑ ≤1/ 1/

1 1

,p qn n

p qi i

i ix f

= =

⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑

anu

( )f x ≤1/

1;

qnq

ii

f x=

⎛ ⎞⋅⎜ ⎟

⎝ ⎠∑

es ki niSnavs, rom 1/

1

.qn

qi

if f

=

⎛ ⎞≤ ⎜ ⎟⎝ ⎠∑ (16.3)

79

axla x–is rolSi vigulisxmoT veqtori, romlis i–uri koordinatia (i=1, 2,…,n)

1

1/

1

sgn.

qi i

i pnq

ii

f fx

f

−

=

=⎧ ⎫⎨ ⎬⎩ ⎭∑

maSin advilad SeiZleba davrwmundeT, rom ||x||=1. amis garda, aseTi x–Tvis

(16.1) tolobidan gveqneba:

f(x)=1

1/1

1

sgnqni i i

pni qi

i

f f f

f

−

=

=

⋅=

⎧ ⎫⎨ ⎬⎩ ⎭

∑∑

11/

1

nq

ii

pnq

ii

f

f

=

=

⎧ ⎫⎨ ⎬⎩ ⎭

∑

∑=

1 1/

1

pnq

ii

f−

=

⎧ ⎫⎨ ⎬⎩ ⎭∑ =

1/

1

qnq

ii

f=

⎧ ⎫⎨ ⎬⎩ ⎭∑ .

aqedan

x =1

sup ( )x

f x=

≥1/

1

qnq

ii

f=

⎧ ⎫⎨ ⎬⎩ ⎭∑ .

maSasadame, (16.3)-is ZaliT adgili aqvs (16.2)-s. 2. vTqvaT H separabeluri hilbertis sivrcea. vigulisxmoT, rom L⊂H

aris H sivrcis Caketili qvesivrce. ganvixiloT L qvesivrcis orTogona-

luri damateba L⊥. me-13 paragrafSi naCvenebi iyo, rom L⊥ aris wrfivi mra-

valsaxeoba da L-is Caketilobis SemTxvevaSi misi L⊥ orTogonaluri dama-

tebac Caketilia. aRsaniSnavia, rom (L⊥)⊥=L. marTlac, ganmartebis Tanaxmad,

(L⊥)⊥ aris H sivrcis yvela im elementTa simravle, romlebic L⊥

-is yoveli

elementis orTogonaluria. kerZod, L-is nebismier elements es Tviseba

gaaCnia. maSasadame, L⊂(L⊥)⊥. piriqiT, vTqvaT x∈(L⊥)⊥⊂ H. 13.2 Teoremis ZaliT

x-Tvis samarTliania warmodgena x=h1+h2, sadac h1∈L, xolo h2∈L⊥. gvaqvs

0=(x,h2)=(h1,h2)+ (h2,h2)=(h2,h2)=2

2h ⇒ h2=0.

amrigad, x=h1∈L, e.i. (L⊥)⊥⊂L. damtkicebulis ZaliT gamarTlebulia L–sa da (L⊥)⊥

–s urTierTorTo-

gonaluri qvesivrceebi vuwodoT.

hilbertis sivrceze ganvixiloT funqcionali f(x)=(x,y), sadac y aris H

sivrcis aranulovani veqtori. cxadia,

|f(x)|=|(x,y)|≤||x||⋅||y||. maSasadame,

|| f ||≤||y||. magram

f(y/||y||)=( y/||y||, y)=(y,y)/||y||=||y||, anu

|| f ||=1

sup ( ) .x

f x y=

≥

es ki niSnavs, rom

|| f ||=||y||. Teorema 16.1 (risi). hilbertis H sivrceze gansazRvrul uwyvet wrfiv

funqcionals aqvs saxe

f(x)=(x,y), (16.4) sadac

|| f ||=||y||.

80

damtkiceba. vTqvaT f wrfivi, uwyveti funqcionalia hilbertis H sivr-

ceze. L–iT aRvniSnoT am sivrcis yvela im x wertilTa simravle, romlis-

Tviasac f(x)=0. f funqcionalis erTgvarovnebidan da adiciurobidan gamom-

dinareobs, rom L aris H–is wrfivi qvesivrce (ratom?), xolo f-is uwyve-tobidan miiReba L–is Caketiloba (ratom?).

vTqvaT L⊥ aris hilbertis H sivrceSi L–is orTogonaluri damateba.

ganvixiloT nebismieri z∈H\ L. maSin 13.2 Teoremis ZaliT z=x0+y0, sadac x0∈L, xolo y0∈L⊥

. cxadia, y0∉L (ratom?). amitom f(y0)=α≠0. vTqvaT y1≡y0/α, ma-Sin y1∉L, amasTan f(y1)=1. axla ganvixiloT nebismieri x∈H veqtori da da-

vuSvaT f(x)=β. maSin f(x)-β f(y1)=0, anu f(x-βy1)=0, saidanac miviRebT x-βy1≡u∈L. amrigad, x=βy1+u. radgan y1∈L⊥, amitom y1⊥ u. maSasadame,

(x, y1)=(βy1+u, y1)=β (y1, y1)= β ||y1||2. aqedan gamomdinareobs, rom

f(x)=β=(x, y1/|| y1||2), an, rac igivea, yoveli x∈H–Tvis f(x)= (x,v), sadac v=y1/|| y1||2. amasTan, rogorc

am Teoremis Camoyalibebamde vaCveneT,

|| f ||=||v||. aRsaniSnavia, rom (16.4) warmodgena erTaderTia Semdegi azriT. Tu f(x)=

=(x, y1) da f(x)=(x, y2), maSin y1=y2. marTlac, gvaqvs

(x, y1)=(x, y2) ⇒ (x, y1-y2)=0. Tu x-is rolSi vigulisxmebT y1-y2 –s, maSin miviRebT

(y1-y2, y1-y2)=0 ⇒ ||y1-y2||2=0 ⇒ y1=y2. damtkicebuli risis Teoremidan gamomdinareobs, rom H sivrceze gan-

sazRvrul wrfiv, uwyvet funqcionalTa simravlesa da H sivrces Soris

arsebobs izometriuli izomorfizmi. rogorc cnobilia, aseT SemTxvevaSi

metrikul (da, kerZod, normirebul sivrceebs) ar vansxvavebT erTmaneTi-

sagan. amitom SeiZleba iTqvas, rom hilbertis sivrcis SeuRlebuli sivr-

ce emTxveva Tavis Tavs: H∗=H. cxadia,ukanaskneli winadadebidan gamomdi-

nareobs wina punqtis ZiriTadi debuleba (n-ganzomilebiani sivrcis Se-

uRlebuli sivrce n-ganzomilebiania)

3. ganvixiloT 0-ken krebad mimdevrobaTa c0 sivrce (ix. §2, punqti 6). aR-saniSnavia, rom c0 sivrcis SeuRlebuli 0c∗

sivrce izomorfuli izometri-

is sizustiT emTxveva l1 sivrces (ix. §2, punqti 4), anu 0c∗= l1. ukanasknel wi-

nadadebas Cven aq ar davamtkicebT. 4. mtkicdeba (Cven mtkicebas ar ganvixilavT), rom izomorfuli izo-

metriis sizustiT 1l∗=m (ix. §2, punqti 5).

5. SeiZleba imis mtkiceba, rom pl∗ = ql izomorfuli izometriis sizus-

tiT, sadac 1 1 1p q

+ = , p>1.

axla ganvixiloT meore SeuRlebuli sivrce da masTan dakavSirebuli

zogierTi sakiTxi.

radgan uwyveti, wrfivi funqcionalebi, romlebic gansazRvrulia E si-vrceze, qmnian normirebul SeuRlebul E∗

sivrces, amitom SeiZleba visa-

ubroT E∗–is SeuRlebul sivrceze, anu e.w. meore SeuRlebul E∗∗

sivrce-

ze. Tavis mxriv, analogiuri msjeloba SeiZleba CavataroT E∗∗–is mimarT,

da a.S.

81

SevniSnoT, rom E sivrcis yoveli x0 elementi E∗–ze gansazRvravs funq-

cionals. marTlac, vTqvaT

0 0( ) ( ),x f f xψ = (16.5) sadac x0 aris E sivrcis fiqsirebuli elementi, xolo f icvleba mTel E∗

SeuRlebul sivrceze. (16.5) tolobis Tanaxmad yovel f funqcionals Se-

esabameba 0( )x fψ ricxvi, anu E∗

–ze ganisazRvreba funqcionali.Ees funqci-

onali wrfivia. marTlac,

0 1 2 1 2 0( ) ( )( )x f f f f xψ α β α β+ = + =

=0 01 0 2 0 1 2( ) ( ) ( ) ( ),x xf x f x f fα β αψ βψ+ = +

amasTan aseTi funqcionali uwyvetia E∗–ze. marTlac,

0 0 0 0( ) ( ) .x f f x f x x fψ = ≤ ⋅ = ⋅

amrigad,

0 0 .x xψ ≤ (16.6)

0xψ funqcionali SemosazRvrulia da, maSasadame, uwyvetic.

Cven avageT E sivrcis asaxva E∗∗ sivrceSi. es asaxva aris wrfivi:

1 2 1 21 2 1 2( ) ( ) ( ) ( ) ( ) ( ).x x x xf f x x f x f x f fα βψ α β α β αψ βψ+ = + = + = +

asaxvas x0→0xψ ewodeba bunebrivi asaxva E sivrcisa E∗∗

sivrceSi. is aRi-

niSneba simboloTi π. es asaxva aris urTierTcalsaxa E sivrcisa E∗∗ -is raRac mravalsaxeobaze (anu ψx-ebis simravleze). marTlac, hanisa da bana-

xis 15.1 Teoremis Sedegi 20-is ZaliT E sivrcis nebismieri ori gansxvavebu-li x′ da x′′ wertilebisaTvis arsebobs iseTi funqcionali f∈E∗

, rom f(x′)≠ ≠f(x′′), anu xψ ′ da xψ ′′ erT f∈E∗

wertilSi mainc aris gansxvavebuli, e.i. xψ ′ ≠ ≠ xψ ′′ .

π asaxva (x→ xψ ) aris uwyveti. marTlac, (16.6) utoloba asec SeiZleba

gadaiweros

0 0 ,x x x xψ − ≤ −

anu

0 0 ,x x x xψ ψ− ≤ −

saidanac gamomdinareobs π asaxvis uwyvetoba. axla vaCvenoT, rom, Tu E aris normirebuli sivrce, maSin bunebrivi

asaxva E sivrcisa E∗∗-Si aris izometria. marTlac, Cven ukve vaCveneT x→ → xψ asaxvis bieqciuroba da (15.6) utolobis samarTlianoba. meore mxriv,

hanisa da banaxis 15.1 Teoremis Sedegi 1-is Tanaxmad nebismieri x-Tvis mo-

iZebneba iseTi aranulovani f funqcionali, rom

xψ (f)=f(x)=||f||⋅||x||, anu

| xψ (f)|=||f||⋅||x||, e.i.

( )sup x

xf E

fx

fψ

ψ∗∈

= ≥ .

ukanaskneli da (16.6) Sefasebidan davaskvniT, rom .x xψ = (16.7)

82

amrigad, bunebrivi asaxva π axorcielebs E normirebuli sivrcis izomet-

riul izomorfizms E∗∗ orjer SeuRlebul sivrcis raRac mravalsaxeoba-

ze, anu

E∗∗⊂E. E normirebul sivrces ewodeba refleqsuri sivrce, Tu π(E)=E∗∗, anu Tu

simravlis saxe bunebrivi asaxvis dros emTxveva E∗∗–s.

radgan normirebuli sivrcis SeuRlebuli sivrce srulia, amitom yo-

veli refleqsuri normirebuli sivrce srulia (ratom?).

sasrulganzomilebiani evkliduri sivrceebi da hilbertis sivrceebi

warmoadgenen refleqsuri sivrceebis umartives magaliTebs. rogorc naC-

venebi iyo, aseTi sivrceebisaTvis E=E∗tolobac ki sruldeba.

normirebul E sivrces ewodeba TviTSeuRlebuli, Tu E=E∗. amrigad, ne-

bismieri hilbertis sivrce TviTSeuRlebulia.

c0 sivrce warmoadgens sruli ararefleqsuri sivrcis magaliTs. marT-

lac, rogorc zemoT iyo aRniSnuli, 0c∗= l1 (ix. punqti 3) da 1l

∗=m (ix. punqti

4). m da c0 sivrceebi ar SeiZleba daemTxves erTmaneTs izometriuli izo-

morfizmis sizustiT, radgan m araseparabeluri sivrcea, xolo c0 – sepa-rabeluri (ix. §3, punqtebi 5 da 6).

C[a,b] sivrce ararefleqsuria, ufro metic, ar arsebobs iseTi normire-

buli sivrce, romlisTvisac C[a,b] sivrce iqneba SeuRlebuli. am faqts aq

ar davamtkicebT.

lpsivrce (p>1) refleqsuria. marTlac, pl∗=lq, xolo ql

∗=lp, e.i. pl∗∗

=lp; amasTan,

Tu p≠2, maSin pl∗ ar emTxveva lp–s.

§17. operatorTa Tanabrad da Zlierad krebadoba.

banaxisa da Steinhauzis Teorema

vTqvaT An (n=1,2,...) aris wrfivi, uwyveti operatori, romelic Ex normi-

rebul sivrces asaxavs Ey normirebul sivrceSi. vityviT, rom aseT opera-

torTa (An) mimdevroba Tanabrad krebadia A: Ex → Ey operatorisaken, Tu

lim 0.nnA A

→∞− =

Tu (An) operatorTa mimdevroba Tanabrad krebadia A operatorisaken, maSin yoveli fiqsirebuli x∈Ex wertilisaTvis

lim 0,nnA x Ax

→∞− =

anu, rogorc amboben, operatorTa mimdevrobis Tanabari krebadoba iwvevs

amave mimdevrobis wertilovan (Zlier) krebadobas. ukanaskneli debulebis

samarTlianoba martivad gamomdinareobs Sefasebidan

( ) .n n nA x Ax A A x A A x− ≤ − ≤ − ⋅

aRsaniSnavia, rom (An) operatorTa mimdevrobis wertilovani krebadoba

ar uzrunvelyofs amave mimdevrobis Tanabar krebadobas. marTlac, ganvi-

xiloT operatorTa (Pn) mimdevroba, romlisTvisac Pn : l2→ l2, amasTan da-

vuSvaT yoveli x=(x1, x2,..., xn,...)∈l2 wertilisaTvis Pn(x)=(x1, x2,..., xn,0,0, ...). gveq-neba

83

2 2 21 2

1( ) (0,0,..., , ,...) .n n n i

i nP x x x x x

∞

+ += +

− = = ∑

radgan x∈l2 amitom

2

1lim 0,in i n

x∞

→∞− +

=∑

e.i.

lim ( ) 0.nnP x x

→∞− =

es ki niSnavs, rom operatorTa (Pn) mimdevroba l2 sivrceze wertilovnad

(Zlierad) krebadia igivuri I operatorisken . cxadia, yoveli Pn Semosaz-

Rvruli operatoria. marTlac,

1 2( ) ( , ,..., ,0,0,...)n nP x x x x= =1/ 2

2

1

n

ii

x=

⎧ ⎫⎨ ⎬⎩ ⎭∑ =

=

1/ 22

1i

i

x∞

=

⎧ ⎫⎨ ⎬⎩ ⎭∑ = ,x

e.i. || Pn||≤1. axla aviRoT x(n)=(x1, x2,..., xn,...)∈l2 wertili, romlisTvisac

1, 1,0, 1.k

k nx

k n= +⎧

= ⎨ ≠ +⎩

Tu

Tu

maSin, cxadia, || x(n)||=1. amis garda, Pn(x(n))=0. amitom yoveli n-Tvis ( ) ( )

1sup ( ) ( ) ( ) ( )n n

n n nx

P I P x I x P x I x=

− = − ≥ − =

=||I(x(n))||=||x(n)||=1.

rogorc vxedavT, SemosazRvrul operatorTa (An) mimdevrobis Zlierad

krebadobidan, sazogadod, ar gamomdinareobs amave mimdevrobis Tanabari

krebadoba, Tumca operatorTa wertilovani (Zlierad) krebadoba iwvevs

ricxviTi (||An||) mimdevrobis SemosazRvrulobas.

Teorema 17.1 (banaxi da Steinhauzi-1). vTqvaT mocemulia SemosazRvru-

li operatorebis (An) mimdevroba, romlebic banaxis sivrces Ex sivrces

asaxavs Ey sivrceSi. vigulisxmoT agreTve, rom yoveli fiqsirebuli x∈Ex wertilisaTvis ricxvTa (||Anx||) mimdevroba SemosazRvrulia (sazogadod,

sxvadasxva mudmiviT). am pirobebSi mimdevroba (||An||) aris SemosazRvruli.

damtkiceba. Tu raime { : ,|| || }xK x x E x a r= ∈ − ≤ birTvSi (Anx) mimdevroba Se-mosazRvrulia erTi da igive C mudmiviT (anu ||Anx||≤ C, x∈K, n=1,2,…), maSin

1 1 2 .n n nx a C C CA A x A a

r r r r r r−⎛ ⎞ ≤ + ≤ + =⎜ ⎟

⎝ ⎠

magram nebismieri y veqtori, romlisTvisac ||y||≤1 SeiZleba warmodgenil

iqnas saxiT y=(x-a)/r, sadac ||x-a||≤r (ratom?). amitom

1

2sup ,n ny

CA A yr≤

= ≤

e.i. mimdevroba (||An||) SemosazRvrulia. Uukanasknelis gamo, Tu davuSvebT,

rom dasamtkicebeli ar aris WeSmariti, maSin am daSvebidan miviRebT, rom

{Anx} veqtorTa simravle ar iqneba SemosazRvruli Ey sivrcis arcerT

birTvSi. kerZod, moiZebneba iseTi x1 veqtori da n1 nomeri, rom 1 1 1.nA x >

1nA

operatoris uwyvetobis gamo ukanaskneli utoloba samarTliani iqneba

84

yvela x-Tvis raRac birTvidan 1 1 1{ : ,|| || }xK x x E x x r= ∈ − ≤ . nebismier birTvSi

(Anx) mimdevrobis SemousazRvrelobis gamo, yovel birTvSi, romelic K1-is

qvesimravlea, moiZebneba K1 birTvis iseTi Siga wertili x2 da nomeri n2>n1, rom

2 2 2.nA x > 2nA operatoris uwyvtobis gamo ukanaskneli utoloba sa-

marTliani iqneba raRac 2 2 2{ : ,|| || }xK x x E x x r= ∈ − ≤ birTvSi, K2⊂ K1. axla, ga-

sagebia, rom ariTmetikuli induqciis meTodis gamoyenebiT miviRebT Cake-

til, Calagebul birTvTa K1⊃K2⊃…⊃Kn⊃… mimdevrobas. amave dros, SeiZ-

leba vigulisxmoT, rom maTi radiusebis mimdevroba (rk) krebadia 0–ken. Calagebul birTvTa Sesaxeb Teoremis ZaliT (ix. Teorema 4.2) arsebobs iseTi wertili, romelic ekuTvnis yvela Kn birTvs. amasTan, agebis Tanax-

mad yoveli naturaluri k-Tvis gveqneba 0 .knA x k> ukanaskneli utoloba

ki ewinaaRmdegeba (Anx0) mimdevrobis SemosazRvrulobas.

Teorema 17.2 (banaxi da Steinhauzi-2). vTqvaT {An}⊂L(Ex,Ey). imisaTvis,

rom (An) mimdevroba Zlierad (wertilovnad) krebadi iyos A∈L(Ex,Ey) ope-ratorisaken aucilebelia da sakmarisi, rom

1. simravle {||An||} iyos SemosazRvruli;

2. An→A Zlierad wrfiv 'xE mravalsaxeobaze, romelic mkvrivia Ex-Si.

damtkiceba. (aucilebloba) Tu yoveli x∈Ex–Tvis (Anx) krebadia Ax-ken, maSin normis Tvisebis gaTvaliswinebiT miviRebT || Anx||→|| Ax||, roca n→∞. ma-

Sasadame, (||Anx||) SemosazRvruli ricxviTi mimdevrobaa, e.i. 17.1 Teoremis

ZaliT (||An||) mimdevroba SemosazRvrulia. amis garda, 2 piroba Sesrule-

bulia, radgan 'xE -is rolSi SeiZleba avirCioT Ex.

(sakmarisoba) vTqvaT x∈Ex\ 'xE . radgan

'xE mkvrivia Ex-Si, amitom nebismi-

eri ε>0 ricxvisaTvis movZebniT iseT x′∈ 'xE wertils, rom ||x- x′||< ε. vTqvaT

C=sup nn

A da vaCvenoT, rom yoveli x∈Ex-Tvis Anx→ Ax, roca n→∞. gvaqvs:

( )( ) ( )n n nA x Ax A x x A x Ax A x x′ ′ ′ ′− = − + − + − ≤

n nA x x A x Ax A x x′ ′ ′ ′≤ ⋅ − + − + ⋅ − ≤

2 .nC A x Axε ′ ′≤ + −

Tu gaviTvaliswinebT, rom Anx′→ Ax′, roca n→∞, davaskvniT: arsebobs

iseTi N(ε) ricxvi, rom roca n>N(ε), maSin nA x Ax′ ′− <ε. maSasadame,

nA x Ax− ≤ (2 1) .C ε+

§18. susti krebadoba normirebul sivrceebSi

vTqvaT E normirebuli sivrcea, xolo {xn}⊂E. vityviT, rom (xn) mimdev-

roba sustad krebadia x∈E elementisken, Tu nebismieri wrfivi, uwyveti

f∈E∗ funqcionalisTvis lim ( ) ( ).nn

f x f x→∞

= Tu (xn) mimdevroba sustad krebadia

x-ken, maSin x∈E elements ewodeba (xn) mimdevrobis susti zRvari. aRsaniS-

85

navia, Tu (xn) mimdevrobis susti zRvaria x∈E, maSin igi erTaderTia. mar-

Tlac, vTqvaT x ∈E aris igive (xn) mimdevrobis susti zRvari, anu lim ( )nn

f x→∞

= ( ).f x

ricxviTi (f(xn)) mimdevrobis zRvris erTaderTobis gamo yovel f∈E∗ funq-

cionalisTvis f(x)=f( x ), e.i. f(x- x )=0. Tu x- x ≠0, maSin hanisa da banaxis 15.1 Teoremis Sedegi 1-is ZaliT arsebobs iseTi f0∈E∗

funqcionali, rom

f0(x- x )=||x- x ||. maSasadame, ||x- x ||=0 ⇔ x= x . vityviT, rom E normirebuli sivr-

cis (xn) mimdevroba Zlierad krebadia x∈E elementisken, Tu || xn-x||→0, n→∞.

Teorema 18.1. Tu E normirebuli sivrcis (xn) mimdevroba Zlierad

krebadia amave sivrcis x elementisaken, maSin es mimdevroba sustad kreba-

dicaa igive x wertilisken.

damtkiceba. ganvixiloT nebismieri uwyveti f∈E∗ funqcionali. gvaqvs

( ) ( ) ( ) .n n nf x f x f x x f x x− = − ≤ ⋅ −

am utolobidan martivad miiReba winadadeba: Tu lim 0,nnx x

→∞− = maSin f(xn)→

→f(x), roca n→∞. (xn) mimdevrobis sustad krebadobidan, sazogadod, ar gamomdinareobs

igive mimdevrobis Zlierad krebadoba. magaliTisaTvis ganvixiloT nebis-

mieri hilbertis sivrce Tvladi e1, e2, …, en,… bazisiT. vTqvaT f∈E∗ uwyveti

funqcionalia. risis 16.1 Teoremis Tanaxmad yoveli x∈H elementisaTvis

gvaqvs warmodgena f(x)= ( , )fx y , sadac yf∈H. kerZod, yoveli n-Tvis f(en)=(en, yf). radgan (en) orTonormirebuli sistemaa, amitom yf elementis furies koe-

ficientTa mimdevroba 0-ken krebadia (ix. beselis (11.4) utoloba), e.i.

(yf,en)→0, roca n→∞, anu lim ( ) 0.nnf e

→∞= amrigad, (en) mimdevroba sustad krebadia

0-ken.

meore mxriv, Tu n≠m, maSin 2 ( , ) ( , ) ( , )n m n m n m n n n me e e e e e e e e e− = − − = − −

- 2 2( , ) ( , ) 2.m n m m n me e e e e e+ = + = amrigad, es mimdevroba ar aris fundamenturi, e.i. ar aris krebadi.

normirebuli E sivrcis M⊂E qvesimravles ewodeba sustad Semosaz-

Rvruli, Tu yoveli f∈E∗ funqcionalisTvis ricxviTi {f(x), x∈M} simravle

SemosazRvrulia.

cxadia, Tu mimdevroba sustad krebadia, maSin is sustad SemosazRvru-

lia.

Tu simravle SemosazRvrulia, maSin is sustad SemosazRvrulia. es

faqti uSualod gamomdinareobs Sefasebidan

| ( ) | ,f x f x≤ ⋅ x∈M. samarTliania, agreTve, sapirispiro debulebac.

Teorema 18.2. normirebul sivrceSi yoveli sustad SemosazRvruli

simravle SemosazRvrulia.

damtkiceba. davuSvaT M⊂E simravle sustad SemosazRvrulia, magram

ar aris SemosazRvruli; e.i. M simravleSi moiZebneba (xn) SemousazRvreli

mimdevroba. aviRoT nebismieri funqcionali f∈E∗ da ganvixiloT warmodge-

na (ix. (16.7)) ( ) ( ),

nn xf x fψ= (18.1)

86

sadac nxψ ∈E∗∗

da

nn xx ψ= . (18.2)

cxadia, (xn) mimdevroba iqneba sustad SemosazRvruli, anu yoveli f∈E∗

funqcionalisTvis ricxviTi ( )( )nx fψ mimdevroba SemosazRvrulia. amitom

banaxisa da Steinhauzis 17.1 Teoremis Tanaxmad ( )nxψ == ( )nx mimdevroba

iqneba SemosazRvruli. miviReT winaaRmdegoba.

Sedegi. Tu mimdevroba sustad krebadia, maSin is SemosazRvrulia (aC-

veneT).

Teorema 18.3. sasrulganzomilebian sivrceSi mimdevrobis krebadoba

da sustad krebadoba tolfasia.

damtkiceba. vTqvaT EmEaris m-ganzomilebiani sivrce da (x(n)) mimdevro-

ba sustad krebadia x(0)-ken. vigulisxmoT, rom e1, e2, …, em bazisia Em sivr-

ceSi. maSin

( ) ( )

1

mn n

i ii

x x e=

= ∑ da (0) (0)

1

.m

i ii

x x e=

= ∑

aviRoT EmEsivrceze gi∈E∗ funqcionali (ix. §16), romlisTvisac gi(ej)=0, Tu

i≠j, xolo gi(ei)=1. maSin Teoremis pirobis ZaliT yoveli i–Tvis (i=1,2,…,m) ( ) ( ) (0) (0)( ) ( ) , .n n

i i i ig x x g x x n= → = → ∞

amrigad, ( )nix → (0)

ix (i=1,2,…,m), roca n → ∞ ; e.i. miviReT koordinatuli kre-

badoba, rac niSnavs, rom (x(n)) mimdevroba Zlierad krebadia x(0)

-ken.

vTqvaT (xn) hilbertis H sivrcis elementTa mimdevrobaa, xolo x0 - am-

ave sivrcis raime wertili. (16.1) Teoremis Tanaxmad yoveli f∈H∗ funqcio-

nalisTvis gvaqvs f(xn)= ( , )n fx y da f(x0)= 0( , ),fx y yf∈H;

anu

f(xn)-f(x0)= f(xn-x0)=(xn-x0, yf). radgan asaxva f→ yf aris bieqcia hilbertis H sivrcisa Tavis TavSi, ami-

tom hilbertis sivrceSi elementTa (xn) mimdevrobis x0∈H –ken krebadoba

niSnavs am sivrcis yoveli y wertilisTvis Semdegi pirobis Sesrulebas:

(xn-x0, y)→0, n → ∞ . Teorema 18.4. vTqvaT normirebuli Ex da Ey normirebuli sivrceebia,

xolo wrfivi operatori A∈L(Ex,Ey). Tu (xn) mimdevroba sustad krebadia x0

wertilisken, maSin (Axn) mimdevroba aseve sustad krebadia Ax0-ken.A

damtkiceba. vTqvaT f∈ yE∗ aris nebismieri funqcionali. f Aψ ≡ kompo-

zicia iqneba wrfivi, uwyveti funqcionali Ex sivrceze (ψ∈ xE∗ ). marTlac,

vTqvaT x∈Ex, maSin

| ( ) | | ( )( ) | | ( ) |x f A x f Axψ = = ≤

.f Ax f A x≤ ⋅ ≤ ⋅ ⋅

amrigad, ψ uwyveti funqcionalia da, amave dros,

.f Aψ ≤ ⋅

gvaqvs

0 0 0

0 0 0

| ( ) ( ) | | ( ) | | ( ( )) || ( )( ) | | ( ) | | ( ) ( ) | .

n n n

n n n

f Ax f Ax f Ax Ax f A x xf A x x x x x xψ ψ ψ

− = − = − == − = − = −

87

radgan ψ uwyveti funqcionalia Ex-ze, amitom ukanasknelis zRvari nulia,

roca n→∞. Teorema 18.5. imisaTvis, rom (xn) mimdevroba (xn∈Ex) sustad krebadi iy-

os x0 elementisken, aucilebeli da sakmarisia, rom

1. ricxvTa {||xn||} simravle iyos SemosazRvruli;

2. xE∗ SeuRlebul sivrceSi raime mkvrivi simravlidan aRebuli yoveli

f funqcionalisaTvis adgili hqondes tolobas

0lim ( ) ( ).nnf x f x

→∞=

damtkiceba. Tu (xn) mimdevroba sustad krebadia, maSin 18.2 Teoremis Sede-

gis ZaliT piroba 1 daculia. cxadia, Sesrulebulia piroba 2-c. piriqiT,

Tu Sesrulebulia piroba 2, es niSnavs (ix. (18.1)) xE∗sivrcis mkvrivi sim-

ravlidan aRebuli yoveli f funqcionalisaTvis ( )( )nx fψ ricxvTa mimdev-

robis krebadobas 0( )x fψ -ken, xolo 1 piroba ganapirobebs (ix. (18.1)) mim-

devrobis ( )nxψ SemosazRvrulobas. amrigad, daculi iqneba banaxisa da

Steinhauzis 17.2 Teoremis pirobebi; e.i. yoveli f∈ xE∗ funqcionalisaTvis

f(xn)= ( )nx fψ = ( )

nx fψ → 0( )x fψ = f(x0), n→∞.

§19. Seqceuli operatori. operatoris

speqtri. operatoris rezolventa

maTematikisa da bunebismetyvelebis mravali problemis gadawyveta da-

kavSirebulia Ax= y operatoruli gantolebis amoxsnasTan. am amocanis am-

oxsna pirdapir kavSirSia A operatoris SeqcevasTan. Tu arsebobs Seqce-

uli A-1operatori, maSin aRniSnuli gantolebis amonaxsni cxadi saxiT

Caiwereba: x=A-1y. didi mniSvneloba aqvs im pirobebis dadgenas, romelTa

Sesrulebisas Seqceuli operatori arsebobs. qvemoT mokled SevexebiT

operatoris Seqcevadobis sakiTxs da SeviswavliT masTan dakavSirebul

zogierT problemas.

vTqvaT Ex da Ey wrfivi sivrceebia da wrfivi A operatori gansazRvru-lia DA wrfiv mravalsaxeobaze (DA⊂Ex). ImA-iT aRvniSnoT A operatoris mniSvnelobaTa simravle, anu ImA aris DA simravlis saxe – A(DA).

A operators ewodeba Seqcevadi, Tu nebismieri y∈ImA elementisaTvis

gantolebas gaaCnia erTaderTi amonaxsni. Tu A operatori Seqcevadia, ma-Sin ImA simravlis nebismier y elements calsaxad SeiZleba SevusabamoT

x∈DA elementi ise, rom Ax=y. operators, romelic axorcielebs am Tanado-

bas ewodeba A operatoris Seqceuli operatori da aRiniSneba simboloTi

A-1. amrigad, Tu Seqceuli A-1

operatori arsebobs maSin A-1: ImA→DA. Teorema 19.1. Tu arsebobs A (A: DA → ImA) wrfivi operatoris Seqceu-

li A-1 operatori, maSin igi wrfivia.

88

damtkiceba. SevniSnoT, rom ImA, anu 1AD − aris wrfivi mravalsaxeoba

(ratom?). vTqvaT y1, y2∈ ImA, xolo α1 da α2 nebismieri ricxvebia. sakmarisia

SevamowmoT Semdegi tolobis samarTlianoba:

A-1(α1y1+α2y2)=α1A-1y1+α2A-1y2. (19.1) vTqvaT Ax1=y1 da Ax2=y2. operatoris wrfivobis gamo gvaqvs

A(α1x1+α2x2)=α1y1+α2y2. (19.2) meore mxriv, Seqceuli operatoris ganmartebis ZaliT, A-1y1=x1 da A-1y2=x2.

Tu ukanasknel tolobebs gavamravlebT, Sesabamisad α1–ze da α2–ze, da

SevkrebT, miviRebT

α1A-1y1+α2A-1y2=α1x1+α2x2. (19.3) Seqceuli operatoris ganmartebis Tanaxmad, (18.2) tolobidan gveqneba

α1x1+α2x2=A-1(α1y1+α2y2). ukanaskneli da (19.3) tolobebidan vRebulobT (19.1)-s.

Seqceuli operatorebis Sesaxeb samarTliania Semdegi umniSvnelovane-

si

Teorema 19.2. (banaxi) vTqvaT SemosazRvruli wrfivi operatoria, ro-

melic urTierTcalsaxad asaxavs banaxis Ex sivrces banaxis Ey sivrceze,

maSin Seqceuli A-1 operatori SemosazRvrulia.

am Teoremis mtkicebas ar ganvixilavT.

sazogadod, SemosazRvruli, wrfivi operatoris Seqceuli ar aris Se-

mosazRvruli. ganvixiloT

magaliTi 1. vTqvaT Ex≡C[0,1] da

Ax≡(Ax)(t)=0

( ) ,t

x dτ τ∫ x∈C[0,1], t∈[0,1].

cxadia, Ax aris [0,1] segmentze uwyvetad diferencirebadi funqcia, amasTan

A SemosazRvruli operatoria. marTlac

[0,1] [0,1]0 0

max ( ) max ( )t t

t tAx x d x dτ τ τ τ

∈ ∈= ≤ ≤∫ ∫

[0,1]max ( ) ,x xτ

τ∈

≤ =

e.i. 1.A ≤ SevniSnoT, rom A operatoris mniSvnelobaTa ImA simravle Sed-

geba yvela iseTi uwyvetad diferencirebadi funqciebisagan, romelTa-

Tvisac y(0)=0 (ratom?). advili saCvenebelia, rom yoveli y∈ImA-Tvis

A-1y= ( ).d y tdt

A-1 SemousazRvrelia. marTlac, ||sinnt||=

[0,1]max | sin |t

nt∈

≤1, amitom

1

1sup sin cos

x

dA x nt n ntdt

−

≤≥ = =

[0,1]max cos .t

n nt n∈

= =

ukanaskneli Sefasebidan miviRebT 1

1sup

xA x−

≤=+∞,

rac A-1 operatoris SemousazRvrelobas niSnavs (ix. §14, gansazRvreba g)).

wrfiv operatorTa TeoriaSi Znelad Tu SeiZleba mivuTiToT ufro

mniSvnelovan cnebaze, vidre operatoris speqtria.

89

vTqvaT A wrfivi operatoria n-ganzomilebian En sivrceSi (A: En→En). λ ricxvs ewodeba A operatoris sakuTrivi mniSvneloba (sakuTrivi ricxvi),

Tu gantolebas Ax=λx (rac igivea, (A-λI)x=0, sadac I erTeulovani operato-

ria) gaaCnia aranulovani amonaxsni (cxadia, A operatoris wrfivobis ga-mo gantolebas Ax=λx yovelTvis aqvs nulovani amonaxsni). sakuTriv mniS-

vnelobaTa simravles ewodeba A operatoris speqtri, xolo ricxvTa da-

narCen mniSvnelobebs - A operatoris regularuli mniSvnelobebi. sxva-

gvarad rom vTqvaT, λ regularuli mniSvnelobaa (regularuli wertilia),

Tu A-λI operatori Seqcevadia, amasTan (A-λI)-1 gansazRvrulia mTel En siv-

rceze. marTlac, Tu λ regularuli mniSvnelobaa, maSin Ax=λx gantolebas

mxolod nulovani amoxsna aqvs. davuSvaT A-λI operatori ar aris Seqceva-di. es imas niSnavs, rom arsebobs iseTi x1 da x2 elementi En sivrcidan, rom

Ax1-λx1=Ax2-λx2, saidanac miviRebT A(x1-x2)-λ(x1-x2)=0; e.i. x1-x2 aris Ax-λx=0 gan-tolebis aranulovani amonaxsni. maSasadame, λ aris A operatoris sakuT-

rivi mniSvneloba da ara – regularuli. piriqiT, Tu A-λI wrfivi opera-tori Seqcevadia, maSin igi En sivrcis mxolod nulovan wertilze iRebs

nulis tol mniSvnelobas. amitom Ax-λx=0 gantolebas mxolod nulovani

amoxsna aqvs. amrigad, λ aris regularuli mniSvneloba. axla vaCvenoT,

rom, Tu λ aris A operatoris regularuli mniSvneloba, maSin (A-λI)-1 gan-sazRvrulia mTel En sivrceze, anu, rac igivea, A-λI operatoris mniSvne-lobaTa simravle emTxveva En–s. ganvixiloT bazisi En sivrceSi: e1, e2,…, en. aviRoT x∈En, maSin gveqneba

x=x1e1+x2e2+…+ xnen,

sadac x1,x2,…, xn skalarebia. radgan A operatori moqmedebs En-Si, amitom

aseTive Tvisebisaa operatori A0= A-λI. A0 wrfivi operatoria, amitom

A0x=x1A0e1+x2A0e2+…+ xnA0en.

amis garda, A0x-Ta simravle (x∈En) qmnis mravalsaxeobas En sivrceSi. rad-

gan λ regularuli mniSvnelobaa, amitom

A0x=x1A0e1+x2A0e2+…+ xnA0en=0 maSin da mxolod maSin, roca x=0, anu roca x1=x2=…=xn=0, e.i. A0e1, A0e2,…, A0en wrfivad damoukidebeli sistemaa. amrigad, aRniSnuli mravalsaxeobis

ganzomileba aris n da igi emTxveva En sivrces. amasTan, radgan (A-λI)-1 wrfivi operatori gansazRvrulia sasrulganzomilebian sivrceze, amitom

SemosazRvrulia (ix. Teorema 10.3). amrigad, sasrulganzomilebiani En sivrcis SemTxvevaSi arsebobs ori

SesaZlebloba:

1. gantolebas Ax=λx gaaCnia aranulovani amonaxsni, e.i. λ aris A opera-toris sakuTrivi mniSvneloba, amasTan (A-λI)-1 ar arsebobs.

2. arsebobs SemosazRvruli operatori (A-λI)-1, gansazRvruli mTel En

sivrceze, anu λ aris regularuli wertili.

Tu operatori usasruloganzomilebian E sivrcezea gansazRvruli (A: E→E), maSin arsebobs mesame SesaZleblobac.

3. operatori arsebobs, anu gantolebas Ax=λx gaaCnia mxolod nulova-

ni amonaxsni, magram (A-λI)-1 operatori ar aris gansazRvruli mTel E

sivrce-ze (SesaZlebelia, SemousazRvrelic iyos E-ze). zogad SemTxvevaSi (rodesac E sivrce ar aris aucileblad sasrul-

ganzomilebiani) SemoviRoT Semdegi gansazRvrebebi.

90

λ ricxvs ewodeba regularuli banaxis sivrceSi moqmedi A operatori-saTvis, Tu Rλ ≡(A-λI)-1 operatori gansazRvrulia mTel E sivrceze da, maSa-sadame, banaxis 19.1 Teoremis Tanaxmad, SemosazRvrulicaa. Rλ operators

ewodeba A operatoris rezolventa, xolo λ ricxvebis yvela sxva mniSvne-

lobaTa simravles - A operatoris speqtri. Tu raime x≠0 wertilisTvis

(A-λI)x=0, maSin (A-λI)-1 ar arsebobs; es niSnavs, rom A operatoris speqtrs miekuTvneba am operatoris yvela sakuTrivi mniSvneloba. yvela im λ ric-xvebis simravle, romelTaTvisac (A-λI)-1 ar arsebobs ewodeba A operato-ris wertilovani speqtri. im λ ricxvebis erTobliobas, romelTaTvisac

(A-λI)-1 arsebobs, magram ar aris gansazRvruli mTel E sivrceze, ewodeba A operatoris uwyveti speqtri. amrigad, λ-s yoveli mniSvneloba aris an

regularuli, an sakuTrivi mniSvneloba, an uwyveti speqtris wertili.

arsebiTi gansxvaveba usasruloganzomilebian sivrceze wrfiv operator-

Ta Teoriisa sasrulganzomilebianTan SedarebiT uwyveti speqtris Sesa-

Zlo arsebobiTac iCens Tavs.

SeiZleba imis mtkiceba, rom regularul wertilTa (mniSvnelobaTa)

simravle Ria simravlea, xolo speqtri, rogorc misi damateba, aris Ca-

ketili.

magaliTi 2. C[a,b] sivrceSi ganvixiloT wrfivi operatori, romelic

gansazRvrulia formuliT:

Ax(t)=tx(t). maSin

(A-λI)x(t)=(t-λ)x(t). A-λI operatori Seqcevadia nebismieri λ–Tvis, radgan tolobidan (t-λ)x(t)=0 gamomdinareobs, rom uwyveti t→x(t) funqcia igivurad udris 0–s. magram, Tu λ∈[a,b], maSin Seqceuli operatori

(A-λI)-1x(t)= 1t λ−

x(t)

ar aris gansazRvruli mTel C[a,b] sivrceze. amasTan L mravalsaxeobaze,

romelzedac es operatoria gansazRvruli SemosazRvrulic ki ar aris.

marTlac, [a,b] segmentze nebismieri uwyveti t→x(t) funqcisaTvis fardoba

( )x tt λ−

ar aris uwyveti; amasTan, SesaZloa t=λ wertilze sasrulic ki ar

iyos. L mravalsaxeobaze operatoris SemousazRvrelobis saCveneblad

ganvixiloT funqcia t→sinn(t-λ). C[a,b] sivrcis normis azriT

[ , ]sin ( ) max sin ( ) 1.

t a bn n tλ λ

∈− = − ≤i

meore mxriv, tolobidan

sin ( )limt

n t ntλ

λλ→

−=

−

davaskvniT

[ , ]

sin ( ) sin ( )max .t a b

n t n t nt t

λ λλ λ∈

− −= ≥

− −

maSasadame,

1 1

1( I) sup ( I) ( )

xA A xλ λ− −

≤− = − ≥

sin ( ) ,n t nt

λλ−

≥ → ∞−

e.i. 1( I)A λ −− =+∞.

91

amrigad, A operatoris speqtri warmoadgens [a,b] monakveTs, amasTan am

operators sakuTrivi mniSvnelobebi ar gaaCnia, e.i. A operators mxolod

uwyveti speqtri aqvs.

magaliTi 3. ganvixiloT l2 sivrceSi A operatori (A: l2 →l2), romlisTvi-

sac A: (x1, x2, …)→(0,x1, x2, …).

am operators ar gaaCnia sakuTrivi mniSvnelobebi. marTlac, gantolebas

Ax=λx gansaxilav SemTxvevaSi eqneba saxe

(0,x1, x2, …)=λ(x1, x2, …); anu

(0,0,…)= (λx1, λx2-x1, λx3-x2, …), e.i.

0=λx1, 0=λx2-x1, λx3-x2=0,… . Tu λ≠0, maSin x1=0, x2=0,… ; Tu λ=0, maSinac x1=0, x2=0,… . amrigad, yoveli λ ricxvisaTvis gantolebas mxolod nulovani amoxsna aqvs.

simartivisaTvis Cven ganvixilavT mxolod SemTxvevas λ=0. aseTi λ-Tvis

(A-λI)-1=A-1 : (0,x1, x2, …)→(x1, x2, …). es operatori SemosazRvrulia. marTlac, l2 sivrceSi normis ganmartebis

Tanaxmad

2 2 21 2

1 1 2 21 2

1 0 ... 1

sup sup ... 1.x x x

A A x x x− −

≤ + + + ≤

= = + + =

cxadia, A-1 operatori gansazRvrulia l2 sivrcis iseT qvesivrceze, romlis

nebismieri elementis pirveli koordinati x1=0. es niSnavs, rom λ=0 ekuT-

vnis uwyvet speqtrs.

⎯⎯⎯⎯⎯⎯⎯

lLi t e r a t u r a

1. Л.А.Люстерник, В.И.Соболев. Элементы функционального анализа, Москва, “Наука”, 1965.

2. А.Н.Колмогоров, С.В.Фомин. Элементы теории функций и функционального анализа, Москва,“Наука”, 1976.

3. В.А.Треногин.Функциональний анализ, Москва,“Наука”, 1980. 4. e.wiTlanaZe. maTematikuri analizis safuZvlebi funqcionalur

sivrceebSi, Tbilisi, Tsu, 1977. 5. В.А.Зорич. Математический анализ, Наука, 1981. 6. И.П.Натансон. Теория функций вещественной переменной, Наука, 1974.

funqcionaluri sivrceebi leqcia ta krsi · li sivrcis cnebamde – tanamedrove matematikis ert-ert...

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