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)|.
samarTliania Semdegi
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)
samarTliania Semdegi
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 funqciis- ken.
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 siv- rce 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 −− =+∞.
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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.
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