hahn-banachòÿ n ÿÝ ØÚ zflkhomepage.fudan.edu.cn/guokunyu/files/2011/08/hahn-banach...stefan...
TRANSCRIPT
![Page 1: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/1.jpg)
Hahn-Banachòÿ½n!ÿÝØÚ`z¯K
H%
EÆêÆÆÆ
1
![Page 2: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/2.jpg)
¼©Û´ïÄÃmƯ§Hahn-Banachòÿ½n´¼©Ûħ3¼©Û9êÆÙ¦©|¥äkÄ
5"§ÌA´Urfmþ5¼òÿ
m§y÷v½¦5¼3"wÌ0
Hahn-Banachòÿ½n1)ê¶2)©Û¶3)AÛ¶4)ÿÀ«"Ó0Hahn-Banachòÿ½n«3ÿÝØ!+Ø!à8AÛ9`z¯K¥A^"
wÌ0¢5m¹§g,í2E¹"
2
![Page 3: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/3.jpg)
Hans Hahn ( 27 September 1879 – 24 July 1934) was
an Austrian mathematician who made contributions
to functional analysis, topology, set theory, the calculus of
variations, real analysis, and order theory.
Stefan Banach ( 30 March 1892–31 August 1945) was
a Polish mathematician who is generally considered one of the
world's most important and influential 20th-century
mathematicians. He was the founder of modern functional
analysis,
and an original member of the Lwów School of
Mathematics. His major work was the 1932 book, (Theory of
Linear Operations), the first monograph on the general theory
of functional analysis.
3
![Page 4: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/4.jpg)
I. Hahn-Banach½nê
X´¢5m, ¡XS5m§XJÙþkS'X≤÷vún
(1)XJx ≤ y, y ≤ z,Kx ≤ z;
(2)XJx ≤ y,Ké∀z ∈ X, x + z ≤ y + z;
(3)XJx ≤ y,Ké∀t > 0, tx ≤ ty.
4
![Page 5: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/5.jpg)
PX+ = x ∈ X : x ≥ 0,K´X+÷v
ex, y ∈ X+,Kx + y ∈ X+;
ex ∈ X+, t > 0,Ktx ∈ X+.=X+´I.
, eP ⊆ X, P÷vÚ, 3Xþ½Â'X“≤” : “x ≤ y” ´y − x ∈ P. Kù'X“≤” ÷vþãún(1) ∼ (3). ÏdS5mÚI5m´,=½S5mX,K(½IX+; , ½IP, KdP(½XþS“≤” ,ù5mÏ~P(X,P).
5
![Page 6: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/6.jpg)
f´S5mþ¢5¼, f´: x ≥ 0, f (x) ≥ 0.eY´Xfm§Y3XSe´Sfm"
e¡´Hahn-Banachòÿ½nS,ÄkdKreinÚRieszuy,¡Krein-Rieszòÿ½n.
½½½nnn[SSS]X´SSS555mmm, Y´Xfffmmm,÷÷÷vvvééézzzx ∈ X, ∃y ∈ Y,¦¦¦y ≥ x. eee f ´Y þþþ555¼¼¼,KKK fòòòÿÿÿX þþþ555¼¼¼.
6
![Page 7: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/7.jpg)
eXþkIP, f´Xþ¢5¼, f´P-, ´µx ∈P, f (x) ≥ 0. eIÚS´d§A^å5B"
½½½nnn[III]eeeXþþþkkkIIIP, Y´Xfffmmm, ÷÷÷vvvééézzzx ∈ X, ∃y ∈ Y, ¦¦¦y − x ∈ P""" f333Yþþþ´P-§§§=== f (y) ≥0, y ∈ Y ∩ P,KKK fòòòÿÿÿX þþþP-555¼¼¼.
555PPPX = R2, Y = (x, 0) ∈ X : x ∈ R,¿-
P = (x, y) ∈ X : y ≥ 0\(x, 0) ∈ X : x < 0,
½Âϕ((x, 0)) = x,@oϕvkòÿ. Ïd½n¥b“y ≥ x”ØUK.
7
![Page 8: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/8.jpg)
A^1. RnþÿÝ
f´½Â3Rnþ¼ê§ f| 8S f = x : f (x) , 0.X´Rnþäkk.| k.¼êN, Y´Rn þäkk.|
Lebesgueÿk.¼êN.-
P = f ∈ X : f ≥ 0,
3Yþ½Â5¼
F( f ) =
∫Rn
f dmn,
KF´. dKrein-Rieszòÿ½n, FòÿXþ5¼, EPF. f ∈ X, F( f )n)/¿Âe0 fÈ©§§´LebesgueÈ©í2"¯¢þ§È©ÚÿÝ´Ú§'uÿݧ·ke(Ø"
8
![Page 9: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/9.jpg)
éRnzk.f8E,½Âµ(E) = F(χE). Kµ÷vµ ≥ 0, µ(∅) = 0;µ´k\,=e½üüØk.8E1, · · · , En,k
µ
n⋃i=1
Ei
= µ(E1) + · · · + µ(En);
E´Lebesgueÿk.8, µ(E) = mn(E).Ïd3Rn¤kk.f8þ,·EÑk\ÿÝ.é∀E ⊆ Rn,½Â
ν(E) = limr→∞
µ(E ∩ Br),
Ù¥Br´±0¥%,»r¥. @o·Eµν´Rn
¤kf8¤σ−êþk\ÿÝ,E´Lebesgueÿ, ν(E) = mn(E).
9
![Page 10: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/10.jpg)
+G35mXþ^´ézg ∈ G, gx ∈ X,÷v
∀x ∈ X,kex = x,Ù¥e´Gü ;
∀x, y ∈ X, α, β ∈ R, g ∈ G,kg(αx + βy) = αg(x) + βg(y);
∀g1, g2 ∈ G, x ∈ X,k(g1g2)x = g1(g2x).
10
![Page 11: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/11.jpg)
½½½nnn[+++](X,P)´SSS555mmm, G´Abel+++, Y´Xfffmmm÷÷÷vvv
ééézzzx ∈ X,333y ∈ Y¦¦¦y − x ∈ P;GY ⊆ Y, GP ⊆ P.eee f ´YþþþP-¼¼¼,
f (gy) = f (y), ∀y ∈ Y, g ∈ G,
KKK fòòòÿÿÿXþþþ¼¼¼F,÷÷÷vvv
F(gx) = F(x), ∀x ∈ X, g ∈ G.
11
![Page 12: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/12.jpg)
A^2. Rþ²£ØCÿÝ
£þA^~f. X´Rþäkk.| k.¼êN, Y´¤kk.| Lebesgueÿk.¼êN§IP´P = f ∈ X : f ≥ 0.3Yþ½Â5¼F( f ) =
∫R
f dx.½Â\+R3Xþ^
(τa f )(x) = f (x − a), ∀x ∈ R.
dué?Ûa ∈ R,kτaY ⊆ Y, τaP ⊆ P,¿
F(τa f ) = F( f ), ∀ f ∈ Y.
ÏdFòÿXþ¼, ÷v+^ØC. ÏdlA^1, 3R¤kf8σ−êRþ3k\ÿÝν´²£ØC,=ν(E + a) = ν(E), ∀a ∈ R.E´Lebesgueÿ,ν(E) = m(E).
12
![Page 13: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/13.jpg)
AOrN,ùÿÝ´“k\”. ¦^ÀJún±y²: Ø3σ−êRþê\²£ØCÿݵ, ÷v0 < µ([0, 1]) <+∞. ¯¢þ,3Rþ½Âd'X“x ∼ y=x − y ∈ Q”. 3ùd'Xe,zda¥áu[0, 1]¤8ÜE(^ÀJún). Kk
[0, 1] ⊆⋃
q∈Q∩[−1,1](E + q) ⊆ [−1, 2].
XJkù²£ØCÿݵ3,Kk
µ([0, 1]) ≤∑
q∈Q∩[−1,1]µ(E+q) =
∑q∈Q∩[−1,1]
µ(E) ≤ µ([−1, 2]) = 3µ([0, 1]),
ù´ØU"LebesgueÿÝ´äk²£ØC5. ÏdLebesgueÿÝØUòÿRþ,ùL²3Lebesgueÿ8.
13
![Page 14: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/14.jpg)
lÑ+G¡^l, XJ3GþØC!k\ÿݵ, µ(G) = 1. ùpØC´éG?Ûf8E,kµ(gE) = µ(E). ±y²R2þfN$Ä+G (²£+^=),lÑ+´^l"Krein-Rieszòÿ½nAbel+g,Uí2^l+/. Ïd3R2¤±f8σ−êR2þ,3k\fN$ÄØCÿÝ., n ≥ 3, Rnþ
fN$Ä+(²£+^=)Ø´^l. ÏdÒÑyBanach-Tarski paradoxy,=R3¥ü ¥©¤5¬,Ù¥“¬”©¤ü ¥, e“n¬” ©¤ü ¥. ù¯¢y²Ñ6ÀJún3Ú/, ù«¹Ø¬u), =ü «mØU©¤k°, ÏL²£, ©¤ü «m. Ó,²¡þü /(ü )ØU©¤k°, ÏL²£^=©¤üü /(ü ).
14
![Page 15: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/15.jpg)
A^3. Abel+þBanach4
G´Abel+(+$P“+”,÷vÆ). f ∈ l∞(G)¡k4L( f ),´∀ε > 0,3Gkf8E,¤á
| f (x) − L( f )| < ε, ∀x ∈ G \ E.
½½½nnn f ∈ l∞(G), a ∈ G, ½Â fa(x) = f (x + a), ∀x ∈ G, K3l∞(G)þ3¼÷v
M( f ) = M( fa), ∀a ∈ G.
fk4M( f ) = L( f ). M( f )¡ fBanach4"
15
![Page 16: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/16.jpg)
yyy²²²G´Ã+.X = l∞(G), Y = f ∈ l∞(G) : fk4, P =
f ∈ l∞(G) : f ≥ 0.½ÂG3Xþ^:
(a · f )(x) = fa(x) = f (x + a), ∀a ∈ G, x ∈ G,
KGY ⊆ Y, GP ⊆ P, ¿∀ϕ ∈ l∞(G), 3ψ ∈ Y, ¦ψ ≥ ϕ.3Yþ½Â¼Φ( f ) = L( f ), ∀ f ∈ Y,KΦ ≥ 0¿Φ( fa) = Φ( f ).d+Krein-Riesz½n, ΦòÿXþ¼, PM,§÷v
M( fa) = M( f ), ∀a ∈ G,
KM÷v¦(Ø.
~~~fffG = N, lù½n, 3l∞(N)þBanach4M÷vMS f = M f ;ùpS´²£S f (n) = f (n + 1).
16
![Page 17: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/17.jpg)
G´G¤kf8¤σ−ê. ke¡(Ø:½½½nnn3Gþ3k\!²£ØCÿݵ,µ(G) = 1"yyy²²²M( f )´l∞(G)þ fBanach4¼"é∀E ∈ G.Pµ(E) = M(χE),Kµ´k\ÿÝ, µ(G) = 1,¿÷v
µ(E + a) = M(χE+a) = M((χE+a)a) = M(χE) = µ(E).
555PPP3Nþk\ÿݵ¦µ(N) = 1. N´^l+. S 2´dü)¤gd+, ±yS 2Ø´^
l. R3\e´ÛÜ;+Ä~f,ÛÜ;+ÌA´3ÙBorel8þ3ØCÿÝ–HaarÿÝ, ù+´R, HaarÿÝÒ´Ï~LebesgueÿÝ,ù!?Ø(JÅiÅéí2ÛÜ;+Ú^l+¹,ùpØ2Kã.
17
![Page 18: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/18.jpg)
II. Hahn-Banach½n©Û
X´¢5m,eXþ¢¼êρ÷v∀x, y ∈ X,kρ(x + y) ≤ ρ(x) + ρ(y);∀t ≥ 0, x ∈ X,kρ(tx) = tρ(x).K¡ρ´Xþg5¼.
XJρ÷v
∀x ∈ X,kρ(x) ≥ 0;∀x, y ∈ X,kρ(x + y) ≤ ρ(x) + ρ(y);∀t ∈ R, x ∈ X,kρ(tx) = |t|ρ(x).K¡ρ´Xþê. Xþêρ÷v“ρ(x) = 0=x = 0”,¡ρ´Xþê.
~~~XJl∞(N)L«k.¢êSN,Kρ(an) = limn→∞
an´l∞(N)þ
g5¼.18
![Page 19: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/19.jpg)
½½½nnn[¼¼¼]ρ´¢¢¢555mmmXþþþggg555¼¼¼, Y´X555fffmmm, f ´Yþþþ¢¢¢555¼¼¼,¿¿¿
f (y) ≤ ρ(y), ∀y ∈ Y.
KKK f±±±òòòÿÿÿXþþþ¢¢¢555¼¼¼F,¦¦¦
−ρ(−x) ≤ F(x) ≤ ρ(x), ∀x ∈ X.
AO, XJX´D5m, Y´X"5fm,f´YþëY5¼,K f±òÿXþ.
19
![Page 20: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/20.jpg)
ù´þã½n+.
½½½nnn[+++]G´Abel+++½½½^lll+++,¿¿¿+++G^333555mmmXþþþ. XXXJJJY´X555fffmmm, ρ´X þþþggg555¼¼¼, f ´Yþþþ555¼¼¼,÷÷÷vvvGY ⊆ Y;ρ(ax) ≤ ρ(x), f (ay) = f (y) ∀a ∈ G, x ∈ X, y ∈ Y,KKK333 f333Xþþþ555òòòÿÿÿF,¦¦¦F(x) ≤ ρ(x), ∀x ∈ X;F(ax) = F(x), ∀a ∈ G, x ∈ X,
20
![Page 21: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/21.jpg)
~~~ X = l∞(N)´¢k.SN, ½ÂN3Xþ^Xe:(τn · f )(m) = f (m + n).½ÂXþg5¼ρ( f ) = lim
n→∞f (n).-Y =
f ∈ X : limn→∞
f (n)3K´τnY ⊆ Y ¿ρ(τn f ) = ρ( f ), ∀n ∈ N.
½ÂYþ5¼λ( f ) = limn→∞
f (n), Kλ(τn f ) = λ( f ). dHahn-
Banach½n+,3λ3Xþ5òÿΛ¦
Λ(τn f ) = Λ( f ), ∀ f ∈ X, n ∈ N,
¿
limn→∞
f (n) ≤ Λ( f ) ≤ limn→∞
f (n) ∀ f ∈ X.
dΛ( f )¡ f (n)Banach4, Ïdz¢ê4UØ3,Banach4o´3.
21
![Page 22: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/22.jpg)
¢¢¢EEE===zzzµµµ XJ f´E5mXþE5¼, f¢Ü´u(x) = Re( f (x)),K
f (x) = u(x) − iu(ix), ∀x ∈ X.
rXw¢5m, Ku´Xþ¢¼, A^c¡¢¼òÿ½n=Hahn-Banachòÿ½nE.XJu´E5mXþ¢5¼, Kdþª½Â¼´XþE¼,¦^¢E=zL§,c¡'(رí2E/.
½½½nnn[EEE] X´EEE555mmm, ρ´Xþþþêêê,Y´X555fffmmm, f ´YþþþEEE555¼¼¼,÷÷÷vvv
| f (y)| ≤ ρ(y), ∀y ∈ Y.
KKK f±±±òòòÿÿÿ¤¤¤XþþþEEE555¼¼¼F,÷÷÷vvv
|F(x)| ≤ ρ(x), ∀x ∈ X.
AO, XJX´ED5m, Y´X"5fm, f´YþEëY5¼,K fòÿXþ.
22
![Page 23: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/23.jpg)
Hahn-Banach½½½nnnÚÚÚKrein-Riesz½½½nnn´ddd
A)^Krein-Riesz½ny²Hahn-Banach½nµ
X´¢5m§ρ(x)´X´þg5¼§Y´Xfm§ f´Yþ¢5¼÷v f (y) ≤ ρ(y), y ∈ Y.-X′ = X ⊕ R,½ÂI
P = (x, t) ∈ X ⊕ R : ρ(x) ≤ t
ÚfmY′ = Y ⊕ R. 3Y′þ½Â5¼ f ′((y, t)) = t − f (y),K f ′3Yþ´"âKrein-Riesz½n, f ′3X′þk5òÿF′. PF(x) = −F′((x, 0)),KF´ fòÿ,Ï(x, ρ(x)) ∈ P,l
F′((x, 0)) + F′((0, ρ(x)) = F′((x, ρ(x))) ≥ 0,
F(x) ≤ F′((0, ρ(x))) = ρ(x)F′((0, 1)) = ρ(x) f ′((0, 1)) = ρ(x).
23
![Page 24: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/24.jpg)
B)^Hahn-Banach½ny²Krein-Riesz½nµ
XþkIP, Y´Xfm, ÷vézx ∈ X, ∃y ∈ Y¦y − x ∈ P§¿3YþkP-5¼ f (y). 3Xþ½Â
ρ(x) = infy∈Y,y−x∈P
f (y),
Kρ´Xþg5¼£y¤, ¿y ∈ Y, f (y) ≤ ρ(y).âHahn-Banach½n, f±òÿXþ,¿÷v
−ρ(−x) ≤ F(x) ≤ ρ(x), ∀x ∈ X.
x ∈ P,´ρ(−x) ≤ 0,x ∈ P§F(x) ≥ −ρ(−x) ≥ 0.
24
![Page 25: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/25.jpg)
III. Hahn-Banach½nAÛ
5mAÛ à8
©Û Minkowski ¼
X´5m§V´X¥áÂà8§=é?Ûx ∈ X,3t > 0¦tx ∈ V,@ow,0 ∈ V"VMinkowski ¼½Âµ
p(x) = inft ≥ 0 : x ∈ t V.
(i) p(x + y) ≤ p(x) + p(y);
(ii) p(tx) = tp(x), t ≥ 0;
(iii) x : p(x) < 1 ⊆ V ⊆ x : p(x) ≤ 1.
25
![Page 26: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/26.jpg)
·lAÛÝ5n)Hahn-Banach½n§§AÛ/ª´µ
½½½nnn[AAAÛÛÛ]X´¢¢¢555mmm§§§V´XáááÂÂÂààà888§§§p(x)´V Minkowski¼¼¼£££ggg555¤¤¤§§§x0 < V, @@@ooo333Xþþþ333¢¢¢555¼¼¼F÷÷÷vvv
F(x) ≤ p(x), x ∈ X, F(x0) = 1.
AAAOOO§§§333Vþþþ§§§F(x) ≤ 1"""
y²µY = tx0 : t ∈ R,3Yþ½Â5¼ f (tx0) = t, t ∈ R. Ïx0 < V, p(x0) ≥ 1, ddN´y3Yþ§ f (y) ≤ p(y). dHahn-Banachòÿ½n§ f òÿXþ5¼F ÷v
F(x) ≤ p(x), x ∈ X, F(x0) = 1.
26
![Page 27: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/27.jpg)
Rnþz5¼k/ªF(x) = a1x1+· · ·+anxn,z²¡´µF(x) = c. ½5mX§Xþ²¡Ò´F(x) = c§ùpF´Xþ5¼§c´¢ê. Hahn-Banachòÿ½nAÛ´`µézáÂà8V§x0 < V,Ñ3Lx0
²¡F(x) = 1§V uù²¡ý"
27
![Page 28: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/28.jpg)
Hahn-Banach½nAÛA^´à8©l½n"à8©l½n´BanachmAÛÆïÄÑu:,3$Ê`znØ¥k2A^"X¥à8V¡²£á§XJ3a ∈ V¦V − a´áÂ"à8Và:x´xØULV¥üØÓ:à|ܧ=ex = ty + (1 − t)z, y; z ∈ V, 0 < t < 1,@ox = y = z.
à8©l½nµV,W´X¥üØàN§´²£á§@o3²¡F(x) = c¦V, W©O u²¡üý§=
supV
F(x) ≤ c ≤ infW
F(x).
28
![Page 29: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/29.jpg)
|:½nµ eV´²£áÂ, D´Và:§@o3LD²¡F(x) = c¦V u²¡ý"
29
![Page 30: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/30.jpg)
IV. Hahn-Banach½nÿÀ
ÿÀ5mVgµ
y3 X ´5m§τ´ X þÿÀ§§÷vµ(i) X ´Hausdorff¶(ii)5$´ëY.¡ (X, τ)´ÿÀ5m.
(ii) )ºXeµa). 'u\“+”ëY´µé x, y ∈ X§±9 x + y ?Û V§3 x V1§y V2§¦
V1 + V2 ⊆ V. b). 'uê¦ëY5´µé α ∈ C§x ∈ X§±9 αx?Û V§3 α W (3E²¡þ)±9 x U§¦ W · U ⊆ V.
30
![Page 31: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/31.jpg)
N´yµéz x0 ∈ X§±9 λ ∈ C§λ , 0§²£ x 7→ x0+x§ê¦ x 7→ λx Ñ´ X ÓN. Ïd§X þ5ÿÀ τ ´
²£ØC. ùL²ÿÀ5mþÿÀd§3:Ä(½. Ä(½§zm8LÄ¥,¤²£¿. ¡ÿÀþm X ´ÛÜà§XJ3 X Ä U§Ù¥z¤´à. ´zD5m´ÛÜà¶DmÙfÿÀ!éómÙ
f ∗- ÿÀÑ´ÛÜàÿÀþm.
31
![Page 32: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/32.jpg)
X´5m§ÙþkÝþd§XJ5$3dÝþe´ëY§·¡(X, d)Ýþ5m¶=÷v
(1) lim(λ,x)→(λ0,x0) d(λx, λ0x0) = 0;
(2) lim(x,y)→(x0,y0) d(x + y, x0 + y0) = 0.
Ýþ5m´ÿÀ5m"ÛÜàÝþ5m¡
Frechetm"
~~~fffµµµΩ ⊆ Rn´m8§C(Ω)L«ΩþëY¼êN";8K1 ⊆ K2 ⊆ · · · ⊆ Ω¦∪nKn = Ω"½ÂÝþ
d( f , g) =∑
n
12n
ρn( f − g)1 + ρn( f − g)
, ùp ρn( f ) = maxx∈Kn
| f (x)|, f ∈ C(Ω).
KC(Ω)´Frechetm§ f : ρn( f ) < 1n, n = 1, 2, · · ·´:Ä"
32
![Page 33: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/33.jpg)
ÿÀ5mþà8©l½nµ
y3·ÄÿÀ5m¹. X´ÿÀ5m,XJ3à8V, 0 ∈ V, VSÜ,@oV´á¿§Minkowski¼P(x)´ëY. dHahn-Banachòÿ½nAÛ,x0 < V,35¼F¦F(x0) = 1¿
−P(−x) ≤ F(x) ≤ P(x), ∀x ∈ X.
ÏdF(x)´ëY"
33
![Page 34: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/34.jpg)
···KKK X´ÿÀ5m, Xþ3"ëY5¼=3²àm8.
yyy²²²X∗L«XþëY5¼N,e0 , f ∈ X∗,K
Ω = x ∈ X : | f (x)| < 1
´²àm8.
,eΩ´Xàm8, Ω , X,a ∈ Ω, Ω′
= Ω − a,KΩ′´
à¿Ù´:,dþ¡ín=3"ëY5¼.
34
![Page 35: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/35.jpg)
þ¡·K`²ÿÀ5mþëY5¼nØÚXþà8m45'.²;~f´Lpm(0 < p < 1),3ùmþØ3"ëY5¼,Ïd3ùm¥Ø3²àm8.
~~~0 < p < 1,-
Lp[0, 1] =
f´[0, 1]þLebesgueÿ¼ê :∫ 1
0| f |pdm < +∞
,¿3Lp[0, 1]þ½ÂÝþ
d( f , g) =
∫ 1
0| f − g|pdm.
N´y²Lp[0, 1]´²£ØCÝþ5m. ùÿÀ5m¥Ø3²àm8,ÏdùmþØ3"ëY5¼.
35
![Page 36: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/36.jpg)
½½½nnn[ààà888©©©lll½½½nnnÿÿÿÀÀÀ]
X´ÿÀ5m, VÚW´X¥Øà8,K
(1)eVSÜV , ∅,K3F ∈ X∗±9γ ∈ R¦
F(x) ≤ γ ≤ F(y), ∀x ∈ V, y ∈ W,
=VÚW u²¡F(x) = γüý;¿x ∈ V, y ∈ W, F(x) < γ ≤ F(y);
(2)XJV´m,@o3F ∈ X∗±9γ ∈ R¦
F(x) < γ ≤ F(y), ∀x ∈ V, y ∈ W,
=²¡F(x) = γòVÚW©l;
36
![Page 37: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/37.jpg)
(3)XJX´ÛÜà, V´;,¿V ∩W = ∅,@o3F ∈ X∗Ú¢êγ1 < γ2¦
F(x) < γ1 < γ2 < F(y), ∀x ∈ V, y ∈ W.
=²¡F(x) = γòVÚWî©l;ùpγ1 < γ < γ2.
(4)eX∗©lX,VÚW´;,@o3F ∈ X∗Ú¢êγ1 < γ2¦
F(x) < γ1 < γ2 < F(y), x ∈ V, y ∈ W.
=²¡F(x) = γòVÚWî©l,ùpγ1 < γ < γ2.
37
![Page 38: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/38.jpg)
yyy²²² (1)cÜ©Hahn-Banach½nAÛ. 5¿ëY5¼´mN§Ïd(1)Ü©F(V)´m8, (ÜcÜ©=(1).
(2)d(1).
(3)3X3:àΩ¦(V + Ω) ∩ W = ∅, 5¿V + Ω´àm8, F(V)´m«mF(V + Ω)¥4«m, ,A^(2)=.
(4)ÄXþfÿÀτ, K(X, τ)´ÛÜà, τfuXþÿÀ,u´3fÿÀe, VÚW´;. A^(3), 3F ∈ (X, τ)∗9¢êγ1 < γ2¦
F(x) < γ1 < γ2 < F(y), x ∈ V, y ∈ W,
du(X, τ)∗ = X∗,£4¤¤á.38
![Page 39: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/39.jpg)
555PPPX´ÿÀ5m,K
(A)0 , f ∈ X∗,K f´mN;(B)V´X¥à8§KVSÜVÚV4VÑ´à¶(C)eM´X5fm,K f (M) = 0½ö f (M) = R.
d5PÚà8©l½nÿÀ(1),·k: eà8VkSÜ, M´X5fm, V ∩ M = ∅,K3F ∈ X∗¦
F|V ≤ 0, F|V < 0, F(M) = 0,
=3LM²¡F(x) = 0,¦V uT²¡ý, VSÜî uTý.
AO, V´àm8, Vî uLM²¡F(x) = 0ý.
39
![Page 40: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/40.jpg)
íííØØØX´ÛÜàÿÀ5m,K
X∗©lX¥:;=Xk///vvvõõõ000ëY5¼;M´X5fm, x0 < M,K3 f ∈ X∗¦ f |M = 0¿ f (x0) = 1;S´Xf8§KSà|Ü4u§f4"
yyy²²². ÄV = x0, W = M, dà8©l½nÿÀ(3),3 f ∈ X∗¦ f (x0) < F(M). d5P(C)§ f (M) = 0, ò f ¦±·~ê=. y²A^à8©l½nÿÀ(3)="
40
![Page 41: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/41.jpg)
½½½nnn[Hahn-Banach½½½nnnÿÿÿÀÀÀ] X´ÛÛÛÜÜÜàààÿÿÿÀÀÀ555mmm, M´X555fffmmm, f ´MþþþëëëYYY555¼¼¼,KKK f±±±ëëëYYYòòòÿÿÿX þþþ.
yyy²²²Øb f , 0,P
M0 = x ∈ M : f (x) = 0,
¿x0 ∈ M¦ f (x0) = 1. â fëY5, x0ØáuM0M−4, u´x0ØáuM0X−4, PM0X−4W. díØ,kg ∈ X∗¦g(x0) = 1¿g|W = 0. N´wÑgÒ´ fXþëYòÿ.
41
![Page 42: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/42.jpg)
V. Krein-Milmanà8à:½nÚ`z¯K
ÿÀ5mþà8©l½nA^Ò´Krein-Milmanà8à:½n.
½½½nnn[ààà:::½½½nnn]X´ÿÀ5m, X∗©lX, K´X;f8,^E(K)L«K¤kà:¤8Ü,K
E(K);
eK´à8,KKuE(K)4à,=K = co(E(K));
eX´ÛÜà, KK ⊆ co(E(K))¶eS ⊆ K, co(S ) = co(E(K)),K
E(K) ⊆ S .
42
![Page 43: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/43.jpg)
;à8~f´Banach-Alaoglu½n,T½n±^Tychonoff½ny².
½½½nnn[Banach-Alaoglu] V´ÿÀ5mX:,KV4
K = F ∈ X∗ : |F(x)| ≤ 1,∀x ∈ V
´X∗w∗−;àf8.
âà:½n, KÒ´E(K)3w∗−ÿÀe4à.
~~~duL1[0, 1]4ü ¥
B =
f ∈ L1[0, 1] :∫ 1
0| f |dm ≤ 1
vkà:. ÏdL1[0, 1]Ø´?ÛD5méóm. , l1(N) = c∗0,ùpc0´Âñ"Sm(êê).
43
![Page 44: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/44.jpg)
~~~H´Hilbertm,KH4ü ¥
B = x ∈ H : ‖x‖ ≤ 1
à:8
E(B) = x ∈ H : ‖x‖ = 1.
N´y,L4ü ¥B>.þz:aÑ3²¡,ÙB:a:,²¡§´〈x, a〉 = 1.
H¢²ÚSml2(N), a = an ∈ l2(N)¿‖a‖2 = 1. ÄÃCþå^5§
∑n∈N
anxn = 1 ∑n∈N
x2n ≤ 1
¦)¯K,þã~fL²5§3å^ek²)x = a = an.
44
![Page 45: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/45.jpg)
Krein-Milman ½n´'u;à8à:|ÜÿÀã§Choquet½nKlÿÝØ*:£ãà8Ú§à:'X"ùü½
nþ´d"
½½½nnn[Choquet]X´ÛÜàÿÀ5m, K´;à8§K(1) E(K)´KBorelf8¶(2)ézx ∈ K;3E(K)þ3BorelVÇÿݵx¦
f (x) =
∫E(K)
f (s)dµx(s), f ∈ X∗.
Choquet½ny²ÄuKrein-Milman½nÚRieszL«½n"
45
![Page 46: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/46.jpg)
`zzz¯KKKµµµ`z¯K98I¼êÚå^,XJ8I¼êÚå^Ñ´5,Ò¡5`z¯K,ÄK¡5`z¯K.
X´¢5m, V ⊆ X, F´Xþ¢5¼,¦
maxx∈V
F(x); minx∈V
F(x)
¯KÒ´`z¯K.dà:½n,XJX´ÛÜà, K´;, F´Xþ¢ëY5¼,Kk
maxx∈K
F(x) = maxx∈E(K)
F(x), minx∈K
F(x) = minx∈E(K)
F(x).
AO, F3Kþ3K>.þ.
46
![Page 47: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/47.jpg)
X = Rn,-
F(x) =
n∑k=1
akxk + b, x = (x1, x2, · · · , xn) ∈ Rn,
Ù¥ai, b ∈ R. XJB´Rn¥k.48,K
maxx∈B
F(x) = maxx∈∂B
F(x), minx∈B
F(x) = minx∈∂B
F(x).
ù´5`z^n.
47
![Page 48: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/48.jpg)
e¡´5`z½n.
½½½nnn X´ÛÜàÿÀ5m, F´Xþ"ëY5¼,K ⊆ X. Ä`z¯K
S max = u ∈ K : F(u) = maxx∈K
F(x)
eK´;à8,K)8S max´;à8,¿S max∩E(K) , ∅;
eX´gBanachm, K´Xk.4à8, K)8S max´
à8,¿S max ∩ E(K) , ∅.
þã½nÄS max¹§éS min(Ø´"
48
![Page 49: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/49.jpg)
~~~ H´¢Hilbertm, Ù4ü ¥B = x ∈ H : ‖x‖ ≤ 1, éu0 , x0 ∈ H,½ÂF(x) = 〈x, x0〉, ∀x ∈ H,K)8
S max = x ∈ B : F(x) = maxy∈B
F(y) =
x0‖x0‖
´ü:8.
~~~1 < p < ∞, 1p + 1
q = 1, 0 , g ∈ Lq(X, dµ)£¢m¤,Ä`z¯K
S max =
f ∈ Lp(X, dµ) : ‖ f ‖ ≤ 1,∫
Xf g dµ = max
‖h‖p≤1
∫X
hg dµ
,N´yd`z¯Kk)
f = |g|q−1‖g‖−
qp
q .
49
![Page 50: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/50.jpg)
X´ED5m, F´XþëY5¼. òXÀ¢Dm,KRe(F)´Xþ¢ëY5¼. N´y
‖F‖ = sup‖x‖≤1
|F(x)| = sup‖x‖≤1
(Re(F))(x)
Ïd¦ê´5`z¯K.AOX´g/§
‖F‖ = max‖x‖≤1
(Re(F))(x).
C´XEf8§x ∈ X, x < C,¦ål
dist(x,C) = infy∈C‖x − y‖
´5`z¯K.AOX´g/§¿C´4à8§`z¯K
miny∈C‖x − y‖ = dist(x,C)
3Cþo´)"50
![Page 51: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/51.jpg)
~~~X´«Ωþ)Û2)ØHilbertm,@oézλ ∈ Ω,D¼
Eλ : X → C, f 7→ f (λ)
´ëY,dRieszL«½n,3Kλ ∈ X,¦
f (λ) = 〈 f ,Kλ〉, ∀ f ∈ X.
duX4ü ¥´f;,k
‖Eλ‖ = sup‖ f ‖≤1
| f (λ)| = max‖ f ‖=1
Re( f (λ)).
´y,d`z¯Kk)
f =Kλ‖Kλ‖
.
ïÄ`z¯K)3595´5`znØØ%¯K.51
![Page 52: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/52.jpg)
···KKK(I) 1 < p < ∞, X = Lp(X, dµ), C´X 4à8§ f ∈ X,¿ f < C,K`z¯K
ming∈C‖ f − g‖ = dist( f ,C)
3Cþk)"
yµeg1, g2 ∈ C)d`z¯K§K12(g1 + g2) ∈ C)d`z¯K,
¿dd´
‖( f − g1) + ( f − g2)‖p = ‖ f − g1‖p + ‖ f − g2‖p.
MinkowskiتL²g1 = g2.
52
![Page 53: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/53.jpg)
£££II¤¤¤eY´X4fm§ f ∈ X \ Y,¿1p + 1
q = 1,K`z¯K
max|
∫f gdµ| : g ∈ Y⊥, ‖g‖q ≤ 1
= dist( f ,Y)
)´
g = c ‖ f − g0‖−p/qp | f − g0|
p( f − g0)−1,
ùpc´~ê§|c| = 1,¿g0´`z¯Kming∈Y ‖ f−g‖ = dist( f ,Y)3Yþ)"yµdåÓ(
X/Y)∗ = Y⊥ =
g ∈ Lq :
∫hgdµ = 0,∀h ∈ Y
,
dist( f ,Y) = max|
∫f gdµ| : g ∈ Y⊥, ‖g‖q ≤ 1
.
duþªm>8´W∗-;§¤±d`z¯Kk), ¿)÷v‖g‖q = 1"
53
![Page 54: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/54.jpg)
g´)§Kk
dist( f ,Y) = ‖ f−g0‖p = |
∫f gdµ| = |
∫( f−g0)gdµ| =
∫| f−g0| |g|dµ.
dHolderت§3~êγ¦|g|q = γ| f − g0|p§(ܯ
¢µXJh ∈ L1÷v|∫
hdµ| =∫|h|dµ, @o31~ês¦
h = s |h|§N´y²3~êc, |c| = 1¦
g = c ‖ f − g0‖−p/qp | f − g0|
p( f − g0)−1.
~~~1 < p < ∞,ıþHardymHp, f ∈ Lp \ Hp,K`z¯K
ming∈Hp
‖ f − g‖p = dist( f ,Hp)
3Hp¥)´",dª(Lp/Hp)∗ = Hq
0,
54
![Page 55: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/55.jpg)
`z¯K
max|
∫f g
dθ2π| : g ∈ Hq
0, ‖g‖q ≤ 1
= dist( f ,Hp)
)ýé1~ê´"
···KKKH´Hilbertm£¢½E¤§C´H4à8§x ∈ H,K`z¯K
miny∈C‖x − y‖ = dist(x,C)
3Cþ)´3§PPCx, ¡ÙxCþÝK"AOC45fm§PC : X → C Ò´Ï~XCÝKf"
55
![Page 56: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/56.jpg)
Hahn-Banachòÿ½n´êÆ¥Ä:½n§§ÚZornÚn!ÀJún´d§3Ä:êÆ¥äk/
"Ïd¡0Hahn-Banach òÿ½nw´(J"3wL§¥§ëþ©z§AOJ
Baggett5Functional analysis6ÚRudin5Functional anal-ysis6§±9·Ö5fnØÄ:6"
56
![Page 57: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/57.jpg)
ë©z:1. W. Rudin, Functional analysis, New York,(1973).2. L. W. Baggett, Functional Analysis, published by Marcel-Dekkerin 1991.3. E. Zeidler, Applied Functional Analysis, (Applications to Mathe-matical Physics), Vol(I, II), Springer, New York, 1995.4. J. Garnett. Bounded Analytic Functions. Graduate Texts in Math-ematics, 236. Springer, New York, 2007.5. H%,fnØÄ:, (EÆêÆïÄ)Æ^Ö)§EÆѧ2014.6. http://en.wikipedia.org/wiki/Portal:Mathematics
Guo’s homepage: http://homepage.fudan.edu.cn/guokunyu/
57
![Page 58: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/58.jpg)
Hahn-Banachòÿ½ny²µ
3ïÄÃm©ÛÆ, ²~^ Zorn Ún§ù~8ÜØ¥ún5É.
A´8§e Amk^S'X “≺”÷v(i) ∀a ∈ A§a ≺ a¶ (g5)(ii)e a ≺ b§b ≺ a§K a = b¶ (é¡5)(iii)e a ≺ b§b ≺ c§K a ≺ c§ (D45)¡ A´ S8.
58
![Page 59: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/59.jpg)
B ´ S8 A f8§ b ∈ A§XJz s ∈ B ÷v s ≺ b§¡ b´ Bþ.. S8 ASf8 S´µ∀a, b ∈ S§o a ≺ b§o b ≺ a. ¡ c´ S8 A4´µØ3A ¥ØÓu c d§¦ c ≺ d.
[Zorn ÚÚÚnnn] A ´ S8§XJ A zSf8kþ.§K Ak4.
59
![Page 60: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/60.jpg)
é5m X ±9 X þ¼ P : X → R§·` P´g5§XJ
P(x + y) ≤ P(x) + P(y), ¿ P(tx) = tP(x), t ≥ 0 x, y ∈ X.
Hahn-Banachòÿ½n:
X ´¢5m§P : X → R´g5¼§Y ´ X fm§¿ f : Y → R´¢5¼§§÷v
f (x) ≤ P(x), x ∈ Y,
@o f òÿ X þ§¦Óتé¤k x ∈ X ¤á.
60
![Page 61: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/61.jpg)
Proofµ x0 ∈ X \ Y§P Y = spanx0,Y = tx0 + y | t ∈ R, y ∈ Y.Äkò f òÿ Y§±Óت§,ÏL ZornÚn¤y².
Ïé?Û x, y ∈ Y§
f (x) + f (y) = f (x + y) ≤ P(x + y) ≤ P(x − x0) + P(x0 + y),
ùÑ
f (x) − P(x − x0) ≤ P(x0 + y) − f (y).
α = infy∈Y
(P(x0 + y) − f (y))§@o
f (x) − α ≤ P(x − x0),¿ f (y) + α ≤ P(x0 + y), x, y ∈ Y.
3 Y þ½Â F(tx0 + y) = tα + f (y)§@o F ´5¿´ f òÿ. lþ¡Øª´y F(x) ≤ P(x), x ∈ Y.
61
![Page 62: Hahn-Banachòÿ n ÿÝ ØÚ zflKhomepage.fudan.edu.cn/guokunyu/files/2011/08/Hahn-Banach...Stefan Banach ( 30 March 1892 ±31 August 1945) was a Polish mathematician who is generally](https://reader033.vdocuments.net/reader033/viewer/2022060922/60ada2c65fd8fc78263a6c74/html5/thumbnails/62.jpg)
S =(H, F) : H ⊇ Y, F´fHþ5òÿ÷vF(x) ≤ P(x), x ∈ H
.
3Sþ½Â S'Xµ(H1, F1) ≺ (H2, F2),XJH1 ⊆ H2¿F2´
F15òÿ"eÏL ZornÚn¤y².
62