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1

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

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

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

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

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

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

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

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

+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

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

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

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

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

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

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

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

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

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

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

~~~ 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

¢¢¢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

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

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

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

·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

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

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

|:½nµ eV´²£áÂ, D´Và:§@o3LD²¡F(x) = c¦V u²¡ý"

29

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

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

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

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

···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

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

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

(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

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

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

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

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

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

;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

~~~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

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

`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

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

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

~~~ 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

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

~~~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

···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

£££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

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

`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

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

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

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

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

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

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

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