Οι χώροι James και James Tree

7/23/2019 James James Tree

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James James Tree

:

I

:

2014


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. .

. - , , . . . -. . . .E .

. - . . , .

, . .

A 2014

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A ,

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........................................................................................................................9

1. X anach Schauder

1.1 ..............................................................................................111.2 B ...............................................................................................191.3 I ........................................................................................221.4 lock ...................................................................................................271.5 I shrinking boundedly complete ......................................31

2. O James

2.1 O J

.................................................................412.2 ........................................................................452.3 .....................................................................492.4 O J 2-saturated...........................................................................................53

3. O James Tree

3.1 O J T ..............................................................573.2 O1 J T......................................................633.3 ................................................69

3.4 J T 2-saturated........................................................................................74B.......................................................................................................................83

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James James Tree. -

Banach . James R.C.James 1954 - , . James Tree R.C.James 20 -, anach 1. - .

Schauder Banach . Schauder

. Hamel , , Schauder -. Schauder, - .

, R.C.James, J. J - 1 c0. James, J .

J - . J . o J, J, 2-saturated 2. J J .

James Tree, J T. , J T 1974 R.C.James Banach - 1. - (tree basis)

. J J T.E J T James. A J T J James. HoBanach anach .

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1

Banach Schauder

1.1 , Banach , -

. , .

(1.1.1)

X Banach {en}nN . {en}nN Schauder x X {n}nN R

x=n=1

nen

(1.1.2)

{en}nN. {en:n N} - en= 0 n N.

A

, o . k1, k2,...,kn N ik1, k2,...,kn R , , k1ek1 +k2ek2 +...+kneknn = 0. 0.

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- Banach Schauder. P.Eno [2].

(1.1.3)

{en}nN. .

D=

n

i=1riei:{ri}ni=1 Q, n N

ToD . A D -

X. x =n=1

nen X. T > 0N N {ri}iN Q

||ni=1

nen x||< 2

n > N |i ri|||ei||< 2i+1

i N

n > N

||ni=1

riei x|| ni=1

|i ri|||ei|| + ||ni=1

iei x||

0 ||x|| |||x||| K||x|| xX

||x|| = limn

||Pn(x)|| |||x|||. , (X,

||||||) Banach.

|| || ||| |||. y(k)kN

Cauchy (X, ||| |||). > 0k0 N

|||y(i) y(j)|||< i, j > k0 ||Pn(y(i)) Pn(y(j))||< i, j > k0, n N

Pn(y(k))kN

Caychy

< e1,...,en >. Pn(y(k))kN n N. zn= lim

kPn(y(k)). T {zn}nN |||| -

. , >0k, N N

||Pn(y(k)) Pn(y(j))||< 3

j > k, n N ||Pn(y(k)) zn||< 3

n N ||Pn(y(k)) Pm(y(k))||<

3 n, m > N

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n, m > N o

||zn zm|| ||zn Pn(y(k))|| + ||Pn(y(k)) Pm(y(k))|| + ||Pm(y(k)) zm||< {zn}nN ||||-Cauchy. (X, || ||) Banach, {zn}nN . z

||||= lim

nzn. m > n N

Pn(zm) = Pn(limk

Pm(y(k))) = lim

kPn(Pm(y

(k))) = limk

Pn(y(k))) = zn. -

Pn - < e1,...,en >.

{i}iN zn =ni=1

aiei n N. , z1 < e1 > 1 R z1 = 1e1. P1(z2) = z1 {en}nN , 2

R z2=1e1+2e2. ,

1, ...n, n+1 .

z= limn

zn, Pn(z) =ni=1

iei = zn n N. >0. k0 N k > k0

sup||zn Pn(y(k))||: n N < sup ||Pn(z) Pn(y(k))||: n N <

|||z y(k)|||< limk

y(k) =z

E (X, ||||||) Banach .

(1.1.5)

.

n N x X. ||Pn(x)|| |||x||| K||x|| .

,

supnN

||Pn||


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n . 1 . n

N en : X

R

en(x) =en(

k=1

kek) =n

x =n=1

en(x)en. T o

(1.1.6)

{en}nN .

n N x =n=1

nenX.

|en(x)| =||en(x)en||

||en|| =||

ni=1

iei n1i=1

iei||

||en|| 2|||x|||||en||

2K

||en|| ||x||

.

T {en}nN {en}nN. , . .

(1.1.7)

Banach {en}nNX. :i) {

en}nN Schauder Xii) :

en= 0 n N [en: n N] =X [en:n N] =< en:n N > K >0 m > n N 1,...,m R

||ni=1

iei|| K||mi=1

iei||

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A

i)ii)m > n N 1,...,m R. T

||ni=1

iei|| =||Pn(x)||=||Pn(Pm(x))|| ||Pn||||Pm(x)|| bc({en}nN)||mi=1

iei||

ii)i)A {en:n N} . -, , n N 1,...,n R 1e1+...+nen = 0. T

|i|||ei||=||i

j=1jej

i1

j=1jej2K||

n

i=1iei||= 0i= 0 i= 1,...,n

n N

pn:< en:n N >< en: n N > pn(mi=1

iei) =

min{n,m}i=1

iei

Opn {en : n N} -

. , ||pn(m

i=1iei)||=||

min{n,m}

i=1iei|| K||

m

i=1iei||

pn ||pn|| K n N. < en : n N > X, pn

pn : X< en : n N >||pn|| K n N. E m > nx X pn(x) = pn(pm(x)). - (1.1.4) x X - {i}iN pn(x) =

ni=1

iei n N.

limn

pn(x) =x. , >0. z=k

i=1iei< en:n N >

||x z||<

K+ 1 . m > kpm(z) =z.

||x pm(x)| | | |x z|| + ||zpm(z)|| + ||pm(z) pm(x)|| (1 + ||pm||)||x z||<

E .

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(1.1.8)

1 < p n N 1,...,m R

||ni=1

iei||= (ni=1

|i|p)1/p (mi=1

|i|p)1/p =||mi=1

iei||

. c0 .

(1.1.9)

Banach {en}nN -K0 m > n N 1,...,m R

||ni=1

iei|| K||mi=1

iei||

.

K o (1.1.7) , bc({en}nN), . - . , K. x X||x|| 1 n N||Pn(x)|| K||Pm(x)|| m > n. n ||Pn(x)|| K||x|| K .

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, - .

(1.1.10)

. x X, x= 0 {xn}nNX sup

nN

||xn||= M 0. N, n0 N

||sN x||< 3 max {M, ||x||} |x

n(sN) x(sN)|0 m > n N 1,...,m R

||ni=1

ixi|| K||mi=1

ixi||

(1.2.3)

Banach {en}nN . T - *. E x [en : n N] x =

n=1

x(en)en.

m > n N 1,...,m R. x X||x|| 1.

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|ni=1

iei (x)| =|

mi=1

iei (Pn(x))|=|Pn(

mi=1

iei )(x)| ||Pn ||||

mi=1

iei || K||

mi=1

iei ||

||ni=1

iei || K||

mi=1

iei ||

{en}nN , x [en : n N] - {n}nN x =

n=1

nen x

(en) = n n N. .

Schauder . . Banach . Mazur.

(1.2.4) (S.azur)

F - . >0 xX ||x||= 1

||y|| (1 +)||y+mx|| yF m R

A F , SF | | | | k N y1,...,ykSF SF

ki=1

S(yi,

2). S(y, )

y . Hahn-Banach, j {1,...,k}yi

SX y

i (yi) =

||yi

|| = 1. O

k

i=1

Ker(yi ) -

1. X xX ||x||= 1 yi (x) = 0 i= 1,...,k. ySF 0<


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m R

||y+mx

|| ||yj+ mx

| || |y

yj

|| yj (yj+ mx)

2

= yj (yj)

2

=

||yj

||

2= 1

2 1

1 + 0 < 0, ySF m R xSX(1 + )||y + mx|| 1. yFy= 0. T >0m R xSX

1(1 +) y||y|| + mx||y||

||y|| (1 +)||y+mx||

T .

(1.2.5) (S.Banach)

Banach . - Banach .

A

>

0. A - {n}nN ln(1 +n)

ln(1 +)

2n n N.

ni=1

ln(1 +n)ln(1 +) n N n=1

(1 +n)1 +

H .x1 X ||x1|| = 1. F1 =< x1 >. A Mazur x2X ||x2||= 1 ||y|| (1 +2)||y+mx2|| yF1, m RF2 =< x1, x2 >. A Mazur x3 X ||x3|| = 1

||y||

(1 +3)||

y+mx3||

y

F2,

m

R. Fn =< x1, x2,...,xn > xn+1 X||xn+1|| = 1||y|| (1 +n+1)||y +mxn+1|| y Fn, m R. {xn}nN X. 1 +. , m > n N 1,...,m R.

yk =ki=1

ieiFk k N

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T

||ni=1

iei|| = ||yn|| (1 +n+1)||yn+n+1xn+1||= (1 +n+1)||yn+1|| (1 +n+1)(1 +n+2)||yn+1+n+2xn+2||= (1 +n+1)(1 +n+2)||yn+3||

m

i=n+1

(1 +i)||ym|| (1 +)||mi=1

iei||

{xn}nN 1 +.

1.3

(1.3.1)

, Banach {xn}nN X, {yn}nN Y . O{xn}nN {yn}nN c, C>0 n N 1,...,n R

c||ni=1

nxn|| ||ni=1

nyn|| C||ni=1

nxn||

(1.3.2)

, Banach {xn}nNX, {yn}nNY . . :i)O{xn}nN {yn}nN .ii) {n}nNR

n=1

nxn , n=1

nyn

.iii) T : [xn:n N][yn:n N] T(xn) =yn n N.

i)ii) {n}nN n=1

nxn . o ni=1

iyi

nN

Cauchy. , > 0N N

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m > n > N

||m

i=n+1ixi|| Tn(i=1

ixi) =ni=1

iyi

Fn:< x1,...,xn >< y1,...,yn > Fn(ni=1

ixi) =ni=1

iyi

T nN Tn =Fn PnPn - (yn)nN. Tn Fn Pn -. , Banach-Steinhauss, T T(x) = lim

nTn(x) xX

T 1-1 y =n=1

nynx =n=1

nxn

T(x) =y . T.

iii) i) c = 1

||T1|| C=||T||.

(1.3.3)

Banach {xn}nN Banach. A {yn}nN c, C>0 n N 1,...,n R

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c||n

i=1

nxn|| ||n

i=1

nyn|| C||n

i=1

nxn||

.

A

{yn}nN , . , m > n N 1, ...m R.T

||ni=1

iyi|| C K||mi=1

ixi|| CKc ||

mi=1

iyi||

- . - .

(1.3.4)

Banach T : X X . (0, 1) xX ||xT(x)|| ||x||, T .

T . ,

||T(x)| || |x|| ||x| | | |T(x)|| (1 +)||x||

||x| || |T(x)|| ||x|| (1 )||x|| ||T(x)||

T , , , T(X) X.E T , T(X) . , Hahn-Banach, x0 SX x0(x) = 0 x T(X). 0 < < 1, x0X ||x0||= 1x0(x0)> . E

||x0 T(x0)|| = sup {x(x0 T(x0)) :||x||= 1} ||x0 T(x0)|| x0(x0 T(x0)) =x0(x0)> =||x0||

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.

.

(1.3.5)(Small Perturbation Lemma)

Banach {xn}nN , {yn}nN {xn}nN K =inf{||xn||: n N}> 0.

n=i

||xn yn||< 3K

{yn}nN {xn}nN

A

o T : X XT(xn) = yn. {yn}nN {xn}nN. m > n1,...,m R

|ni=1

iyi|| ||T|||ni=1

ixi|| ||T||K||mi=1

ixi|| ||T||K||T1||mi=1

iyi||

{yn

}nN . H

n N 1, ...n R 1

||T1|| ||ni=1

ixi|| ||ni=1

iyi|| ||T||||ni=1

ixi||

n N. T x[xn:n N]

|xn(x)|||xn||=||ni=1

xi (x)ei n1i=1

xi (x)ei||||xn|| 2K||x|| ||xn|| 2K

n N o, Hahn-Banach, xnX - . xX

n=1

xn(x)(ynxn) X Banach . ,

n=1

|xn(x)|||yn xn|| 2K

||x||

n=1

||yn xn||< 23||x||

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Oo

T :X

X T(x) =x +

n=1

xn(x)(yn

xn)

T oT , T(xn) =yn n N

||x T(x)||=||n=1

xn(x)(yn xn)|| ||x||n=1

||xn||||yn xn||0

n N k, m N :n < k < m ||sk sm|| () () {pn}nN N

||sp2n sp2n1|| n N

un=sp2nsp2n1 un w0 ||un|| n N. {zn}nNzn =

un

||un|| Y.

1.4 Block block -

. block .

O (1.4.1)

Banach {en}nN. {un}nN block {i}iN

{ni

}iN k

N

uk =

nk+1i=nk+1

iei

(1.4.2)

Banach {en}nN. . T block .

{un}nNblock . Tun= 0 n N m > k N1,...,m R

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||k

j=1

juj|| = ||k

j=1

j

nj+1i=nj+1

iei||=||k

j=1

nj+1i=nj+1

jiei||

K||mj=1

nj+1i=nj+1

jiei|| K||mj=1

juj||

{un}nN K.

(1.4.3) (Sliding hump argument )

Banach {en}nN {xn}nN

= infnN ||xn||

>0 limn

ek(

xn) = 0

kN

. >

0 -

{xn}nN block{un}nN{en}nN n=1

||xnun||2 < 2.

||xn|| = 1 n N

x

n

nN

block {bn}nN {en}nNn=1

||xn bn||2 < 2

o limn

Pk(yn) = 0

k

N. , k

N >0.

N N i= 1,...,k |ei (xn)|||ei|| N

n > N

||Pk(xn)||=||ki=1

ei (xn)ei|| ki=1

|ei (xn)|||ei||< ()

o 0< < . () o .k1= 1n1 = 0. xk1 =

i=1

(1)

i

ei n2

N n2 > n1

||

n=n2+1

(1)n en|| <

2

||n=1

(1)n en n2n=1

(1)n en|| <

2

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u1 =n2

n=n1+1

(1)n en ||xk1 u1||2 k1

xk2 =

i=1

(2)i ei

||n2i=1

(2)i ei|| n2

||

i=n3+1

(2)i ei|| 0

{xn}nN block {un}nN {en}nNn=1

||xn un||< . -

=

3K , K {en}nN, {xn}nN

block {un}nN .

A

-

. =

3K, Small

Perturbation Lemma, {xn}nN .

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(1.4.5)

Banach {xn}nN 1. T block {xn}nN 1. block .

A 1 X, {yn)}nN X 1. m, M >0 m ||yn|| M n N. k N |xk(yn)| ||xk||M n N {xk(yn)}nN k N. - {yn}nN {yn}nN {xk(yn)}nN k

N. zn = yn+1

yn. k

N

limn

xk(zn) = 0. E c, C > 0 m N 1,...,m R

c

mi=1

|i| ||mi=1

izi|| Cmi=1

|i| infnN

{||zn||}> 0

E {zn}nN . sliding hump argument - {zn}nN block{un}nN {xn}nN -. {un}nN 1.

1.5 I shrinking boundedly complete

O (1.5.1)

X Banach {en}nN. H shrinking X = [en:n N]. X Banach {xn}nN X. {xn}nN boundedly complete {n}nN

supnN ||ni=1

ixi||


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i) ii) block{un}nN||un|| = 1 n N. un w 0. x X x =

n=1

nen. T >0k0

k > k0 :

||

n=k+1

nen||< |

n=k+1

nen(x)|< ||x|| 1 ()

uk =

nk+1i=nk+1

iei m N nm > k0, k > m

nk > k0 (

)

|x(uk)|=|i=1

iei (uk)|=|

i=nk+1

iei (uk)|< uk w0

ii)i) {en}nN shrinking. Tx / [en : n N] ||x|| = 1. Pn(x) =

ni=1

x(ei)ei

{Pn(x)}nN x. E > 0 {mk}kN

||x Pmk(x)|| 2 k N ()

n1 = m1. () x1 =n=1

(1)n en X, ||x1|| = 1

|x(

i=n1+1

(1)i ei)| 2. o k2 Nmk2 > n1

n2=mk2

||x

||||

i=n1+1

(1i )ei

n2

i=n1+1

(1)i ei

||<

|x(

i=n1+1

(1)i ei)

|

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()x2 =n=1

(2)n en X ||x2||= 1, |x(

i=n2+1

(1)i ei)| 2.

k3N

mk3 > n2 n3=mk3

||x||||

i=n2+1

(2)i ei

n3i=n2+1

(2)i ei||< |x(

i=n1+1

(1)i ei)|

block{uk}kN |x(uk)| > k N.H {uk}kN . {uk}kN ||uk||=||Pnk+1(xk)Pnk(xk)|| 2K||xk||= 2K, . zn=

un

||un|| , .

(1.5.3)

Banach shrinking {en}nN . -x X

1

KsupnN

||ni=1

x(ei )ei|| ||x|| supnN

||ni=1

x(ei )ei||

||x||= limn

||ni=1

x(ei )ei||.

A

n N. T ||n

i=1

x(ei )ei||= sup|n

i=1

x(ei )y(ei)|: y BX

y BX . shrinking y =n=1

y(en)en.

|ni=1

x(ei )y(ei)| = |x(

ni=1

y(ei)ei )| ||x||||

ni=1

y(ei)ei )|| K||x||||y|| K||x||

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||ni=1

x

(e

i )ei|| K||x

|| n N 1

KsupnN ||ni=1

x

(e

i )ei|| ||x

||

x BX x =n=1

x(en)en. T

|x(x)| = |n=1

x(en)x(en)|= lim

n|ni=1

x(ei)x(ei )| sup

nN

|x(ni=1

x(ei )ei)|

supnN

||n

i=1

x(ei )ei|| ||x|| supnN

||n

i=1

x(ei )ei||

.

(1.5.4)

c0 boundedly complete.

A

{enk

}kN c0.

{enk

}kN -

. supkN

|| ki=1

eni|| = 1 0 ||zn|| M n N. E en

w 0, zn w 0 sliding humpargument block{un}nN X {znk}kN {zn}nN . {un}nN boundedly

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complete {xnk}kN . T {enk}kN boundedly complete .

(1.5.6)

Banach boundedly complete {en}nN . Y = [en, nN].E .

A

OJ : X Y J(x)(y) = y(x) y Y J . J ., xX. T

|J(x)(y)|=|y(x)| ||x| | | |y|| yY ||J(x)|| ||x||

o J .

ox =n=1

nen. xn =nn=1

nen x= limn

xn.

||

xn||

K

||J(xn)

|| n

N J

. , n N. .Hahn-Banach x X ||x|| = 1 |x(xn)| =||xn||. x Pn< e1,...,en > Y xn=Pn(xn)

||xn||=|(x Pn)(xn)|=|J(xn)(x Pn)| ||x| | | |Pn| | | |J(xn)|| K||J(xn)||

E - ||J(x)|| =||x|| x X. M J . y Y. o

n

i=1y(ei )ei

nN

X.

K2||y||. , n N

||ni=1

y(ei )ei|| K||J(ni=1

y(ei )ei)||= K||ni=1

y(ei )J(ei)||

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A o ||ni=1

y(ei )J(ei)|| K||y||. , y =n=1

y(en)enY

||y||

1,

|ni=1

y(ei )J(ei)(y)| = |y(ni=1

y(ei)ei )| ||y||||

ni=1

y(ei)ei || ||y||K||y|| K||y||

E boundedly complete i=1

y(ei )ei . A

x=n=1

y(en)en y =J(x) .

(1.5.7)(R.C.James)

Banach {en}nN . -:i)H{en}nN shrinking boundedly complete.ii) .

i)

ii) shrinking X = [en, n

N]. A

, , J , X X .ii) i) shrinking. x X. T Pn(x

) =ni=1

x(ei)ei

w x n . X w- w- X

Pn(x)

wx X =< en, n N >w

=< en, n N >||.||

Mazur. shrinking.

boundedly complete. {n}nN R sup

nN

||

ni=1

iei||

=M. n=1

nen. A

xn =ni=1

iei E X , MBX w. ,

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X , x MBX {xnk}kN {xn}nNx = w lim

kxnk . E i N ei w

ei (x) =ei (w limk

xnk) = limk

ei (xnk) =i i N

Ex=n=1

en(x)en =n=1

nen .

, - Banach boundedly complete . - .

(1.5.8)

X X .

T :X Y Y =T[X] X. E Y =X X =T[X] X.

oT : X Y T(x)(y) =y(x) y Y . H . .

||T(x)

|| = sup

{|T(x)(y)

|: y

BY

}= sup

{|y(x)

|: y

BY

} sup

{|x(x)

|: x

BX

}= sup {|x(x)|: xBX}=||x||E

||T(x)||= sup {|y(x)|: yBY} sup {|x(x)|: x BX}=||x|| T . T[X] - Y. , X Banach T[X] Banach Y. Q: Y X Q(y) =y , : X X X X. H Q . P : Y

Y P = T

Q. HP -

. . , Q T = IY IY o Y. ,(Q T)(x)(x) =T(x)(x) = x(x) =x(x) xX. P2 =T Q T Q=T Q = P. M P[Y] = T[X]. A Q.

, x X. T y = xY

Y. TQ(y) = x. E- T[X] Y Y =T[X] KerP.

37


38/83

KerP = X. y X P(y)(y) = T(Q(y) = T(y )(y) = y(y ) = 0 y Y X

KerP. , y

KerP T(y

) = 0 T

11 y = 0 y X KerP X. T Y = T[X] X. To Y =X

(1.5.9)

Banach . o (X/Y) - Y.

A

oT : (X/Y) Y T(x)(x) = x(x+Y). xY T(x)(x) = x(Y) = 0. xX |T(x)(x)|=|x(x + Y)| ||x||||x + Y|| ||x||||x|| T . . T . x Y. x :X/Y R x(x+Y) =x(x). T xX

|x(x+Y)| = |x(x)|=|x(x y)| yY ||x| | | |x y|| yY |x(x+Y)| ||x| | | |x+Y||

x (X/Y). T T(x) =x . x (X/Y).

||T(x)|| = sup {|T(x)(x)|: xBX} sup|T(x)(x)|: x+YBX/Y

= sup|x(x+Y)|: x+YBX/Y =||x||

xX ||x+Y|| 1. T |x(x+Y)| = |T(x)(x)|=|T(x)(x y)| yY ||T(x)| | | |x y|| yY

|x

(x+Y)| ||T(x

)| | | |x +Y|| ||T(x

)||

E ||T(x)||=||x|| T .

Banach boundedly complete .

38


39/83

(1.5.10)

nach boundedly complete Y = [en, n N]. X

= X Y

. dim(X

/X) =dimY

=dim(X

/Y)

.

Y = T[Y] Y. - Y = J[X] Y = J[X]. J[X] =T[Y] J[Y] = Y.OX = X Y. (X/X)=Y (X/Y) dim(X/X) =dimY =dim(X/Y).

39


40/83

40


41/83

2

James

2.1 J n, m N n m [n, m] ={k N :nkm}. T

, , . n N [n, ) ={k N :nk}. T .

J =

x={xn}nN R :sup m

i=1|nIi

xn|2


42/83

T

(m

i=1 |

nIi

(xn

+yn

)|2)1/2 = [(

nI1

xn

+ nI1

yn

)2 +...+ (nIm

xn

+ nIm

yn

)2]1/2

(mi=1

|nIi

xn|2)1/2 + (mi=1

|nIi

yn|2)1/2 ||x|| + ||y|| ||x+y|| ||x|| + ||y||

inkowski. Banach.

x(n)nN

Cauchy J. T > 0N m > n > N ||x(m) x(n)|| < . i N Ii ={i} |x(m)i x(n)i |< m > n > N.

x

(n)

inN i N. xi = limn x

(n)

i . x ={xi}iN. o x J x = lim

nx(n). > 0 M N

m > n M ||xn xm|| < 2 . o {Ii}ki=1

(ki=1

|jIi

(x(m)j x(n)j )|2)1/2 nM m

(k

i=1

|jIi

(x(n)j xj)|2)1/2

2 < nM ()

n = M () x(M) x J x J x= lim

nx(n)

, n N, en =X{n}

(2.1.2)

x J

. k

N sk =k

i=1

xiei.

||sk||2 + ||x sk||2 ||x||2

42


43/83

A

{Ii}mi=1 l {1,...,m 1} Ii[1, n] i= 1,...,lIi[n+ 1, +) i= l + 1,...,m. T

li=1

|nIi

sn|2 +m

i=l+1

|nIi

(x sn)|2 =mi=1

|nIi

xn|2 ||x||2

||sk||2 + ||x sk||2 ||x||2

(2.1.3)H{en}nN= (X{n})nN Schauder J.

A

en = 0 n N||en|| = 1 n N. J = [en : n N]. x ={xn}nN J. sn =

ni=1

xiei.

x = limn

sn., > 0. T

{Ii

}m

i=1

mi=1

|nIi

xn|2 >||x||2 2 ()

n0 = max

n: n

mi=1

Ii

()

||x||2 ||sn||2 < 2 nn0 nn0 ||xsn||2 =||xsn||2+||sn||2||sn||2 ||x||2||sn||2 < 2 . x = lim

n

sn.T k, nN

k > n1,...,k R o {Ii}mi=1Ii[1, n] i= 1,...,m.

(mi=1

|jIi

j|2))1/2 ||ki=1

iei|| ||ni=1

iei|| ||ki=1

iei||

43


44/83

(2.1.4)

H J boundedly complete. c0 - J.

{n}nN supnN

||ni=1

iei|| 0

n0 N m > n > n0 ||mi=1

iei||2 ||ni=1

iei||2 < 2 E (2.1.2)

||m

i=n+1

iei||

||mi=1

iei||2 ||ni=1

iei||2 <

J Banach, n=1

nen.

(2.1.5)

A J < c00(N) > c00(N) Banach Hamel . E J < c00(N)> . J.

(2.1.6)

I . I =nI

en. I J.

I. x J > 0 n0 N m > n > n0 |

mi=n+1

ei (x)| < .

44


45/83

K nI

en(x) x J. I :J R

I(x) = nI

en(x)

x

J, I

Banach-Steinhauss I J. I w=nI

en||I|| = 1

I. s w

=n=1

en.

E s / [en : n N]. , s

||.||=

n=1

en 0 <


46/83

(2.2.1)

{dn}nN J d1=e1dn=en en1, n >1. {dn}nN Schauder J.A

< dn, n N >=< en, n N > [dn, n N] =J.nN 1,...,n, n+1 R. {Ii}mi=1[1, n]. An /

mi=1

Ii

(m

i=1 |

Ii(n

i=1

idi)

|2)1/2 = (

m

i=1 |

Ii(n+1

i=1

idi)

|2)1/2

||

n+1

i=1

idi|| ||

n

i=1

idi|| ||

n+1

i=1

idi||

Im = [i, n] -in.

(mi=1

|Ii(ni=1

idi)|2)1/2 = (m1i=1

|Ii(ni=1

idi)|2 + |Im(ni=1

idi)|2)1/2

= (m1

i=1|Ii(

n+1

i=1idi)|2 +2i )1/2 ||

n+1

i=1idi|| ||

n

i=1idi|| ||

n+1

i=1idi||

||ni=1

idi|| ||n+1i=1

idi|| {dn}nN- .

(2.2.2)

H{dn}nN shrinking. .

I A{un}nN block {dn}nN M = supnN

||un|| n=1

un

n .

A

46


47/83

m > k N ||mi=k

ui

i||2 5M2

mi=k

1

i2 -

J

Banach. A

uk =

nk+1i=nk+1

idi,

mi=k

ui

i = nk+1

k enk +

nk+1 nk+2k

enk+1+...+nk+11 nk+1

k enk+11+

+ (nk+1

k nk+1+1

k+ 1 )enk+1+

nk+1+1 nk+1+2k+ 1

enk+1+1+

+ ...+

+ (nm1

m 1nm+1

m

)enm+ ...+nm+11 nm+1

m

enm+11+nm+1

m

enm+1

{Ij}j=1. To{1,...,}- .

Fk = {j {1,...,}: Ij[nk, nk+1 1]}Fs = {j {1,...,}: Ij[ns+ 1, ns+1 1]} , s= k + 1,...,m 1

Fm = {j {1,...,}: Ij[nm+ 1, nm+1]}

F =m

i=kFi

U = {j {1,...,}:s1 < s2 {k+ 1,...,m}: Ij supp(us1)=, Ij supp(us2)=}

F, U {1, ...}. s= k, k+ 1,...,mFs=,

jFs

|Ij (mi=k

ui

i)|2 = 1

s2

jFs

|Ij (us)|2 M2

s2

jF

|Ij (mi=k

ui

i)|2 M2

mi=k

1

i2

A jU, sj,1 = min

{i

{1,...,m

}: Ij

supp(ui)

=

}sj,2 = max {i {1,...,m}: Ij supp(ui)=}

|Ij (mi=k

ui

i)|2 = |Ij (

usj,1sj,1

+usj,2

sj,2)|2 2M

2

s2j,1+

2M2

s2j,2 4M

2

s2j,1

47


48/83

E |U| m k+ 1

jU

|I

j (

mi=k

ui

i )|2

4M2mi=k

1

i2

||mi=k

ui

i||2 5M2

mi=k

1

i2.

n=1

un

n. A x =

n=1

un

n.

A

A (dn)nN shrinking, (1.5.2) x X, > 0 (un)nN block (dn)nNx(un) >

n N. Tx(x) =n=1

x(un)

n >

n=1

1

n = +, .

(2.2.3)

J = [en, n N] < s >. J 1 J.

A

< dn

, n

N >=< en

, n

N >

< s >.

J = [dn, n N] =< en, n N >< s > [en, n N]< s >= [en, n N]< s >

dim < s >= 1< +. J = [en, n N]< s >

(2.2.4)

T 1 c0 J. , J , unconditional (.[7]unconditional ).

48


49/83

2.3

J .

- J. J J R.C.James - . J .

(2.3.1)

Ye J \ J en we [en, n N] = [e]

A

o {en}nNw-Cauchy. , x J. A (2.2.3) x =

n=1

nen+s

.

x(en) =n+n

nn 0. anach-Steinhauss e J en w

e - e /

J. , e w

e(s) =n=1

e(en) = 0. e(s) = lim

ns(en) = 1. T -

[en, n N] = [e]. x =n=1

nen [en, n N]. T

e(n=1

nen) =

n=1

ne(en) = 0< e >[en:n N]

A, x [en: n N] x J.

y [en:n N] R :x =y +s x(x) = x(s) =x(s)e(x)x = x(s)e

[enn: N] < e >. [en, n N] =< e >

49


50/83

(2.3.2)

J = J [e]. A - J quasi-reexive -1dim(J/J) = 1

A

{en}nN boundedly complete , (1.5.10) .

O James, J | | | |1 | | | |2. ,

J (

J,|| ||

1) (J

,|| ||

2) . J . O

J ={xn} c0 : sup (xp1 xp2)2 + (xp2 xp3)2 +...+ (xpm1 xpm)2


51/83

A

(1.3.2) o n N 1,...,nR ||

ni=1

idi|| = ||ni=1

iei||1. , m N 1 p1 p1 < p2 p2 < . . . < pm pm n. Ij = [pj, p

j ] j = 1,...,m n+1 = 0

mj=1

|Ij (ni=1

idi)|2 = (p1 p1+1)2 + (p2 p2+1)2 +...+ (pm pm+1)2

||n+1

i=1iei)||21 = ||

n

i=1iei)||21 ||

n

i=1idi)|| ||

n

i=1iei)||1

mN p1 < ... < pmn + 1, Ij = [pj , pj+1 1], j= 1,...,m

(p1 p2)2 +...+ (pm1 pm)2 =mj=1

|Ij (ni=1

idi)|2 ||ni=1

idi||2

||ni=1

iei||1 ||ni=1

idi||

.

(2.3.4)

{en}nN shrinking (J, | | | |1) (J, | | | |2).

A

A ||n

i=1 idi|| =||n

i=1 iei||1 1,...,n R, n N || ||1 || ||2 o (2.2.2) {un}nN block {en}nN

n=1

un

n

. (2.2.2).

51


52/83

(2.3.5)

x J {x(en)}nN.

A

x J. ||||1. {en}nN - shrinking, (1.5.3) ||x||1= lim

n||

ni=1

x(ei )ei||1

n N xn=ni=1

x(ei )ei. T N N

||x||21 ||xN||21 < 2 ()E k N p1 < ... < pkN+1 ||xN||1=[x(ep1) x(ep2)]2 +...+ [x(epk1) x(epk)]2.T m > nN+ 2 |x(em) x(en)|< , (x(en))nN Cauchy . m N ||xm||1 >||x||1 ||x||1 =sup

nN||

ni=1

x(ei )ei||1.

(2.3.6)

O(J, | | | |2) .

o :J J (x) = (, x(e1) , x(e2) , ...) =limn

x(en), (2.3.5).

. 1=,...,n=x(en1) (x) =

n=1

nen

||(x)||2 =||n=1

nen||2 = supnN

||ni=1

iei||2=supnN

||ni=1

x(ei )ei||2 =||x||2

{en}nN shrinking (J, | | | |2). . x={xn}nN J.TsupnN

||ni=1

(xi+1 x1)ei||2


53/83

J , x J {yn}nN {yn}nN yn wx x(en) =xn+1x1 n N.E (x) =x.

(2.3.7)

J (J, | | | |2)o J - (J, | | | |2) .

2.4 J 2-saturated J 2-saturated -

2. J .

(2.4.1)

{un}nN block {en}nN. m N 1,...,m R

(m

i=1

2i

)1/2

||

m

i=1

i

ui||

A

X m N 1,...,m R mi=1

2i = 1. 1 ||mi=1

iui||., ||ui|| = 1 i = 1,...,m {un}nN block {Ij}j=1 1 < 1 < ... < m1 < 0= 1m =

{Ij

}i

j=i1 supp(ui)

i= 1,...,m

ij=i1

|Ij (ui)|2 = 1 i= 1,...,m

j=1

|Ij (mi=1

iui)|2 =mi=1

2i = 11 ||mi=1

iui||

53


54/83

(2.4.2)

{un}nN block {en}nN limn

s(un) = 0. T >0

{un}nN m N 1, ...m R

(1 2

)(mi=1

2i )1/2 ||

mi=1

iui|| (

5 +

2)(

mi=1

2i )1/2

A

A uk =

nk+1i=nk+1

iei, yk = uk s(uk)enk+1.

||yk

|| 1 +

|s(uk)

| k

N lim

k ||uk

yk

|| = 0, -

{yn}nN (yn)nN {un}nN {un}nN

k=1

||uk yk||2 < 2

16 ||yk||< 1 +

2

128 k N

m N 1,...,m R mi=1

2i = 1. T

1 2


55/83

sj,2 =max {i {1,...,m}: Ij supp(yi)=}.T

jU

|Ij (mi=1

iyi)|2 =

jU

|Ij(sj,1ysj,1) +Ij(sj,2ysj,2)|2

jU

(22sj,1 |Ij(ysj,1)|2 + 22sj,2|Ij(ysj,2)|2)4 + 2

32

X F, U {1,...,},

||mi=1

iy

i||< 5 + 2

16 < 5 +

4

||mi=1

iui||

mi=1

|i|||ui yi|| + ||mi=1

iyi|| (

mi=1

||ui yi||2)1/2 +

5 +

4

5 +

2

(2.4.3)O J 2-saturated. E J

{xn}nN > 0 {xn}nN{xn}nN m N 1,...,m R

(1 )(mi=1

2i )1/2 ||

mi=1

ixi|| (

5 +)(

mi=1

2i )1/2

A

J 1 (1.3.7) {xn}nN J . sliding hump argument block{un}nN {xn}nN {xn}nN

n=1

||xn un||2 < 2

4

55


56/83

m N 1,...,m R mi=1

2i = 1

||mi=1

ixi||

mi=1

|i|||xi ui|| + ||mi=1

iui||

5 +

||mi=1

iui|| mi=1

|i|||xi ui|| ||mi=1

ixi|| 1 ||

mi=1

ixi||

To (1.3.3).

56


57/83

3

O James Tree

3.1 O J T Cantor

o . O - . J T - . J , 2.

O (3.1.1)

O2


58/83

i= j. - 1,...,n.

I

2


59/83

sup {Ii}mi=1. J T - .

x J T o||x|| = sup mi=1

|tIi

x(t)|2

1/2

, sup

- {Ii}mi=1

(3.1.3)

O(J T, | | | |) Banach.

H (2.1.1).

. || || . . x, y J T {Ii}mi=1 . T

(mi=1

|tIi

(x(t) +y(t))|2)1/2 = [(tI1

x(t) +tI1

y(t))2 +...+ (tIm

x(t) +nIm

y(t))2]1/2

(

m

i=1 |tIi

x(t)

|2)1/2 + (

m

i=1 |tIi

y(t)

|2)1/2

||x

||+

||y

|| ||x+y

|| ||x

||+

||y

|| inkowski. Banach.

{xn}nN Cauchy J. T > 0N m > n N ||xm xn|| < . t 2 n > N. {xn(t)}nN t2 n M ||xnxm|| < 2

.

o {Ii}k

i=1

(ki=1

|tIi

|xm(t) xn(t))|2)1/2 < 2

m > nM m

(ki=1

|tIi

|xn(t) x||2)1/2 2

< nM ()

59


60/83

n = M() xM x J T x J T x= lim

nxn

(3.1.4)

x JT. k N sk =ki=1

xiei.

||sk||2 + ||x sk||2 ||x||2

(2.1.2)

(3.1.5)H{en}nN Schauder J T.

A

en = 0 n N ||en|| = 1 n N. o J T = [en :nN]. x J T. sn =

ni=1

x(ti)ei.

x = limn

sn., > 0. T

{Ii

}m

i=1 ,

mi=1

|tIi

x(t)|2 >||x||2 2 ()

n0 = max

|t|: t

mi=1

Ii

()

||x||2 ||sn||2 < 2 n2n0+1 n2n0+1 ||xsn||2 =||xsn||2 + ||sn||2||sn||2 =||x||2||sn||2 < 2 . x= lim

nsn. T n N

1,...,n, n+1 R o {Ii}mi=1tn+1 /

mi=1

Ii.

(mi=1

|tjIi

j|2))1/2 ||n+1i=1

iei|| ||ni=1

iei|| ||n+1i=1

iei||

60


61/83

.

(3.1.6)H{en}nN boundedly complete . c0 -

J T.

(2.1.4).

(3.1.7)

J T < c00(2 .

(3.1.8)

K (2.1.6), I ,

I =sI

es JT. A, I , I w

=sI

es JT.

||I|| = 1 I.E, 2N

w

=n=1

e|n. I /[en : n N], I, {en}nN shrinking J T . , x

J T,

||x||= sup mi=1

|Ii(x)|21/2

{Ii}mi=1 , , .

(3.1.9)

2N

o[e|n

, n

N] J.

T :< e|n, n N > J T(ni=1

ie|i) = (1,...,n, 0,...). H T -

. N, I1= [n1, n1], ...Im=

61


62/83

[nm, nm] 2

0

N N m > n > N ||m

i=n+1

ie|i||=||m

i=n+1

iei||<

E

n=1

ne|n T(

n=1

ne|n) =

n=1

nen. E

.

(3.1.10)

O J T .

A

:2N 1-. 1=2 s2


63/83

(e|k)kNwCauchy. x J T x

[e|n :

nN]

[e|n :n N] k N

x(e|k) =n=1

nen(T(e|k))+s

(T(e|k)) =n=1

nen(ek)+s

(ek) =k+ k

kk 0. JT =w lim

ke|k.

/ J T.I () = limn

(e|n) = 1. A o

J T w () =

n=1(e|n) = 0

. .

3.2 O 1 J T 1 J T.-

J J -, J T .

{In}nN 2


64/83

x J T. K BJT sup {k(x) :k K} ||x||. n N xn=

ni=1

ei (x)ei. T

{Ii}mi=1 ||xn|| 1n < (mi=1

|Ii(xn)|2)1/2.

i= Ii(xn)

(mi=1

|Ii(xn)|2)1/2 k =

mi=1

iIiKk(xn) = (

mi=1

|Ii(xn)|2)1/2

o ||xn|| 1n < k(xn)sup {k(xn) :k K}n ||x|| sup {k(x) :k K}

. H .

(3.2.2)

To K w- J T

A

J T , (BX, w) , K . kn = w

n=1i,nI

i,n

K. n N|i+1,n| |i,n| i N . .

{In}nN . T {Ikn}nN IIkn

wI. {Ikn}nN

Ikn(es)

nN

s

2


65/83

M [N], {i}iNB2i,n

nM i i N {Ii}iN Ii = w limnM

Ii,n. {Ii}iN

. k = w n=1

iIi K.

k = w limnM

kn. s 2 0. N N

(

i=N+1

2i )1/2

limsupnL

L2(un)+liminf

nLL

2(un)>2

2 () k N k

2

4 > M2.

L0 [M]1 2N (). T lim sup

nL0

12(un) lim inf

nL01

2(un) 2

4.

L1[L0] limnL1

12(un)>

2

4 22N (). -

1

=2. Lk

Lk1

, ...,

L1

N

1,...,k2N limnLi

i2(un) >

2

4

i = 1,...,k limnLk

i2(un) >

2

4 i = 1,...,k. E N Lk

i = 1,...,k i2(un) >

2

4 n Lk : n NE 1,...,k

{un}nN block

n0 Lk n0 N ||un0||2 ki=1

i2(un0) > k

2

4 > M2

, .

(3.2.7)

K J T wCauchy .

68


69/83

1 J T 1- Rosenthal.

(3.2.8)

J T 1c0. unconditional.

3.3

J T . - . R.Haydon - .(.[3])

(3.3.1)(R.Haydon)

Banach 1. T ow-

KX convw

(K) =conv||.||(K)

M J T.

(3.3.2)

J T = [I :I 2


70/83

.

2(2N) =

f : 2N R :supF

|f()|2 :F 2N =sup

F

f()g() :F 2N 1/2

2(2N) Hilbert. - Riesz Hilbert S : 2(2

N) 2(2N)S(f)(g) =< f, g > g2(2N) .

(3.3.3)

f 2(2N). T {n}nN 2 {n}nN 2N f=

n=1

nX{n}

A

suppf =n=1

2N :|f()|> 1

n

. suppf

.

2N :|f()|> 1

n

n

N. n

N k N 1,...,k 2N |f(i)| > 1n i = 1,...,k.

k

n2 m1. y2Ym2 . , ykYmk mk+1N yk+1Ymk+1, -mk+1 =max

mk, m

k+1

+ 1. E {mk}kN

. dist(x, Ymk) d(x, yk) k N limm

dist(x, Ym) =

limk

dist(x, Ymk) limk

d(x, yk) = d(x, y) limm

dist(x, Ym) dist(x, Y). - dist(x, Y) = lim

mdist(x, Ym).

Y = [es :s N]. -Q:J T J T/Y Q(x) =x +Y . E

J T/Y = Q[J T] =Q(< I :I >Q[< I :I >]= [I +Y :I ]

J T/Y = [I +Y :I ].

(3.3.5)O J T/Y 2(2N).

A < I + Y :I 2 . E I , S

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I + Y =S + Y (I) =(S) I S Y . E- U :< I +Y : I 2 2(2N)

U(

ni=1

iIi + Y) =

ni=1

iX{(Ii)}. HU . . X , I1,...,In

1,...,n Rni=1

2i = 1.A ||ni=1

iIi+Y||= 1.

T

|ni=1

iIi(x)| (

ni=1

2i )1/2(

ni=1

Ii2(x))1/2 ||x|| x J T

||

n

i=1

iIi || 1 ||

n

i=1

iIi +

Y|| 1

, N I1,...,In.

Ym =< es :|s| m >. T Y =

m=1

Ym. m N,

xm=ni=1

ie(Ii)|m+1 . ||xm||= 1 mN.T mNy Ym

|ni=1

iI

i(xm) y

(xm)|=|ni=1

iI

i(xm)|= 11 ||ni=1

iI

i y

|| y

Ym

1dist(ni=1

iIi, Ym) mN ||

ni=1

iIi +Y||= dist(

ni=1

iIi, Y) =

= limm

dist(ni=1

iIi, Ym)1 (3.3.4)

K U J T/Y U. M . f=

n=1

nX{n}= limk

U(kn=1

nn+Y).

E f U[J T/Y]. E J T/Y Banach, U[J T/Y] Banach . U[J T/Y] 2(2N)f U[J T/Y]. U .

J.

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(3.3.6)

J T quasi-reexive J T/ J T .

J T boundedly complete (1.5.10) dim(J T/ J T) =dim((J T/Y)) =dim(2(2N)) =dim(2(2N)) = +

(3.3.7)

J T = J T [ : 2N]. E, x JT x = x +

n=1

nn x J T, {n}nN2

{n}nN2N.

A

, (3.1.11), J T [ :2N] ={0}. J T boundedly complete (1.5.10) y Y y =

n=1

nn {n}nN2

{n}nN 2N. TU (1.5.9) (3.4.5) S 2(2N) . A L = T US : 2(2N) Y L . L L(X{}) = 2N. 2N I 2


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(3.3.8)

O J T w J T x J T {xn}nN J T x =w limn xn.

A

x = x+j=1

jj . x1 = x. n > 1kn

1,...,n. xn= x +n

j=1

jej |kn . {xn}nN

{n}nN 2. E, (1.1.10) I(xn) n

j=1

jj (I) I. ,

I. A i N (I) = i n n0 I(xn) = i =

n=1

nn (I

). I(xn) = 0 =j=1

jj (I

) n N.

I I(xn)n

j=1

jj (I

)

.

3.4 O J T 2-saturated J T 2saturated,

2. - [1].

. - Banach. , M , M(2) =

{(n, m) :n, m

M, n < m

} [M]

M.

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(3.4.1) (.Ramsey)

A1,...,Ak N N(2)

=

ki=1 A

i. T M [N] i {1,...,k} M(2) Ai.

L [N]n N {n}(2)L ={n, m) :mL,n < m}. M1= N m1M1. M2[M1] i1 {1,...,k} {m1}(2)M2 Ai1 .m2 M2 m2 > m1. T M3 [M2] i2 {1,...,k} {m2}(2)M3 Ai2 . , -

{mn

}nN

N,

{Mn

}nN

[N]

{in

}nN

{1,...,k} mp Mn p n{mn}(2)Mn+1 Ain n N.

N =ki=1

n N :{mn}(2)Mn+1 Ai

. i {1,...,k}

L =

n N :{mn}(2)Mn+1 Ai

.

M ={mn : nL} [N]. T mn, mp M mn < mp n < p.Omp Mn+1 (mn, mp) Ai. M(2) Ai .

Ramsey , .

(3.4.2)

{xn}nN block J T limn

I(xn) = 0

I. T > 0 {xn}nM S, in(S) =, |S(xn)| nM, n(S)M, S.

> 0. Y K = supnN

{||xn||}. n N n = min {supp(xn)}. Qn =

t2 Stin(S) =t

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T n N

2

|Qn|< tQn |S

t(xn)|2

||xn||2

K2

|Qn| K2

2 n N

N {0} L [N]|Qn| = n L. A = 0 (xn)nL, Qn = n L. 1. nL Qn ={ti,n: 1i}. i, j {1,...,}

Ai,j ={n, mL : n < m S ti,n tj,m, |S(xn)|> }

A A = N(2)

\ 1i,j

Ai,j . N(2)

= A 1i,j

Ai,j . ,

Ramsey M [N]M(2) AM(2) Ai,j i, j {1,...,}. , i, j {1,...,} M(2) Ai,j . n, kM n + 1< k, (n, k), (n + 1, k)M(2). ti,nti,n+1. , S1 ti,n S2 ti,n+1 tj,k, |S1(xn)| > |S2(xn+1)| > . , - ti,n+1 S1. Sn,n+1 ti,n ti,n+1.

|Sn,n+1(xn)

|=

|S1(xn)

|> . I=

n=1

Sn,n+1. E

I |I(xn)|=|Sn,n+1(xn)|> nM, lim

nI(xn) = 0. M(2) A.

(xn)nM . , -S in(S) = n, m Mn < m |S(xn)| > |S(xm)| > . t1 S |t1|= n t2 S |t2|= m, QnQm, oi, j {1,...,} t1 = ti,nt2 = tj,m.(n, m)Ai,j , A

1i,j

Ai,j =.

block (xn)nN

J T, > 0, -

M [N] S in(S) = |S(xn)| nM n(S), .

(3.4.3)

block {yn}nN J T limn

I(xn) = 0

I. T > 0, {yn}nN n N

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1,...,n R

(

ni=1

2

i )1/2

||ni=1

iy

i|| 2(1 +)(ni=1

2

i )1/2

> 0. {k}kN R. , {Mk}kN [N] {yn}nMk k k N. k N S in(S) =, |S(yn)| k n Mk n(S) Mk. n N, n = min {supp(xn)} mn = 2n . - {pn}nN {kn}nN

pnMkn n N n=1

mn(

l=n+1

2kl)< 2

k1= 1 p1Mk1 . En n 0L1[N]rL1

2r < 2

2m1

k2 L1 k2 L1 p2 Mk2 p2 > p1. En nL1 0L2[L1]

rL2

2r < 2

22m2

k3 L2 k3 > k2 p3 Mk3 p3 > p2. {kn}nN {pn}nN, pn Mkn n N, {Ln}nN [N] kn Ln

rLn2r n}. AA(S)=, (S) = minA(S)., (S) = +.

n=(S)

|S(yn)|2 ni(S)

2n

A(S) =

{n1 < n2 < ... < nj < nj+1 < ...

}. E

(S) = n1 i(S) n1. j N (yn)nnj nj |S(ynj)|> nj |S(ynj+1| nj . E n=(S)

|S(yn)|2 =

ni(S):n=n1

|S(yn)|2 =j=2

|S(ynj )|2 +

ni(S):n/A(S)

|S(yn)|2

j=1

2nj+

ni(S):n/A(S)

2n=ni(S)

2n

nN 1,...,n

R

n

i=1

2i = 1. M

(2.4.1) 1 ||ni=1

iyi||.

, U . S U, S=S0 S,

S0 =

tS :|t|< bi(S)

S =

tS :bi(S) |t|

. , , , . S0, S S . S =S1 S

S1 =

tS:bi(S) |t|< bi(S)+1

S

=

tS:bi(S)+1 |t|

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ToS

S

=S

1 S2 S3

S

1 = tS:bi(S)+1 |t|< b(S)S

2 =

tS:b(S) |t|< b(S)+1

S

3 =

tS:b(S)+1 |t|

N (S

) = + b(S) = +. S = S0 S1 S1 S2 S3 . x=

n

i=1iy

i

SU

|S(x)|2 =SU

|S0(x) +S1(x) +S1 (x) +S

2 (x) +S

3 (x)|2

4SU

(|S0(x)|2 + |S1(x)|2 + |S2 (x)|2)) + 4

SU

|S1 (x) +S3 (x)|2

R S0, S1, S

2 , R(x) = iR(yi)

i {1,...,n}. |R(x)|2 =ni=1

2i |R(yi)|2. E

SU

(|S0(x)|2 + |S1(x)|2 + |S2 (x)|2)) =

SU

ni=1

(2i (|S0(yi)|2 + |S1(yi)|2 + |S2 (y

i)|2))

=ni=1

2i (SU

(|S0(yi)|2 + |S1(yi)|2 + |S2 (y

i)|2))

ni=1

2i ||yi||=ni=1

2i = 1

E S U,

|S1 (x) +S

3 (x)

|2 =

|

n

i=1,i=(S

)

iS(yi)

|2

(

n

i=1,i=(S

)

2i )(n

i=1,i=(S

) |

S(yi)

|2)

n=(S)

|S(yn)|2

ni(S)

2n

k N Uk ={S U :i(S) =k}. S Uk i(S) = k + 1

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SU|

S

1 (x) +S

3 (x)

|2 =

k=1

SUk |

S

1 (x) +S

3 (x)

|

=k=1

SUk

ni(S)

2n =k=1

SUk

nk+1)

2nk=1

mk(

n=k+1

2n)< 2

, U

SU|S(x)|2 < 4 + 42 0 {xn}nN n N 1,...,n R

(1 )(ni=1

2i )1/2 ||

ni=1

ixi || (2 + 3)(

ni=1

2i )1/2

A

Y JT. A 1 - J T, (1.3.7), {xn}nN Y . > 0. slidinghump argument {xn}nN block (yn)nN

n=1

||xn yn||2 < 2 ()

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I(xn) n 0 () ||xn yn|| n 0.

I

|I(yn)| |I(xn)| + ||I||||xn yn|| n N lim

nI(yn) = 0

E, , > 0 - {yn}nN n N 1,...,n R

(ni=1

2i )1/2 ||

ni=1

iyi|| 2(1 +)(

ni=1

2i )1/2

{xn

}nN

{xn

}nN. n

N 1,...,n R,

||ni=1

ixi ||

ni=1

|i|||xi yi|| + ||ni=1

iyi||

(ni=1

2i )1/2(

ni=1

||xi yi||2)1/2 + 2(1 +)(ni=1

2i )1/2

(2 + 3)(ni=1

2i )1/2

||ni=1

iyi||

ni=1

|i|||xi yi| | | |ni=1

ixi ||

(ni=1

2i )1/2 (

ni=1

2i )1/2(

ni=1

||xi yi||2)1/2 ||ni=1

ixi ||

(1 )(ni=1

2i )1/2 ||

ni=1

ixi ||

(1 )(ni=1

2i )1/2 ||

ni=1

ixi || (2 + 3)(

ni=1

2i )1/2

(1.3.3).

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[1] I.Amemiya, T.Ito, Weakly null sequences in James spaces on trees, KodaiMathematical Journal 4, 3, 418425, 1981.

[2] P.Eno,A counterexapmle to the approximation property in Banach spaces, Acta

Math, 130, 309-317, 1973.

[3] R. Haydon, Some more characterizations of Banach spaces containing l1,Mathematical Proceedings of the Cambridge Philosophical Society, 80, 269-276, 1976.

[4] R.C. James,A non-reexive space isometric with its second conjugate space, Proc. Nat.Acad. Sci. U.S.A. 174-177, 1951

[5] R.C. James,A separable somewhat reexive Banach space with non-separable dual,Bull. Amer. Math. Soc., 80, 738-743, 1974.

[6] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain 1and whose duals are non separable, Studia Math., 54, 81-105, 1975.

[7] J. Lindenstrauss and L. Tzafriri,Classical Banach spaces I and II, Springer, 1996.

[8] W.Rudin,Real and Complex Analysis, McGraw-Hill, New York, 1966.

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