Οι χώροι James και James Tree
Transcript of Οι χώροι James και James Tree
-
7/23/2019 James James Tree
1/83
James James Tree
:
I
:
2014
-
7/23/2019 James James Tree
2/83
2
-
7/23/2019 James James Tree
3/83
. .
. - , , . . . -. . . .E .
. - . . , .
, . .
A 2014
3
-
7/23/2019 James James Tree
4/83
4
-
7/23/2019 James James Tree
5/83
A ,
5
-
7/23/2019 James James Tree
6/83
6
-
7/23/2019 James James Tree
7/83
........................................................................................................................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
7
-
7/23/2019 James James Tree
8/83
8
-
7/23/2019 James James Tree
9/83
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 .
9
-
7/23/2019 James James Tree
10/83
10
-
7/23/2019 James James Tree
11/83
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.
11
-
7/23/2019 James James Tree
12/83
- 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
13
-
7/23/2019 James James Tree
14/83
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||
-
7/23/2019 James James Tree
15/83
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||
15
-
7/23/2019 James James Tree
16/83
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 .
16
-
7/23/2019 James James Tree
17/83
(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 .
17
-
7/23/2019 James James Tree
18/83
, - .
(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.
19
-
7/23/2019 James James Tree
20/83
|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<
-
7/23/2019 James James Tree
21/83
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
21
-
7/23/2019 James James Tree
22/83
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
22
-
7/23/2019 James James Tree
23/83
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
23
-
7/23/2019 James James Tree
24/83
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||
24
-
7/23/2019 James James Tree
25/83
.
.
(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||
25
-
7/23/2019 James James Tree
26/83
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
27
-
7/23/2019 James James Tree
28/83
||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
28
-
7/23/2019 James James Tree
29/83
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 .
30
-
7/23/2019 James James Tree
31/83
(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||
-
7/23/2019 James James Tree
32/83
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)
|
32
-
7/23/2019 James James Tree
33/83
()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||
33
-
7/23/2019 James James Tree
34/83
||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
34
-
7/23/2019 James James Tree
35/83
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)||
35
-
7/23/2019 James James Tree
36/83
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. ,
36
-
7/23/2019 James James Tree
37/83
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
40/83
40
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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 <
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
57/83
3
O James Tree
3.1 O J T Cantor
o . O - . J T - . J , 2.
O (3.1.1)
O2
-
7/23/2019 James James Tree
58/83
i= j. - 1,...,n.
I
2
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
-
7/23/2019 James James Tree
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
71
-
7/23/2019 James James Tree
72/83
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.
72
-
7/23/2019 James James Tree
73/83
(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
-
7/23/2019 James James Tree
74/83
(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.
74
-
7/23/2019 James James Tree
75/83
(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
75
-
7/23/2019 James James Tree
76/83
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
76
-
7/23/2019 James James Tree
77/83
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|
78
-
7/23/2019 James James Tree
79/83
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
79
-
7/23/2019 James James Tree
80/83
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 ()
80
-
7/23/2019 James James Tree
81/83
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).
81
-
7/23/2019 James James Tree
82/83
82
-
7/23/2019 James James Tree
83/83
[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.