R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O Sé ria I : P R A C E M A T EM A T Y C Z N E X X I I I (1983)
Marek Wôjtowicz (Poznan)
On the James space J ( X ) for a Banach space X
Abstract. For X an arbitrary Banach space, we consider the natural analogue J (X) of the famous James space J — J (В ). Our main result is that J (X)** is isomor
phic to J(X **)@ X **. In particular, if X is reflexive, then J (X)**/J(X) is isomorphic to X .
The Banach space constructed by В . C. James [3] in 1950, now com
monly denoted by J , was the first example of a quasi-reflexive space (i.e. of finite codimension in its bidual) of order 1 : dim = 1 . In 1971, J . Lindenstrauss [4] (see also [5], Theorem l.d.3) modified James, construction so as to prove that for every separable Banach space X there exists a Banach space Z such that Z**\Z is isomorphic to X . The same holds for weakly compactly generated spaces X , as shown by Davis, Figiel, Johnson and Felczynski [1] in 1974; this class contains all separable and all quasi-reflexive Banach spaces. It seems to be an open question whether any restrictions on X are necessary at all. Our result thus shows that for X a reflexive Banach space we may take Z = J (X) to have Z**/Z isomor
phic to X.
Our terminology and notation are mostly standard (e.g., as in [5]).
We write X ^ Y or X Y if the Banach spaces X and Г are isometric or isomorphic, respectively. Let Y — (Y, ||-||) be a Banach space, Г a finite set and l < p < oo. Then 1р (Г, Y) denotes the Banach space consisting of all families у — (уу)уеГ with yY e T, under the norm ||-||р defined by the formula
\\y\\p = { V*r
if p — oo.
We have
under a natural isometry (cf. [2 ], Chap. П, §2, (11)).
184 M. W ô j t o w i c z
Let X = (X , ||*||) be a Banach space. Then the Banach space J (X )
= (J(X ), 111*111) is defined as follows:
J ( X ) = {x = (х{) e X N: |||æ|j| = sup<rp(®) < oo},
where & is the set of all strictly increasing sequences p = (^(1 ), ...
...,р ( 2 Ь + 1)) in N , k e N u { 0}, and к
Gp{pC) ( ll*^p(2f) Xp{2i— l ) l l 2 “b 11^(2*4-1)11 j *
t’ = l
I t is easily seen that for each x = (x{) e J (X) the limit x0 = lim ^ exists
г->оо
in X. The James space J (X) = ( J (X)] |||*|||) is the closed subspace of J (X) consisting of those x e J (X) for which lim xi = 0. I t is clear that the sub-
i-*r OO
space X of all constant sequences is isometric in a natural manner to X and is the range of the norm 1 projection Q in J (X) defined by Q{x)
= (x0, x0, ...). Moreover, Q~1(0) = J (X). Hence
(1) J ( X ) = X @ J ( X ) .
Consider the space X x J (X) equipped with the norm Ц-lb defined by IK^oj ®)lli = ЩН-HI N11»
and define T: X x J ( X )- > J {X ) by
T(xQ, x) — (х0) #?i, x2y...).
Then T 'is an algebraic isomorphism between X x J ( X ) and J ( X ) , and
||(a?0, a?) ||x < 2 111 T(x0, ж)|||. Hence T~l is continuous, and so is P b y theBanach isomorphism theorem (this can also be checked directly). Thus T is a topo
logical isomorphism between X x J ( X ) and J ( X) , whence X x J ( X ) ^ J { X ) .
Hence and from (1) we have
(2 ) J ( X ) ^ J ( X ) .
For each n e N the map P n: x ь-> (0, . . . , 0, xn, 0, ...) is a projection in
CO
J (X) of norm 1. Moreover, P nP m — 0 if n Ф m, and x = 2 -P»®* Y x e J { X ) .
71—1
This means that the sub spaces P n( j (X)), n = 1, 2, ..., form a Schauder decomposition of J (X), with P 1? P 2, ... as the associated sequence of projections. Write
= -P1+-P2+ •••
J n(X) = Sn(J(X)) = {CO = ( x J e J i X ) : x{ = 0 V i > » },
and observe that 8n is a projection in J (X) and
\\ SJ =1 ( n e N ) .
Similarly as in the case of the usual James space J — J (JR) (cf. [3]), it can be shown that the just described decompo sition of J (X) is shrinking, i.e., for every f e J ( X ) * ,
* OO
(3) / — J ? P * f = lim 8 * f (norm convergence),
n~=\ oo
so that thé sequence of subspaces P * ( J (X)*), n = 1, 2 , ..., forms a Schau- der decomposition of J (X)*, with P *, P *, ... as the associated sequence of projections.
Since X and P n[ J (X)) are isometric via the natural injection, for every / e J (X)* there is a unique f n e X* such that
( Kf)(x) = V æ e J ( X ) .
We may thus identify every / e J (X)* with the corresponding sequence (/») c X *; thus
' OO
f(x) = ^fn(X n), V x e J ( X ) . n~l
Similarly, for every F e J ( X ) * * there is a unique sequence F = (Fn) in X ** such that for all n e N and / = (fn) e J (X )*,
OO
( P T F ) ( f ) - F n(fn) and F ( f ) = n—1 We shall now prove our main result.
Th e o r e m. The map Fi->F = (F n) is an isometry between J(X )* * and J(X **).
Proof. For each p = (p( 1), p(2k + l)) e & consider the operator Bp : J(X )-> lz(X ) defined by
Bp (%>) = (%p{2) •••? ®р(2к) ^р(2к— 1)? ^p(2k-\-lp •*•)•
Then *
H-BpMII = ap (x) <|||®|||,
and hence |]Бр|| = 1 . It is easily seen that B** : J(X )* * -* l2(X)** = l2(X**) is given by
B p *(F ) = (Fp(2)—Fp(i)i ••• / Fp(2k)~Fp(2k-i)i —F P(2k+i)i h, • ••)•
Hence
\\b;*(F )\ \ =op(é),
186 M. W ô j t o w i c z
and since ||J8j*|| = ||2?p|| = 1, we have
, ■/ ap (F)^\\F\\.
I t follows that
( 4 ) l ! | # | i | < m , V F e J ( X Г .
The proof of the converse inequality requires some preparations.
For each n e N let 3?n be the set of those p e g ? whose last term is < n - f 1.
Define
^ n — ^oo { ^ n l ^2 ( - ^ ) )
and note that
(under a natural isometry). I t is clear that the map An: J n(X)-~+3Fn defined by
A n ( æ ) = ( B p i æ ) ) ^
is am isometric embedding. This and the Hahn-Banach theorem imply that for every y> e J n (X)* there is g = {gp)pe£?n e 3C*n such that
Ml = Ml = 2 IM
P&n and y> = A*g, i.e.,
v “ (BP\jn(x )f (gP) •
P ^ n
In particular, if / e J(X )* and y = (S*f)\Jn{X), then ||y|| = \\S*J\\ and so we have
(5) . iis :/n - 2 " w and s«f = 2 Biop-
p e & n p e & n
F ix F e J (X )** and le t/ e J (X)*. Then F ( f ) = lim F ( 8 * f ) by (3), and hence
n-yoo
(6) \F(f)\ < sup \F(8*f)\.
П Using (6), we get
•p(«:/) = Z F (Ku,>) = S ( K * F )(sP),
p e P n P e & n
and hence
|.F(Slft|<max||£;*F||- У Ifell = max ||В**Л|• | Ю Pe^n
< sup ||B” B||-Ii/I| = infill • ll/il.
p e #
In view of (6) we have thus shown that Ж infill.
Combined with (4) this proves
{7) ' 111*111 = P 4 I < oo, V F e J ( X ) * * .
Now suppose Ф = (Fn) e J (X**), i.e., |||Ф|Ц < oo. For each n e N let Фп = ( F1, . . . , F n, 0, ...) and define F n e J ( X ) * * by
П
F ( f ) = ^ F A f i ) , У/ = (/<) 6 J ( I ) | t=l
elearly, F 11 = Фп, and so \\Fn\\ = |||ФП||| < Ц|Ф||| by (7). Moreover, F n(f)
— F n(8*f), and if m < n, then
n
I 2 w t)I = i^ ‘(«„*/)--p“( O i <
г = т + 1
< 11|Ф|1НЙ/-£т/1К0 as m ,n-+oo by (3).
Therefore the formula
OO
F ( } ) = lim *"*(/) = f = (/«) e J ( X f ,
n-*0О t-= l
defines a linear functional F on J (X)*, which is continuous, because
|J?(/)K sup|i"(/)|< 111ФЦ1-11Я - П
Clearly Ф — F , and this completes the proof of the theorem.
Using (1) and (2 ) for X** instead of X , we now have COEOLLAEY 1. J ( X f * 9* J(X **) = J(X **)@ X ** J(X **).
Coeollaey 2. I f X is reflexive, then
J(X )** g * J { X ) & J ( X ) ; hence
J ( X ) * * / J ( X ) ъ X .
E e m a r k s. (1) If X Ф {0}, then the natural decomposition of J ( X ) is not boundedly complete (by an argument similar to that used for J , cf.
188 M. W o j t o w i c z
[3]), and hence J (X) is not reflexive. This conclusion is based on the result of Sanders [6 ] stating that a Banach space Z with a Schauder decomposi
tion (Zn) is reflexive iff the decomposition is shrinking and boundcdly complete and all Zn are reflexive.
(2) Corollary 2 for more general James-Orliez spaces has been recently announced by P.Y. Semenov [7]. (The results of the present paper were obtained independently.)
The author wishes to thank Professor L. Drewnowski for his valuable assistance and helpful suggestions.
References
[1] W. J . D a v is, T. F ig ie l, W. B. Jo h n s o n and A. P e lc z y h s k i, Factoring weakly compact operators, J . Funct. Anal. 17 (1974), 311-327.
[2] M. M. D a y , Normed linear spaces, 3rd ed., Springer-Verlag, Berlin-Heidelberg- New York 1973.
[3] R. C. J a m e s , Bases and reflexivity of Banach spaces, Ann. of Math. 52 (1950)*
518-527.
[4] J . L in d e n s tr a u s s , On James' paper “ Separable conjugate spaces”, Israel J . Math.
9 (1971), 279-284.
[5] J . L in d e n s tr a u s s and L. T z a f r ir i, Classical Banach spaces I , Springer-Verlag, Berlin-Heidelberg-New York 1977.
[6] B. L . S a n d e rs, Decompositions and reflexivity in Banach spaces, Math. Annalen 153 (1964), 199-209.
[7] P. V. S em en o v , James-Orlicz spaces, Uspehi Mat. Nauk 34 (4) (1974), 209-211 (in Russian).
IN S T I T U T E O F M A T H E M A T IC S A . M IC K IE W IC Z U N I V E R S I T Y P oznan
