ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X III (1983) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X X III (1983)
Marek W ôjtowicz (Poznan)
On the James spaceJ(X ) for a Banach space X
Abstract. For X an arbitrary Banach space, we consider the natural analogue J (X) of the famous James spaceJ — J (В ). Our main result is that J (X)** is isomor phic to J(X**)@X**. In particular, if X is reflexive, thenJ (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. Wôjtowicz
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