Coreflexive and somewhat reflexive Banach spaces

Abstract

A coreflexive Banach space is shown to have many of the same properties as a quasi-reflexive space. An infinite dimensional reflexive subspace of a Banach space with boundedly complete basis and separable dual is constructed, and it is noted that somewhat reflexive Banach spaces need not be coreflexive. 0. Introduction. Let X be a Banach space and let X* and X** denote the first and second conjugate spaces of X. If Q denotes the canonical map of X into X* *, then [2] X is quasi-reflexive of order n if dim X**/Q(X)=n. We say that X is coreflexive if X**/Q(X) is reflexive and that X is complemented coreflexive if Q(X) is complemented by a reflexive subspace of X*8. A somewi1hat reflexive Banach space [7] is a Banach space in which each infinite dimensional closed subspace contains an infinite dimensional reflexive subspace. Herman and Whitley [7] give an example of a somewhat reflexive space which is coreflexive. In this paper we investigate the conjecture that a necessary and sufficient condition for somewhat reflexivity is coreflexivity (complemented coreflexivity). In 62 we develop some properties of coreflexive spaces, many of which hold for quasi-reflexive spaces. In ?3 we give a proof that every Banach space with boundedly complete basis and separable dual contains an infinite dimensional reflexive subspace. This theorem is a consequence of a theorem in [9], however our proof involves an interesting constructive process so we include it here. Using results of [9] we are able to prove that if X is a Banach space such that X**/Q(X) is separable, then Xand X* are somewhat reflexive. Finally we note that somewhat reflexive spaces need not be coreflexive and pose some unanswered questions. Presented to the Society, November 28, 1970 under the title An example concerning somewhat reflexivity; received by the editors October 26, 1971. AMS (MOS) subject class/ilcations (1970). Primary 46B99; Secondary 46B05, 46B10, 46B15.