Quasi-reflexive spaces

Abstract

1. Introduction. For a Banach space X, let ir denote the canonical isomorphism of X into X**. An example was given by James [S] with X**/ir(X) finite dimensional. The example has the further property that X and X** are isomorphic. We consider the class of spaces which we call quasi-reflexive spaces of order n in which X**/ir(X) is (finite) w-dimensional. We characterize such spaces by the compactness in a suitable weak topology of the unit ball under an equivalent norm. Quasi-reflexive spaces are rath conjugate spaces for all integers m.We show that if E is a closed linear manifold in a space X, then X is quasi-reflexive of order n if and only if E and X/E are quasireflexive of orders m and p respectively and n = m+p. The case w = 0 would then be the corresponding result for reflexive spaces [6], It is further shown that the additivity of dimension holds in a natural interpretation even when X**/ir(X) is infinite dimensional. Any weakly complete quasi-reflexive space is reflexive. Every nonseparable quasi-reflesive space has a nonseparable reflexive subspace.

Operators on a Quasi-Reflexive Banach Space

“…Next we will show that 2Ii is a right annihilator algebra. By the proof of Theorem 3.1 [2] we see that there exists in isomorphism a of w2X2 onto V* such that a(w2x2)(v) =v(x2) for all x2EX2, vE V.…”

“…Finally we notice that if 9Ii is a left annihilator algebra it is an annihilator algebra and so are (33(F))* and 33(F), consequently by [l] F is a reflexive Banach space and so is V*. But F* is isomorphic with X2 by proof of Theorem 3.1 [2 ] ; so X2 is reflexive and by Lemma 3 we conclude that X is reflexive. If X is reflexive then U= V+ = (ir2X2) + = (0) which implies 33,= (0) for i = 1, 2 and 3.…”

Onw∗-sequential convergence and quasi-reflexivity

“…Proof. If X is quasi-reflexive of order <=n and 7 is a closed subspace of X, then Y is also quasi-reflexive of order <,n [1] and hence Y has property P n by Theorem 2. Conversely, if every normclosed separable subspace Y of X has property P n , then every such Y is quasireflexive of order <*n by Theorem 2, and hence X is quasireflexive of order ^n by a theorem of Singer [6].…”

“…If X has property P n and Si is a norm-closed subspace of X*, then S 1 e^' n (S 1 ) and hence C^iSJ = 1. If S is an arbitrary subspace of X* and S 2 the norm-closure of S, then Cχ ] (S) = C^Si) and therefore Q (%) (X) = 1.…”

Onk-shrinking andk-boundedly complete basic sequences and quasi-reflexive spaces

“…Civin and Yood have shown [2]: Proof. By [6, p. 546] We note that we have also shown that each Y k has a basis.…”