No 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.

No abstract

No abstract

No abstract