Vol. 23, No. 1, 1967

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On w-sequential convergence and quasi-reflexivity

Ralph David McWilliams

Vol. 23 (1967), No. 1, 113–117

This paper characterizes quasi-reflexive Banach spaces in terms of certain properties of the w-sequential closure of subspaces. A real Banach space X is quasi-reflexive of order n, where n is a nonnegative integer, if and only if the canonical image JXX of X has algebraic codimension n in the second dual space X∗∗. The space X will be said to have property Pn if and only if every norm-closed subspace S of X has codimension n in its w-sequential closure KX(S). By use of a theorem of Singer it is proved that X is quasireflexive of order n if and only if every norm-closed separable subspace of X has property Pn.A certain parameter Q(n)(X) is shown to have value 1 if X has property Pn and to be infinite if X does not have Pn. The space X has P0 if and only if w-sequential convergence and w-sequential convergence coincide in x. These results generalize a theorem of Fleming, Retherford, and the author.

Mathematical Subject Classification
Primary: 46.10
Received: 15 June 1965
Published: 1 October 1967
Ralph David McWilliams