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.
