Subspaces of Asplund Banach spaces with the point continuity property
A space X is called somewhat quasireflexive if every non-reflexive subspace has a subspace which is quasireflexive of order one. X has the point continuity property (PC) if every weakly closed bounded subset of X has a point of weak to norm continuity. A Banach space X is an Asplund space if every separable subspace of X has a separable dual space. \textit{W. B. Johnson} and \textit{H. P. Rosenthal} [Studia Math. 43, 77-92 (1972; Zbl 0231.46035)] showed that if \(X^{**}\) is separable, then X is somewhat reflexive. \textit{G. A. Edgar} and \textit{R. F. Wheeler} [Pac. J. Math. 115, 317-350 (1984; Zbl 0506.46007)] extended the result of Johnson and Rosenthal to the case when X has (PC) and is an Asplund space (PCA). In this paper, the author proves that if X is a non-reflexive Banach space with separable dual and (PC) and it is Asplund, then X is somewhat quasireflexive. The main result of this paper has many implications, so that we think that the work is a very interesting one.
scientific article; zbMATH DE number 4050263
scientific article