Dual Kadec-Klee norms and the relationships between Wijsman, slice, and Mosco convergence

Title: Dual Kadec-Klee norms and the relationships between Wijsman, slice, and Mosco convergence
Creator: Borwein, Jon; Vanderwerff, Jon
Relation: The Michigan Mathematical Journal Vol. 41, Issue 2, p. 371-387
Publisher Link: http://dx.doi.org/10.1307/mmj/1029005003
Publisher: University of Michigan. Department of Mathematics
Resource Type: journal article
Date: 1994
Description: In this paper we study the relationships between the three most fundamental forms of set convergence. In particular, it is shown that Wijsman and slice convergence coincide precisely when the weak-star and norm topologies agree on the dual sphere. Consequently, a weakly compactly generated Banach space admits a dense set of norms for which Wijsman and slice convergence coincide if and only if it is an Asplund space. We also show that Wijsman convergence implies Mosco convergence precisely when the weak-star and Mackey topologies coincide on the dual sphere. A corollary of these results is that, given a fixed norm on an Asplund space, Wijsman and slice convergence coincide if and only if Wijsman convergence implies Mosco convergence.
Subject: Kadec-Klee norms; Banach spaces; Wijsman convergence; Mosco convergence; Asplund space
Identifier: http://hdl.handle.net/1959.13/1041205
Identifier: uon:13882
Identifier: ISSN:0026-2285
Language: eng
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