Existence of nearest points in Banach spaces

Title: Existence of nearest points in Banach spaces
Creator: Borwein, Jonathan M.; Fitzpatrick, Simon
Relation: Canadian Journal of Mathematics Vol. 61, Issue 4, p. 702-720
Publisher Link: http://dx.doi.org/10.4153/CJM-1989-032-7
Publisher: University of Toronto Press
Resource Type: journal article
Date: 1989
Description: This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.
Subject: Banach spaces; Kadec spaces; subderivatives
Identifier: http://hdl.handle.net/1959.13/941078
Identifier: uon:13175
Identifier: ISSN:0008-414X
Language: eng
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