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Existence Of Nearest Points In Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Jonathan M. Borwein  and
Simon Fitzpatrick
Jonathan M. Borwein
Affiliation:
Dalhousie University, Halifax, Nova Scotia
Simon Fitzpatrick
Affiliation:
University of Auckland, Auckland, New Zealand
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 4 , 01 August 1989 , pp. 702 - 720
Copyright
Copyright © Canadian Mathematical Society 1989

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