Antiproximinal Sets in Banach Spaces of Continuous Vector-Valued Functions

Abstract

A closed nonvoid subset Z of a Banach space X is called antiproximinal if no point outside Z has a nearest point in Z. The aim of the present paper is to prove that, for a compact Hausdorff space T and a real Banach space E, the Banach space C T E , of all continuous functions defined on T and with values in E, contains an antiproximinal bounded closed convex body. This extends a result proved by V. S. Balaganskii (1996, Mat. Zametki 60, 643-657) in the case E = .  2001 Academic Press

Antiproximinal Norms in Banach Spaces

“…=sup {||x i ||: i ¥ I}= max {||x(i)||: i ¥ I}. Cobzas gave a proof in [6] of the existence of an isomorphic companion norm to the supremum norm || · || . in (; i X i ) c 0 , whenever X i =X, for every i ¥ I.…”

On Quasi-Chebyshevity Subsets of Unital Banach Algebras

“…A problem which has been intensively studied is to check whether a Banach space X does or does not contain bounded closed non-proximinal sets. The results in general Banach spaces can be found in [1,5,6]. A subset W of a Banach space X is called quasi-Chebyshev if P W (x) is a non-empty and compact set in X for every x ∈ X (see [10]).…”