Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

HOME ALL ARTICLES View

J. Korean Math. Soc. 1995; 32(3): 415-425

Printed September 1, 1995

Copyright © The Korean Mathematical Society.

A study about the converse of R.B.Holmes' theorem

Sung Ho Park and Hyang Joo Rhee

Sogang University and Sogang University

Abstract

R.B.Holmes proved that in a uniformly convex space $X$, for any $\varepsilon > 0$ there exists $\delta (\varepsilon ) > 0 $ such that for any $ x, y $ in the unit ball of $X$ with $ \|x-y\| < \delta $ imply that $$ \| P_M(x) - P_M(y) \| < \varepsilon $$ for every proximinal subspace $M$ of $X$. In $[2]$, F.R.Deutsch gave the open problem which is the converse of the R.B.Holmes' Theorem. In this article, we will give an example which the answer of the open problem is no.

Keywords: Best appproximation, proximinal, uniformly convex, metric projection, uniformly equicontinuous

MSC numbers: 41A65, 46B20