J. Korean Math. Soc. 1995; 32(3): 415-425
Printed September 1, 1995
Copyright © The Korean Mathematical Society.
Sung Ho Park and Hyang Joo Rhee
Sogang University and Sogang University
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
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