R.B.Holmes proved that in a uniformly convex space $X$, for any $arepsilon > 0$ there exists $delta (arepsilon ) > 0 $ such that for any $ x, y $ in the unit ball of $X$ with $ |x-y| < delta $ imply that $$ | P_M(x) - P_M(y) | < arepsilon $$ 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.
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机译:RBHolmes证明,在均匀凸空间$ X $中,对于任何$ varepsilon> 0 $,都存在$ delta( varepsilon)> 0 $,从而对于$ X $的单位球中的任何$ x,y $用$ | xy | < delta $表示$$ | P_M(x)-P_M(y) | < varepsilon $$ for $ X $的每个近邻子空间$ M $。在$ [2] $中,F.R。Deutsch提出了开放性问题,这是R.B. Holmes定理的反面。在本文中,我们将给出一个示例,说明开放问题的答案是否定的。
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