The Problems of Best Approximation in $\beta$-Normed Spaces~($0<\beta<1$)
Received:April 21, 2006  Revised:August 28, 2006
Key Words: locally $\beta$-convex space   $\beta$-normed space   normed conjugate cone   the best approximation.  
Fund Project:the Foundation of the Education Department of Jiangsu Province (No.05KJB110001).
Author NameAffiliation
WANG Jian Yong Department of Mathematics, Changshu Institute of Technology, Jiangsu 215500, China 
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Abstract:
      This paper deals with the problems of best approximation in $\beta$-normed spaces. With the tool of conjugate cone introduced in [1] and via the Hahn-Banach extension theorem of $\beta$-subseminorm in [2], the characteristics that an element in a closed subspace is the best approximation are given in Section 2. It is obtained in Section 3 that all convex sets or subspaces of a $\beta$-normed space are semi-Chebyshev if and only if the space is itself strictly convex. The fact that every finite dimensional subspace of a strictly convex $\beta$-normed space must be Chebyshev is proved at last.
Citation:
DOI:10.3770/j.issn:1000-341X.2008.02.012
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