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Characterizations of simultaneous farthest point in normed linear spaces with applications. (English) Zbl 1154.90620

Summary: In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space \(X\). We study simultaneous farthest point in \(X\) by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball \({B_{X^{*}}}\) of \(X^{*}\), where \(X^{*}\) is the dual space of \(X\). We establish the characterizations of simultaneous farthest points for bounded sets in \({C_{\mathbb{R}}(Q)}\), the space of all real-valued continuous functions on a compact topological space \(Q\) endowed with the usual operations and with the norm \({\| x \|=\max_{q\in Q}\mid x(q) \mid}\). It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Bucharest Academy of the Socialist Republic of Romania; Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0197.38601), Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in \(X\) which specifically concentrates on the Hahn-Banach Theorem in \(X\).

MSC:

90C48 Programming in abstract spaces

Citations:

Zbl 0197.38601
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References:

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