Abstract
This paper deals with Quasi-Chebyshevity in the Bochner function spaces , where X is a Banach space. For W a nonempty closed subset of X and x ∊ X, an element w0 in W is called to x from W, if , for all w in W. All best approximation points of x from W form a set usually denoted by PW (x). The set W is called proximinal in X if PW (x) is non empty, for each x in X. Now, W is said to be in X whenever, for each x in X, the set PW (x) is nonempty and compact in X. This subject was studied in general Banach spaces by several authors and some results had been obtained. In this work, we study Quasi-Chebyshevity in the Bochner Lp- spaces. The main result in this paper is that: given W a Quasi-Chebyshev subspace in X then Lp(μ, W) is Quasi-Chebyshev in , if and only if L1 (μ, W) is Quasi-Chebyshev in L1(μ, X). As a consequence, one gets that if W is reflexive in X such that X satisfies the sequential KK-property then Lp(μ, W) is Quasi-Chebyshev in .
Highlights
Let X be a Banach space and W nonempty subset in X
Keywords Quasi-Chebyshev Set, Proximinal Set, Compact Set, Reflexive Subspace, Sequential KK-property approximations to x ∈ X has the notation: PW(x)
Let x an element in X, a best approximation point to x from W say w0∈W is that satisfies the following:
Summary
Let X be a Banach space and W nonempty subset in X. Keywords Quasi-Chebyshev Set, Proximinal Set, Compact Set, Reflexive Subspace, Sequential KK-property approximations to x ∈ X has the notation: PW(x). The concept of Quasi-Chebyshevity can be considered as compact proximinality. A proximinal subset W of X is Quasi-Chebyshev in X if PW(x) (which is nonempty) is a compact set, See [1].
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