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Proximinality in Banach spaces. (English) Zbl 1138.46008
Let \(X\) be a Banach space and let \(\tau\) be the norm or the weak topology of \(X.\) The paper is concerned with various \(\tau\)-notions related to the approximation properties of \(\tau\)-closed subsets of \(X\). For instance, a \(\tau\)-closed subset \(K\) of \(X\) is called \(\tau\)-strongly proximinal if, for every \(\tau\)-neighborhood \(V\) of \(0\in X\), there exists \(\delta>0\) such that \(P_K(x,\delta)\subset P_K(x)+V\), where \(P_K(x)\) is the metric projection of \(x\) on \(K\) and \(P_K(x,\delta)=\{z\in K:\| x-z\|\leq d(x,K)+\delta\}\). The set \(K\) is called \(\tau\)-strongly Chebyshev (approximatively \(\tau\)-compact) if, for any \(x\in X\setminus K\), every minimizing sequence for \(d(x,K)\) is \(\tau\)-convergent (contains a \(\tau\)-convergent subsequence). Strongly proximinal subspaces, in the case when \(\tau\) is the norm-topology, were defined and studied by G. Godefroy and V. Indumathi [Rev. Mat. Complut. 14, No. 1, 105–125 (2001; Zbl 0993.46004)].
The authors study the relations of these notions with the geometric properties of the Banach space \(X\) (as, for instance, the \(\tau\)-almost local rotundity, a notion considered by P. Bandyopadhyay, D. Huang, B.–L. Lin and S. L. Troyanski [J. Math. Anal. Appl. 252, No. 2, 906–916 (2000; Zbl 0978.46004)]) and with various continuity properties of the metric projection. For instance, for \(x^*\in S_{X^*},\) the subspace \(\ker x^*\) is approximatively \(\tau\)-compact (respectively, \(\tau\)-strongly Chebyshev) iff \(x^*\) is a \(\tau\)-strongly support (respectively, a \(\tau\)-strongly exposing) functional of the unit ball of \(X\) (Theorem 2.10). The paper ends with some stability results for \(\ell^p\)-sums of Banach spaces.

MSC:
46B20 Geometry and structure of normed linear spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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