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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)
##### Citations:
Zbl 0993.46004; Zbl 0978.46004
Full Text:
##### References:
 [1] Bandyopadhyay, Pradipta; Huang, Da; Lin, Bor-Luh, Rotund points, nested sequence of balls and smoothness in Banach spaces, Comment. math. prace mat., 44, 163-186, (2004) · Zbl 1097.46009 [2] Bandyopadhyay, P.; Huang, Da; Lin, Bor-Luh; Troyanski, S.L., Some generalizations of locally uniform rotundity, J. math. anal. appl., 252, 906-916, (2000) · Zbl 0978.46004 [3] Brown, A.L., A rotund reflexive space having a subspace of codimension two with a discontinuous metric projection, Michigan math. J., 21, 145-151, (1974) · Zbl 0275.46016 [4] Cobzaş, Stefan, Geometric properties of Banach spaces and the existence of nearest and farthest points, Abstr. appl. anal., 2005, 259-285, (2005) · Zbl 1100.46005 [5] Deutsch, F., Existence of best approximations, J. approx. theory, 28, 132-154, (1980) · Zbl 0464.41016 [6] Efimov, N.V.; Stechkin, S.B., Approximative compactness and Chebyshev sets, Soviet math. dokl., 2, 1226-1228, (1961) · Zbl 0103.08101 [7] Fan, Ky; Glicksberg, Irving, Some geometric properties of the spheres in a normed linear space, Duke math. J., 25, 553-568, (1958) · Zbl 0084.33101 [8] Godefroy, G.; Indumathi, V., Strong proximinality and polyhedral spaces, Rev. mat. complut., 14, 105-125, (2001) · Zbl 0993.46004 [9] Godefroy, G.; Indumathi, V.; Lust-Piquard, F., Strong subdifferentiability of convex functionals and proximinality, J. approx. theory, 116, 397-415, (2002) · Zbl 1022.49017 [10] Greim, Peter, Strongly exposed points in Bochner $$L^p$$-spaces, Proc. amer. math. soc., 88, 81-84, (1983) · Zbl 0524.46021 [11] Harmand, P.; Werner, D.; Werner, W., M-ideals in Banach spaces and Banach algebras, Lecture notes in math., vol. 1547, (1993), Springer-Verlag Berlin · Zbl 0789.46011 [12] Hu, Zhibao; Lin, Bor-Luh, Strongly exposed points in lebesgue – bochner function spaces, Proc. amer. math. soc., 120, 1159-1165, (1994) · Zbl 0820.46010 [13] Kottman, Clifford A.; Lin, Bor-Luh, The weak continuity of metric projections, Michigan math. J., 17, 401-404, (1970) · Zbl 0206.43803 [14] Narayana, Darapaneni; Rao, T.S.S.R.K., Transitivity of proximinality and norm attaining functionals, Colloq. math., 104, 1-19, (2006) · Zbl 1085.41027 [15] Singer, Ivan, Best approximations in normed linear spaces by elements of linear subspaces, Grundlehren math. wiss., vol. 171, (1970), Springer-Verlag New York · Zbl 0197.38601
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