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Ball proximinality of equable spaces. (English) Zbl 1184.46015

Let \(X\) be a Banach space and let \(X_1\) denote its closed unit ball. For a closed subspace \(Y \subset X\), it is easy to see that if \(Y_1\) is a proximinal subset of \(X\), then \(Y\) is a proximinal subspace. Motivated by an example, due to F. B. Saidi [Proc. Am. Math. Soc. 133, No. 9, 2697–2703 (2005; Zbl 1071.41033)], of a Banach space \(X\) and a closed proximinal subspace \(Y\subset X\) such that \(Y_1\) is not a proximinal set, the reviewer and his collaborators have initiated the study of ball proximinal subspaces, i.e., subspaces \(Y\) for which \(Y_1\) is a proximinal set.
In this interesting paper, using the notion of equability, due to D. Yost, the author shows that any equable subspace is strongly ball proximinal and the metric projection is Hausdorff metric continuous (H.m.c). She then applies these ideas to the space of continuous functions valued in a uniformly convex Banach space \(X\) to conclude that \(C(Q,X)\) is strongly ball proximinal in its bidual and the metric projection map is H.m.c. This substantially improves on some earlier results of the reviewer.
The paper also contains an example, due to G. Godefroy, of a Banach space \(X\) for which \(X_1\) is not a strong proximinal set. The geometry of spaces where this can happen is waiting to be discovered.
Reviewer’s remark: The reference to article [2] on page 79 is incorrect.

MSC:

46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Citations:

Zbl 1071.41033
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References:

[1] Pr. Bandyopadhyay, B.L. Lin, and T.S.S.R.K. Rao, Ball proximinality in Banach spaces,Banach spaces and their applications in analysis, 251–264, Conference in Honor of Nigel Kalton’s 60th Birthday, Berlin, 2007. · Zbl 1135.41011
[2] J. Blatter,Grothendieck Spaces in Approximation Theory, American Mathematical Society120, Providence, 1972. · Zbl 0236.46027
[3] G. Godefroy and V. Indumathi, Strong proximinality and polyhedral spaces,Rev. Mat. Complut. 14 (2001), 105–125. · Zbl 0993.46004
[4] P. Harmand, D. Werner, and W. Werner,M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics,1574, SpringerVerlag, Berlin, 1993. · Zbl 0789.46011
[5] V. Indumathi, S. Lalithambigai, and B.L. Lin, Ball proximinality of closed * subalgebras inC(Q), Extracta Math. 22 (2007), 1–17. · Zbl 1161.46004
[6] K.S. Lau, Approximation by continuous vectorvalued functions,Studia Math. 68 (1980), 291–298 · Zbl 0455.41015
[7] J. Mach, Best simultaneous approximation of bounded functions with values in certain Banach spaces,Math. Ann. 240 (1979), 157–164. · Zbl 0388.41014 · doi:10.1007/BF01364630
[8] C. Olech, Approximation of setvalued functions by continuous functions,Colloq. Math. 19 (1968), 285–293. · Zbl 0183.13603
[9] T.S.S.R.K. Rao, Some remarks on proximinality of higher dual spaces,J. Math. Anal. Appl. 328 (2007), 1173–1177. · Zbl 1122.46006 · doi:10.1016/j.jmaa.2006.06.026
[10] F.B. Saidi, On the proximinality of the unit ball of proximinal subspaces in Banach spaces: a counterexample,Proc. Amer. Math. Soc. 133 (2005), 2697–2703. · Zbl 1071.41033 · doi:10.1090/S0002-9939-05-08152-9
[11] D. Yost,Intersecting balls and proximinal subspaces in Banach spaces, Ph.D. thesis, University of Eidenburgh, 1979. · Zbl 0407.41015
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