Best approximation in polyhedral Banach spaces. (English) Zbl 1229.41027
J. Approx. Theory 163, No. 11, 1748-1771 (2011); corrigendum ibid. 182, 110-112 (2014).
The authors study conditions under which the metric projection of a polyhedral Banach space \(X\) onto a closed subspace \(Y\) is Hausdorff lower or upper semicontinuous.
The paper is organized as follows. Section 0 is an introduction. Section 1 contains notation concerning Banach spaces, followed by definitions and preliminary facts on polyhedral Banach spaces. Section 2 is a collection of preliminaries concerning metric projections onto closed subspaces. Sections 3 and 4 deal with (semi)continuity properties of the metric projection \(P_Y\) under polyhedrality-type assumptions on \(X\). Section 5 contains results on the proximinality of subspaces and polyhedrality of quotients. Sections 6, 7 and 8 contain one example each. These examples illustrate the importance of some of the hypotheses in the main results.
The paper is organized as follows. Section 0 is an introduction. Section 1 contains notation concerning Banach spaces, followed by definitions and preliminary facts on polyhedral Banach spaces. Section 2 is a collection of preliminaries concerning metric projections onto closed subspaces. Sections 3 and 4 deal with (semi)continuity properties of the metric projection \(P_Y\) under polyhedrality-type assumptions on \(X\). Section 5 contains results on the proximinality of subspaces and polyhedrality of quotients. Sections 6, 7 and 8 contain one example each. These examples illustrate the importance of some of the hypotheses in the main results.
Reviewer: Richard A. Zalik (Auburn)
MSC:
| 41A50 | Best approximation, Chebyshev systems |
References:
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