Best approximation in polyhedral Banach spaces |
| |
Authors: | Vladimir P Fonf Joram Lindenstrauss Libor Veselý |
| |
Institution: | aDepartment of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel;bEinstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel;cDipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy |
| |
Abstract: | In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (∗) (a geometric property stronger than polyhedrality) and Y⊂X is any proximinal subspace, then the metric projection PY is Hausdorff continuous and Y is strongly proximinal (i.e., if {yn}⊂Y, x∈X and , then ).One of the main results of a different nature is the following: if X satisfies (∗) and Y⊂X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y⊥ attains its norm. Moreover, in this case the quotient X/Y is polyhedral.The final part of the paper contains examples illustrating the importance of some hypotheses in our main results. |
| |
Keywords: | Polyhedral Banach space Metric projection Proximinal subspace |
本文献已被 ScienceDirect 等数据库收录! |
|