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manuscripta mathematica

, Volume 7, Issue 4, pp 375–386 | Cite as

Zur Stetigkeit der mengenwertigen metrischen Projektion in endlichdimensionalen Räumen

  • Rudolf Wegmann
Article
  • 16 Downloads

Abstract

Let X be a real normed linear space, X* its dual, V a linear subspace of X and S(V) the unit sphere in the orthogonal space\(v^ \bot : = \{ x* \in \chi * :x*(v) = O\forall v \in V\} .\) In this note we prove in the case of finite-dimensional X the following sufficient condition for the continuity of the set-valued metric projection\(P_V (x): = \{ v_O \in V:\parallel x - v_O \parallel \leqslant \parallel x - v\parallel \forall v \in V\} \) in terms of the mapping\(T(\begin{array}{*{20}c} o \\ x \\ \end{array} *): = \{ x \in ^o x:\parallel x\parallel \leqslant 1\) and\(x*(x) = \parallel x*\parallel \} \): If the restriction of T to S(V) is lower semi-continuous then PV is lower semi-continuous.

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Literatur

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    BERGE, C.: Espaces topologiques, fonctions multivoques. 2nd ed. Paris, Dunod 1966.Google Scholar
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    SINGER, I.: Best approximation in normed linear spaces by elements of linear subspaces. Berlin-Heidelberg-New York, Springer 1970.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • Rudolf Wegmann
    • 1
  1. 1.Max-Planck-Institut für Physik und AstrophysikMünchen 23

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