# Semicontinuity of metric projections in \infty -direct sums.

Lalithambigai, S and Narayana, Darapaneni (2006) Semicontinuity of metric projections in \infty -direct sums. In: Taiwanese Journal of Mathematics, 10 (5). pp. 1245-1259.

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## Abstract

Let {Xi : i ∈ N} be a family of Banach space and let Yi ⊆ Xi be a closed subspace in Xi for each i ∈ N such that at least two Y i s are non-trivial. Consider X = (⊕c0Xi)i∈N and Y = (⊕c0Yi)i∈N. We show that Y is strongly proximinal in X if and only if PY is upper Hausdorff semi-continuous on X if and only if Yi is strongly proximinal subspace in Xi for each i ∈ N. This shows that in [9, Theorem 3.4], strong proximinality of Yi’s is a necessary assumption. We also show that lower semi-continuity of metric projections is stable in c0-direct sums.

Item Type: Journal Article The copyright belongs to National Tsing Hua University. Proximinal;Metric projection;Lower Semi-Continuity;Upper Hausdorff Semi-Continuity Division of Physical & Mathematical Sciences > Mathematics Mr. Ramesh Chander 12 Nov 2007 19 Sep 2010 04:41 http://eprints.iisc.ac.in/id/eprint/12439