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A note on proximinality in C(S × T) with the L1-norm

Published online by Cambridge University Press:  20 January 2009

W. A. Light
W. A. Light
Affiliation:
Department of Mathematics, University of Lancaster, Lancaster, LA1 4YL
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Let S and T be compact Hausdorff spaces and G and H finite-dimensional subspaces of C(S) and C(T) respectively. Suppose μ and ν are regular Borel measures on S and T respectively such that μ(S)= ν(T)= 1. The product measure μ × ν will be denoted by σ. Set U = G⊗C(T), V =C(S)⊗H and W = U + V. If G and H possess continuous proximity maps, then U and V are proximinal subspaces of C(S × T) when this linear space is equipped with the L1-norm, [4, Lemma 2]. That is, every z∈C(S × T) possesses at least one best approximation from U and from V. A metric selection Au:C(S × T →U is a mapping which associates each z ∈ C(S × T) with one of its best approximations in U.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 3 , October 1986 , pp. 335 - 341
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

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