Journal of Convex Analysis 24 (2017), No. 3, 955--958
Copyright Heldermann Verlag 2017
Coproximinality in Spaces of Bochner Integrable Functions
T. S. S. R. K. Rao
Theoretical Statistics and Mathematics Division, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India
tss@isibang.ac.in
[Abstract-pdf]
In this short note we use a new way of applying von Neumann's selection theorem for obtaining best coapproximation in spaces of measurable functions. For a coproximinal closed subspace $Y$ of a Banach space $X$, we show that if $Y$ is constrained in a weakly compactly generated dual space, then the space $L^1(\mu,Y)$ of $Y$-valued Bochner integrable functions is coproximinal in $L^1(\mu,X)$. This extends a result of M. R. Haddadi, N. Hejazjpoor and H. Mazaheri [{\it Some results about best coapproximation in $L^P(S,X)$}, Anal. Theory Appl. 26 (2010) 69--75], proved when $Y$ is reflexive.
Keywords: Coproximinality, constrained subspaces, weakly compactly generated spaces, spaces of Bochner integrable functions, von Neumann's selection theorem.
MSC: 41A50; 46B20, 46E40
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