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AI InsightThe object of this paper is to give sufficient conditions for the proximinality of L1(I,H)+L1(I,G) in L1(I,X), where H and G are two proximinal subspaces of X which include as a special case the proximality of L1(I) \stackrel{\wedge}{\otimes} G + H \stackrel{\wedge}{\otimes} L1(I) in L1(I \times I).
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Özet
Let X be a Banach space and let (I,W ,m) be a measure space. For 1 \leq p < \infty, let Lp(I,X) denote the space of Bochner p-integrable functions defined on I with values in X. The object of this paper is to give sufficient conditions for the proximinality of L1(I,H)+L1(I,G) in L1(I,X), where H and G are two proximinal subspaces of X which include as a special case the proximality of L1(I) \stackrel{\wedge}{\otimes} G + H \stackrel{\wedge}{\otimes} L1(I) in L1(I \times I).
Abstract
Let X be a Banach space and let (I,W ,m) be a measure space. For 1 \leq p < \infty, let Lp(I,X) denote the space of Bochner p-integrable functions defined on I with values in X. The object of this paper is to give sufficient conditions for the proximinality of L1(I,H)+L1(I,G) in L1(I,X), where H and G are two proximinal subspaces of X which include as a special case the proximality of L1(I) \stackrel{\wedge}{\otimes} G + H \stackrel{\wedge}{\otimes} L1(I) in L1(I \times I).