• T S S R K Rao

      Articles written in Proceedings – Mathematical Sciences

    • L1 (μ,X) as a complemented subspace of its bidual

      T S S R K Rao

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      We show that for a Banach spaceX, if the space ofX-valued Bochner integrable functions is complemented in some dual space, then it is complemented in the space ofX-valued countably additive, μ-continuous vector measures.

    • Extremely strict ideals in Banach spaces

      T S S R K RAO

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      Motivated by the notion of an ideal introduced by Godefroy {\it et al.} ({\it Studia Math.} {\bf 104} (1993) 13–59), in this article, we introduce and study the notion of an extremely strict ideal. For a Poulsen simplex $K$, we show that the space of affine continuous functions on $K$ is an extremely strict ideal in the space of continuous functions on $K$. For injective tensor product spaces, we prove a cancelation theorem for extremely strict ideals. We also exhibit non-reflexive Banach spaces which are not strict ideals in their fourth dual.

    • Two results on strong proximinality

      T S S R K Rao

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      Let $Y \subseteq X$ be a closed subspace. By a simple argument, we show that $Y^{\bot\bot} X^{**}$ is strongly proximinal at points of $X$ if and only if $Y$ is a strongly proximinal subspace of $X$. This substantially improves the main result of Jayanarayanan and Paul (J. Math. Anal. Appl. 426 (2015) 1217--1231). As a consequence we get an easy proof of a classical result of Alfsen and Effros (Ann. Math. 98 (1972) 98--173), that $M$-ideals are proximinal subspaces and a result of Dutta and Narayana (Function Spaces,Contemporary Mathematics, vol. 435, American Mathematical Society, Providence(2007) pp. 143--152), that $M$-ideals are strongly proximinal subspaces.

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