T. S. S. R. K. Rao, Weak coproximinality for Banach spaces, Vol. 2017 (2017), Article ID 13, pp. 1-10

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DOI: 10.23952/jnfa.2017.13

Received June 19, 2016; Accepted December 28, 2016; Published January 5, 2017

Abstract. In this paper, we introduce a weaker form of the classical notion of coproximinality for Banach spaces. This is intended as a new tool to make the metric projection map linear. We show that if a weakly coproximinal subspace $Y \subset X$ is a semi- $M$ -ideal in $X$ , then the associated metric projection map is linear and $Y$ is a $M$ -ideal in $X$ . This is also linked to the classical problem of identifying Banach spaces as a quotient space $X/Y$ , where $Y$ has certain non-linear geometric properties in $X$ . We give a counterexample to the 3-space problem for weak coproximinality. We also study its stability properties for spaces of vector-valued continuous functions.

How to Cite this Article:

T. S. S. R. K. Rao, Weak coproximinality for Banach spaces, Journal of Nonlinear Functional Analysis, Vol. 2017 (2017), Article ID 13, pp. 1-10.