Simultaneous strong proximinality in Banach spaces

Abstract

Several researchers have discussed the problem of strong proximinality in Banach spaces. In this paper, we generalize the notion of strong proximinality and define simultaneous strong proximinality. It is proved that if W is a simultaneously approximatively compact subset of a Banach space X then W is simultaneously strongly proximinal and the converse holds if the set of all best simultaneous approximations to every bounded subset S of X from W is compact. We show that simultaneously strongly Chebyshev sets are precisely the sets that are simultaneously strongly proximinal and simultaneously Chebyshev. It is also proved that if F and W are subspaces of a Banach space X such that F is simultaneously strongly proximinal, W is finite dimensional and F +W is closed then F +W is simultaneously strongly proximinal in X . How simultaneous strong proximinality is transmitted to and from quotient spaces is discussed in this paper.

On the sum of simultaneously proximinal sets

“…In this section, we consider the simultaneous proximinality of τ -closed subsets in a Banach space. It was shown that a reflexive subspace or a weakly compact subset of a Banach space X is simultaneously proximinal [11,16], and a finite dimensional subspace is simultaneously approximatively compact [7]. We will show that every τ -compact subsets are τ -simultaneously approximatively compact, and we shall characterize reflexive spaces from simultaneous proximinality point of view.…”

“…Gupta and Narang [7] showed that a closed subset C of X is simultaneously approximatively compact if and only if C is simultaneously strongly proximinal and P C (A) is compact for every bounded subset A of X. When C is weakly closed, we have the following result.…”

“…This can be regarded as a localized version as the above conclusions. Furthermore, Gupta and Narang [7] generalized Rawashdeh's result to the sum of simultaneously strongly proximinal subspaces. For the recent development of this topic, we refer to [14,15] and references therein.…”

“…The notions of simultaneous approximative compactness and simultaneous strong proximinality were introduced by Gupta and Narang [7]. We extend those as following.…”