On best simultaneous approximation in quotient spaces

Abstract

We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W + M and the quotient space W /M.

Simultaneous strong proximinality in Banach spaces

“…The results proved in this section are motivated by the corresponding results proved for proximinality in [3], simultaneous proximinality in [8,11], and strong proximinality in [13].…”

“…In this section, we discuss simultaneous strong proximinality for the sum and quotient subspaces of X and see how simultaneous strong proximinaliy is transmitted to and from quotient spaces and to sum of subspaces. The results proved in this section are motivated by the corresponding results proved for proximinality in [3], simultaneous proximinality in [8,11], and strong proximinality in [13].…”

“…A closed subset W of a Banach space X is called simultaneously quasi-Chebyshev (see [8]) if L W (S) is nonempty and compact for every bounded subset S of X . Therefore, we have Corollary 2.5 Let W be a closed subset of a Banach space (X, ∥.∥); then W is simultaneously approximatively compact if and only if W is simultaneously strongly proximinal and simultaneously quasi-Chebyshev.…”

“…We require the following lemma in the proof of the next theorem, which shows that if W + F is simultaneously strongly proximinal then so is its quotient space. Lemma 3.5 [8] Let W be a proximinal subspace of a Banach space X ; then for any bounded subset S of X we have d(S, W ) = sup s∈S inf w∈W ∥s − w∥.…”

On Simultaneous Weakly-Chebyshev Subspaces

“…Throughout this paper, we suppose that S is either a bounded or a finite set. Also, we shall use the following Lemma in the sequel which has been proved in [1] and [5]. …”