In this paper, we characterize classes of matrix transformations from BK spaces into spaces of bounded sequences and their
subclasses of infinite matrices that define compact operators. Furthermore, using these results and the solvability of certain
infinite linear systems we give necessary and sufficient conditions for A to be a compact operator on spaces that are strongly α-bounded or summable.
In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.
We define two properties of sequences in Banach spaces that may be related to measures of noncompactness of subsets of these
spaces. The first one concerns properties of sequences related to the strong topology, and the second one is related to the
weak topology. Given a Banach space X, we introduce a new Banach space such that we can find a subset E in it that may be identified with the balls in the first one. We use compactness in this new space to characterize our sequential
properties. In particular, we prove a general form of the Eberlein-Smulian theorem.