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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.