The Eberlein-Smulian Theorem states that a subset of a Banach space is weakly compact if and only if it is sequentially weakly compact, meaning that every sequence in the set has a subsequence that converges weakly to a limit within the set. This theorem provides a crucial connection between weak and weak* convergence in the context of functional analysis, particularly in studying compactness properties of subsets of Banach spaces.
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