Eberlein–Šmulian theorem

In a Banach space, a set is weakly compact if and only if every sequence in it has a subsequence converging weakly to a point in the set.

Weak topology: The coarsest topology on a Banach space making every continuous linear functional continuous. Compactness in this topology is the theorem’s topological condition.
Banach–Alaoglu theorem: The closed unit ball of a dual space is compact in the weak-star topology. Its compactness supplies a central tool in proofs using the dual space.
Robert Eberlein: An American mathematician who worked in functional analysis and proved a sequential criterion for weak compactness. His 1947 work established one half of the theorem’s history.
Reflexive Banach space: A Banach space whose canonical map onto its double dual is surjective. Combined with Banach–Alaoglu, the theorem yields weak sequential compactness of bounded sequences.
Weak convergence: A net or sequence converges weakly when every continuous linear functional converges on it. The theorem tests sets using subsequences that converge in this sense.
Krein–Šmulian theorem: A theorem characterizing weak-star closed convex sets in a dual Banach space by their bounded slices. It helps turn bounded weak-star arguments into global closure statements.
Witold Śmulian: A Polish mathematician whose work established a sequential criterion for weak compactness. His independent result supplied the other name attached to the theorem.
Weak sequential completeness: The property that every weakly Cauchy sequence converges weakly in the space. Sequential compactness arguments often distinguish this property from weak compactness.
Compact space: A topological space in which every open cover has a finite subcover. Weak compactness is ordinary compactness applied to the weak topology.
Eberlein compactum: A compact space homeomorphic to a weakly compact subset of a Banach space. The theorem gives such spaces a sequential characterization.

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