 Eberlein–Šmulian theorem

For a subset of a Banach space, relative weak compactness is equivalent to every sequence having a weakly convergent subsequence. In particular, weak compactness and weak sequential compactness coincide.

The Eberlein–Šmulian theorem is a result in functional analysis that characterizes weak*-compactness in the dual space of a Banach space. Specifically, it provides a criterion for when a subset of the dual space \( X^* \) (the space of continuous linear functionals on a Banach space \( X \)) is weak*-compact.