Weakly compact set

A subset of a Banach space is weakly compact when it is compact in the weak topology; it is relatively weakly compact when its weak closure is weakly compact.

A set is weakly sequentially compact when every sequence in it has a subsequence that converges weakly to a point of the set.

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 closed convex hull of a weakly compact subset of a Banach space is weakly compact.