Eberlein–Šmulian theorem

In the mathematical field of functional analysis, the Eberlein–Šmulian theorem is a result relating three different kinds of weak compactness in a Banach space. The three kinds of compactness for a subset "A" of a topological space are:
* Compactness (or Lindelöf compactness): Every open cover of "A" admits a finite subcover.
* Sequential compactness: Every sequence from "A" has a convergent subsequence whose limit is in "A".
* Limit point compactness: Every infinite subset of "A" has a limit point in "A".The Eberlein–Šmulian theorem states that the following conditions on a subset "A" of a Banach space "X" are equivalent:
* "A" is weakly compact.
* "A" is weakly sequentially compact.
* "A" is weakly limit point compact.These properties hold for subsets of a metric space; however the weak topology is not metrizable unless the space "X" is finite dimensional. Thus the Eberlein–Šmulian theorem asserts a certain property on the (non-metrizable) weak topology on a Banach which is usually reserved for metric spaces.

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