Eberlein–Šmulian theorem


In the mathematical field of functional analysis, the Eberlein–Šmulian theorem is a result that relates three different kinds of weak compactness in a Banach space.

Statement

Eberlein–Šmulian theorem: If X is a Banach space and A is a subset of X, then the following statements are equivalent:

  1. each sequence of elements of A has a subsequence that is weakly convergent;
  2. each sequence of elements of A has a weak cluster point;
  3. the weak closure of A is weakly compact.
A set A can be weakly compact in three different ways:
The Eberlein–Šmulian theorem states that the three are equivalent on a weak topology of a Banach space.
While this equivalence is true in general for a metric space, the weak topology is not metrizable in infinite dimensional vector spaces, and so the Eberlein–Šmulian theorem is needed.

Applications

The Eberlein–Šmulian theorem is important in the theory of PDEs, and particularly in Sobolev spaces. Many Sobolev spaces are reflexive Banach spaces and therefore bounded subsets are weakly precompact by Alaoglu's theorem. Thus the theorem implies that bounded subsets are weakly sequentially precompact, and therefore from every bounded sequence of elements of that space it is possible to extract a subsequence which is weakly converging in the space. Since many PDEs only have solutions in the weak sense, this theorem is an important step in deciding which spaces of weak solutions to use in solving a PDE.