# Eberlein–Šmulian theorem

In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in a Banach space.

## Statement of the theorem

### Types of weak compactness

A set A can be weakly compact in three different ways:

### The Eberlein–Šmulian theorem

The Eberlein–Šmulian theorem states that the three are equivalent on 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[clarification needed] that bounded subsets are weakly precompact, and therefore have limits in the weak sense[clarification needed]. Since most 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.