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:

• 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

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 Kakutani's theorem. Thus the theorem implies that bounded subsets are weakly precompact, and therefore have limits in the weak sense. 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.

References

• Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5.
• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
• Whitley, R.J. (1967), "An elementary proof of the Eberlein-Smulian theorem", Mathematische Annalen, 172 (2): 116–118, doi:10.1007/BF01350091.