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

Eberlein–Šmulian theorem: [1] If X is a Banach space and A is a subset of X, then the following statements are equivalent:

each sequence of elements of A has a subsequence that is weakly convergent;
each sequence of elements of A has a weak cluster point;
the weak closure of A is weakly compact.

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

Compactness (or Heine-Borel 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 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.

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gollark: All integers in base 0 start with 0.

References

Conway 1990, p. 163.

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.

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[FOOTNOTEConway1990163-1] Conway 1990, p. 163.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Eberlein–Šmulian theorem

Statement

Applications

See also

References