Banach–Alaoglu theorem

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In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.[1] A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.

Since the Banach–Alaoglu theorem is proven via Tychonoff's theorem, it relies on the ZFC axiomatic framework, in particular the axiom of choice. Most mainstream functional analysis also relies on ZFC.

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[edit] Generalization: Bourbaki–Alaoglu theorem

The Bourbaki–Alaoglu theorem is a generalization by Bourbaki to dual topologies.

Given a separated locally convex space X with continuous dual X ' then the polar U0 of any neighbourhood U in X is compact in the weak topology σ(X ',X) on X '.

In the case of a normed vector space, the polar of a neighbourhood is closed and norm-bounded in the dual space. For example the polar of the unit ball is the closed unit ball in the dual. Consequently, for normed vector space (and hence Banach spaces) the Bourbaki–Alaoglu theorem is equivalent to the Banach–Alaoglu theorem.

[edit] Proof

For any x in X, let

D_x=\{z\in\mathbb{C}: |z|\leq \|x\|\},

and

D=\Pi_{x\in X} D_x .

Since each Dx is a compact subset of the complex plane, D is also compact in the product topology by Tychonoff theorem.

We can identify the closed unit ball in X*, B1(X*), as a subset of D in a natural way:

f \in B_1(X^*) \mapsto (f(x))_{x \in X} \in D.

This map is injective and continuous, with B1(X*) having the weak-* topology and D the product topology. Its inverse, defined on its range, is also continuous.

The claim will be proved if the range of the above map is closed. But this is also clear. If one has a net

(f_{\alpha}(x))_{x \in X} \rightarrow (\lambda_x)_{x \in X}

in D, then the functional defined by

g(x) = \lambda_x \,

lies in B1(X*).

[edit] Consequences

If X is a reflexive Banach space, then every bounded sequence in X has a weakly convergent subsequence. (This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of X; or, more succinctly, by applying the Eberlein–Šmulian theorem.) For example, suppose that X=Lp(μ), 1<p<∞. Let fn be a bounded sequence of functions in X. Then there exists a subsequence fnk and an f ∈ X such that

\int f_{n_k} g\,d\mu \to \int f g\,d\mu

for all g ∈ Lq(μ) = X* (where 1/p+1/q=1). The corresponding result for p=1 is not true, as L1(μ) is not reflexive.

[edit] Notes

  1. ^ Rudin, section 3.15.

[edit] See also

[edit] Further reading

  • John B. Conway (1994). A course in functional analysis, 2nd edition, Berlin: Springer-Verlag. ISBN 0-387-97245-5.  See Chapter 5, section 3.
  • Rudin, W. (1991), Functional Analysis (2nd ed.), Boston, MA: McGraw-Hill, ISBN 0-07-054236-8 . See section 3.15, p.68.