Banach–Alaoglu theorem

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. However, the theorem does not rely upon the axiom of choice in the separable case (see below): in this case one actually has a constructive proof.

This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any states can be written as a convex linear combination of so-called pure states.

The theorem

Let X be a normed space, the dual X* is hence also a normed space (with the operator norm).

The closed unit ball of X* is compact with respect to the weak* topology. (cf. also section "dual" in the article "topological vector space")

This is a motivation for having different topologies on a same space since in contrast the unit ball in the norm topology is compact if and only if the space is finite-dimensional, cf. Riesz lemma

Sequential Banach–Alaoglu theorem

A special case of the Banach–Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak* topology. In fact, the weak* topology on the closed unit ball of the dual of a separable space is metrizable, and thus compactness and sequential compactness are equivalent.

Specifically, let X be a separable normed space and B the closed unit ball in X^∗. Since X is separable, let {x_n} be a countable dense subset. Then the following defines a metric for x, y ∈ B

\rho(x,y)=\sum_{n=1}^\infty \, 2^{-n} \, \frac{\left|\langle x-y, x_n\rangle\right|}{1 + \left|\langle x-y, x_n\rangle\right|}

in which $\langle \cdot ,\cdot \rangle$ denotes the duality pairing of X^∗ with X. Sequential compactness of B in this metric can be shown by a diagonalization argument similar to the one employed in the proof of the Arzelà–Ascoli theorem.

Due to the constructive nature of its proof (as opposed to the general case, which is based on the axiom of choice), the sequential Banach–Alaoglu theorem is often used in the field of partial differential equations to construct solutions to PDE or variational problems. For instance, if one wants to minimize a functional $F: X^* \to {\Bbb R}$ on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence $x_1, x_2, \ldots \in X^*$ which approaches the infimum of F, use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit x, and then establish that x is a minimizer of F. The last step often requires F to obey a (sequential) lower semi-continuity property in the weak* topology.

When X^∗ is the space of finite Radon measures on the real line (so that $X = C_0({\Bbb R})$ is the space of continuous functions vanishing at infinity, by the Riesz representation theorem), the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.

Generalization: Bourbaki–Alaoglu theorem

The Bourbaki–Alaoglu theorem is a generalization^[2]^[3] by Bourbaki to dual topologies on locally convex spaces.

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.

Proof

For any x in X, let

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

and

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

Since each D_x 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\left(X^*\right) \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 theorem 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*).

Consequences

In a Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded sequence has a weakly convergent subsequence (Hilbert spaces are reflexive.)
As norm-closed, convex sets are weakly closed (Hahn–Banach theorem), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact.
Closed and bounded sets in B(H) are precompact with respect to the weak operator topology (the WOT is weaker than the ultraweak topology which is in turn the weak-*-topology with respect to the predual of B(H), the trace class operators.) Hence bounded sequences of operators have a weak accumulation point.

As a consequence, B(H) has the Heine–Borel property, if equipped with either the weak operator or the ultraweak topology.

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=L^p(μ), 1<p<∞. Let f_n be a bounded sequence of functions in X. Then there exists a subsequence f_{n_k} and an f ∈ X such that

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

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

It should be cautioned that despite appearances, the Banach–Alaoglu theorem does not imply that the weak-* topology is locally compact. This is because the closed unit ball is only a neighborhood of the origin in the strong topology, but is usually not a neighbourhood of the origin in the weak-* topology, as it has empty interior in the weak* topology, unless the space is finite-dimensional. In fact, it is a result of Weil that all locally compact Hausdorff topological vector spaces must be finite-dimensional.

Notes

↑ Rudin 1991, section 3.15.
↑ Köthe 1969, Theorem (4) in §20.9.
↑ Meise & Vogt 1997, Theorem 23.5.

References

Rudin, W. (1991). Functional Analysis (2nd ed.). Boston, MA: McGraw-Hill. ISBN 0-07-054236-8. See section 3.15, p. 68.
Meise, Reinhold; Vogt, Dietmar (1997). Introduction to Functional Analysis. Oxford: Clarendon Press. ISBN 0-19-851485-9. See Theorem 23.5, p. 264.
Köthe, Gottfried (1969). Topological Vector Spaces I. New York: Springer-Verlag. See §20.9.

Functional analysis

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