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.

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 in 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 }{\frac {\langle x-y,x_{n}\rangle }{2^{n}(1+\langle x-y,x_{n}\rangle )}}

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.

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 $F:X\to {\mathbb {R}}$ on 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 lower semi-continuity property in the weak* topology.

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.

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 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}(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*).

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=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 fg\,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.

Notes

^ Rudin, section 3.15.