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. 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.
Contents
- The theorem
- Sequential BanachAlaoglu theorem
- Generalization BourbakiAlaoglu theorem
- Proof
- Consequences
- References
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 {xn} be a countable dense subset. Then the following defines a metric for x, y ∈ B
in which
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
When X∗ is the space of finite Radon measures on the real line (so that
Generalization: Bourbaki–Alaoglu theorem
The Bourbaki–Alaoglu theorem is a generalization 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
and
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:
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
in D, then the functional defined by
lies in B1(X*).
Consequences
As a consequence, B(H) has the Heine–Borel property, if equipped with either the weak operator or the ultraweak topology.
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.