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. 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.

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

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