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. 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.

[edit] 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 '.

This does generalise the Banach-Alaoglu theorem, since in a Banach space, the polar of the unit ball is the unit ball in the dual space.

[edit] Formal 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 $D x$ is a compact subset of $\mathbb{C}$ , $D$ is compact in product topology by Tychonoff theorem. We prove the theorem by finding a homeomorphism that maps the closed unit ball $B_{X^*}$ of $X *$ onto a closed subset of $D$ . Define $\Phi_x:B_{X^*}\to D_x$ by $Φ x (f) = f (x)$ and $\Phi:B_{X^*}\to D$ by $\Phi=\Pi_{x\in X}\Phi_x$ , so that $\Phi(f)=(f(x))_{x\in X}$ . Obviously, $Φ$ is one-to-one, and a net $(f α)$ in $B_{X^*}$ converges to $f$ in weak-* topology of $X *$ if, and only if, $Φ(f α)$ converges to $Φ(f)$ in product topology, therefore $Φ$ is continuous and so is its inverse $\Phi^{-1}:\Phi(B_{X^*})\to B_{X^*}$ .

It remains to show that $\Phi(B_{X^*})$ is closed. If $(Φ(f α))$ is a net in $\Phi(B_{X^*})$ , converging to a point $d=(d_x)_{x\in X}\in D$ , we can define a function $f:X\to \mathbb{C}$ by $f (x) = d x$ . As $\lim_\alpha \Phi(f_\alpha(x))=d_x$ for all $x\in X$ by definition of weak-* convergence, one can easily see that $f$ is a linear functional in $B_{X^*}$ and that $Φ(f) = d$ . This shows that $d$ is actually in $\Phi(B_{X^*})$ and finishes the proof.