Banach-Alaoglu theorem

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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.

[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 '.

[edit] Reference

J. B. Conway, Chapter 5, section 3, A Course in Functional Analysis, Springer-Verlag, 2nd edition.