# Alaoglu–Bourbaki Theorem

Let $X$ be locally convex space and let $U⊂X$ be a neighborhood of zero. Let $X_{′}$ denote the continuous dual of $X$. Recall that there is a canonical pairing

$X×X_{′}→C,(x,f)↦⟨x,f⟩=f(x).$The weak topology on $X_{′}$ with respect to this pairing is called weak* topology. It is the weakest topology on $X_{′}$ such that all evaluation maps $⟨x,⋅⟩:X→C$ are continuous. The polar of $U$ is the subset $U_{∘}⊂X_{′}$. The theorem asserts that $U_{∘}$ is compact in the weak* topology.

Alaoglu–Bourbaki Theorem

The polar of a neighborhood of zero in a locally convex space is weak* compact.