Alaoglu–Bourbaki Theorem

Let $X$ be locally convex space and let $U\subset X$ be a neighborhood of zero. Let $X^{\prime }$ denote the continuous dual of $X$. Recall that there is a canonical pairing

$$X\times X^{\prime }\to \mathbb{C},(x,f)\mapsto ⟨x,f⟩=f(x).$$

The weak topology on $X^{\prime }$ with respect to this pairing is called weak* topology. It is the weakest topology on $X^{\prime }$ such that all evaluation maps $⟨x,\cdot ⟩:X\to \mathbb{C}$ are continuous. The polar of $U$ is the subset $U^{\circ }\subset X^{\prime }$. The theorem asserts that $U^{\circ }$ is compact in the weak* topology.

Alaoglu–Bourbaki Theorem

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