---
title: Alaoglu–Bourbaki Theorem
parent: Locally Convex Spaces
grand_parent: More Functional Analysis
nav_order: 1
---

# {{ page.title }}

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

$$
X \times X' \to \CC, \quad (x,f) \mapsto \angles{x,f} = f(x).
$$

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 $\angles{x,\cdot} : X \to \CC$ are continuous.
The polar of $U$ is the subset $U^{\circ} \subset X'$.
The theorem asserts that $U^{\circ}$ is compact in the weak\* topology.

{% theorem * Alaoglu–Bourbaki Theorem %}
The polar of a neighborhood of zero in a locally convex space is weak\* compact.
{% endtheorem %}