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+---
+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 %}