pages/functional-analysis-basics/banach-alaoglu-theorem.md


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---
title: Banach–Alaoglu Theorem
parent: Functional Analysis Basics
nav_order: 3
---

# {{ page.title }}

{% theorem * Banach–Alaoglu Theorem %}
The closed unit ball in the dual of a normed space is weak\* compact.
{% endtheorem %}

{% proof %}
{% endproof %}

The {{ page.title }} is a special case of the following result:

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

See [Alaoglu–Bourbaki Theorem]({% link pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md %}) for more information.