Banach–Alaoglu theorem

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and related branches of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Ball_(mathematics)/f9vpMVMp">unit ball</a> of the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> is <a href="/facts/Compact_space/d0cgXJH7">compact</a> in the <a href="/facts/Weak*_topology/XJASE9Up">weak* topology</a>. 
A common proof identifies the unit ball with the weak-* topology as a closed subset of a <a href="/facts/Cartesian_product/BfqBWzdL">product</a> of compact sets with the <a href="/facts/Product_topology/0hg02IaX">product topology</a>. 
As a consequence of <a href="/facts/Tychonoff%2527s_theorem/PhVArhKr">Tychonoff's theorem</a>, this product, and hence the unit ball within, is compact.
</p><p>This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states.
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Banach–Alaoglu theorem open-in-new

Banach–Alaoglu theorem