In functional analysis, the Banach-Alaoglu theorem (also sometimes called Alaoglu's theorem) is a result which states that the norm unit ball of the continuous dual of a topological vector space is compact in the weak-* topology induced by the norm topology on .
More precisely, given a topological vector space and a neighborhood of in , the Banach-Alaoglu theorem says that the so-called polar of is weak-* compact (i.e., is compact in the above-mentioned weak-* topology of ) where
and where denotes the magnitude of the scalar in the underlying scalar field of (i.e., the absolute value of if is a real vector space or its complex modulus if is a complex vector space).
The proof for a general topological vector space was proved by Alaoglu in the 1940s though the special case for separable was proved by Banach in the 1930s. Since then, the theorem has been generalized to other miscellaneous contexts, most notably by Bourbaki into the language of dual topologies, and has a number of significant corollaries. For example, the theorem implies that every bounded sequence in a reflexive Banach space (e.g., when is a Hilbert space) has a weakly convergent subsequence and hence that the norm-closures of bounded convex sets in such spaces are weakly compact.
Worth noting is that the Banach-Alaoglu theorem has a sort of converse which is also true. In particular, if is a Banach space with dual , if denotes the closed unit ball in , and if is a convex set in for which the intersection is weak-* compact for every , then is necessarily weak-* closed.