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Banach-Alaoglu Theorem


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 X^* of a topological vector space X is compact in the weak-* topology induced by the norm topology on X.

More precisely, given a topological vector space X and a neighborhood V of 0 in X, the Banach-Alaoglu theorem says that the so-called polar K=K(V) of V is weak-* compact (i.e., is compact in the above-mentioned weak-* topology of X^*) where

 K(V)={Lambda in X^*:|Lambdax|<=1 forall x in V}

and where |Lambdax| denotes the magnitude of the scalar Lambdax in the underlying scalar field of X (i.e., the absolute value of Lambdax if X is a real vector space or its complex modulus if X is a complex vector space).

The proof for a general topological vector space X was proved by Alaoglu in the 1940s though the special case for X 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 X (e.g., when X 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 X is a Banach space with dual X^*, if B^* denotes the closed unit ball in X^*, and if E is a convex set in X^* for which the intersection E intersection (rB^*) is weak-* compact for every r>0, then E is necessarily weak-* closed.


See also

Dual Normed Space, Norm Topology, Topological Vector Space, Vector Space Polar, Weak Topology, Weak-* Topology

This entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Cite this as:

Stover, Christopher. "Banach-Alaoglu Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Banach-AlaogluTheorem.html

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