In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.
According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe the most important fact about the weak-* topology—[that] echos throughout functional analysis.”[2] In 1912, Helly proved that the unit ball of the continuous dual space of
[3] In 1932, Stefan Banach proved that the closed unit ball in the continuous dual space of any separable normed space is sequentially weak-* compact (Banach only considered sequential compactness).
[3] The proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
According to Pietsch [2007], there are at least twelve mathematicians who can lay claim to this theorem or an important predecessor to it.
(not necessarily Hausdorff or locally convex) with continuous dual space
This proof will use some of the basic properties that are listed in the articles: polar set, dual system, and continuous linear operator.
To start the proof, some definitions and readily verified results are recalled.
Specifically, this proof will use the fact that a subset of a complete Hausdorff space is compact if (and only if) it is closed and totally bounded.
(endowed with its usual operator norm) is compact with respect to the weak-* topology.
is an infinite dimensional normed space then it is impossible for the closed unit ball in
This is because the unit ball in the norm topology is compact if and only if the space is finite-dimensional (cf.
It should be cautioned that despite appearances, the Banach–Alaoglu theorem does not imply that the weak-* topology is locally compact.
In fact, it is a result of Weil that all locally compact Hausdorff topological vector spaces must be finite-dimensional.
but, since tuples are technically just functions from an indexing set, it can also be identified with the space
depending on whichever notation is cleanest or most clearly communicates the intended information.
was defined in the proposition's statement as being any positive real number that satisfies
is closed can also be reached by applying the following more general result, this time proved using nets, to the special case
A special case of the Banach–Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak-* topology.
in this metric can be shown by a diagonalization argument similar to the one employed in the proof of the Arzelà–Ascoli theorem.
Due to the constructive nature of its proof (as opposed to the general case, which is based on the axiom of choice), the sequential Banach–Alaoglu theorem is often used in the field of partial differential equations to construct solutions to PDE or variational problems.
use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit
to obey a (sequential) lower semi-continuity property in the weak* topology.
is a compact subset of the complex plane, Tychonoff's theorem guarantees that their product
It will now be shown that the image of the above map is closed, which will complete the proof of the theorem.
The Banach–Alaoglu may be proven by using Tychonoff's theorem, which under the Zermelo–Fraenkel set theory (ZF) axiomatic framework is equivalent to the axiom of choice.
Most mainstream functional analysis relies on ZF + the axiom of choice, which is often denoted by ZFC.
In the general case of an arbitrary normed space, the ultrafilter Lemma, which is strictly weaker than the axiom of choice and equivalent to Tychonoff's theorem for compact Hausdorff spaces, suffices for the proof of the Banach–Alaoglu theorem, and is in fact equivalent to it.
However, the Hahn–Banach theorem is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space[6] in which the conclusion of compactness (in the weak-* topology of the closed unit ball of the dual space) is replaced with the conclusion of quasicompactness (also sometimes called convex compactness); Weak version of Alaoglu theorem[6] — Let
denote the closed unit ball of its continuous dual space