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. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

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.

History

According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a "very important result—maybe most important fact about the weak-* topology—[that] echos throughout functional analysis." is countably weak-* compact. 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). 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.

The Bourbaki–Alaoglu theorem is a generalization of the original theorem by Bourbaki to dual topologies on locally convex spaces. This theorem is also called the Banach–Alaoglu theorem or the weak-* compactness theorem and it is commonly called simply the Alaoglu theorem.

Statement

will denote the algebraic dual space and these two spaces are henceforth associated with the bilinear forms a dual system called the .

is a topological vector space (TVS) then its continuous dual space always holds. Denote the weak-* topology The weak-* topology is also called the topology of pointwise convergence and a net

Proof involving duality theory

is a normed vector space (with the usual dual norm). Consequently, this theorem can be specialized to:

is an infinite dimensional normed space then it is has its usual norm topology. This is because the unit ball in the norm topology is compact if and only if the space is finite-dimensional (cf. F. Riesz theorem). This theorem is one example of the utility of having different topologies on the same vector space.

It should be cautioned that despite appearances, the Banach–Alaoglu theorem does imply that the weak-* topology is locally compact. This is because the closed unit ball is only a neighborhood of the origin in the strong topology, but is usually not a neighborhood of the origin in the weak-* topology, as it has empty interior in the weak* topology, unless the space is finite-dimensional. In fact, it is a result of Weil that all locally compact Hausdorff topological vector spaces must be finite-dimensional.

Elementary proof

The following elementary proof does not utilize duality theory and requires only basic concepts from set theory, topology, and functional analysis. What is needed from topology is a working knowledge of net convergence in topological spaces and familiarity with the fact that a linear functional is continuous if and only if it is bounded on a neighborhood of the origin (see the articles on continuous linear functionals and sublinear functionals is identified as the Cartesian product and the relationship between pointwise convergence, the product topology, and subspace topologies they induce on subsets such as the algebraic dual space An explanation of these details is now given for readers who are interested.

Identification of functions with tuples

-indexed tuples as is now described:

• -indexed) "

(or conversely). However, the Cartesian product, being the (categorical) product in the category of sets (which is a type of inverse limit), also comes equipped with associated maps that are known as its (coordinate) .

The ".

partitions canonically identifies these two Cartesian products; moreover, this map is a homeomorphism

Notation for nets and function composition with nets

A net from a non-empty directed set Every sequence Similarly for function composition

Topology

is assumed to be endowed with the product topology. It is well known that the product topology is identical to the topology of pointwise convergence and a net converges in the product topology if and only if

this happens if and only if

This proof will also use the fact that the topology of pointwise convergence is preserved when passing to topological subspaces is some (topological) subspace is equal to the subspace topology

denotes the supremum and the identity map).

The essence of the Banach–Alaoglu theorem can be found in the next proposition, from which the Banach–Alaoglu theorem follows. Unlike the Banach–Alaoglu theorem, this proposition does to endowed with any topology.

is a topological vector space is a neighborhood of the origin).

1. Observation is a closed in the topology of pointwise convergence.

   Proof of observation

denote

as desired.

The lemma above is a Hausdorff complete uniform space) is closed if and only if it is complete.

is a complete Hausdorff locally convex topological vector space, the same is true of the product space

(such as any absorbing subset is a weak-* compact

As a side note, with the help of the above elementary proof, it may be shown (see this footnote) are unique; indeed, this infimum formula can be used to define them.

This implies (among other things the unique least element this may be used as an alternative definition of this (necessarily convex and balanced is a seminorm is replaced by the convex balanced hull is replaced by its closure

Sequential Banach–Alaoglu theorem

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 fact, the weak* topology on the closed unit ball of the dual of a separable space is metrizable, and thus compactness and sequential compactness are equivalent.

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 to obey a (sequential) lower semi-continuity property in the weak* topology.

is the space of continuous functions vanishing at infinity, by the Riesz representation theorem), the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.

Consequences

Consequences for normed spaces

is a normed space with the usual dual norm.

is weak-* compact. is infinite dimensional then its closed unit ball is necessarily compact in the norm topology by F. Riesz's theorem (despite it being weak-* compact). A Banach space is reflexive-compact; this is known as James' theorem. is a reflexive Banach space; or, more succinctly, by applying the Eberlein–Šmulian theorem is the space Lp space is not reflexive.

Consequences for Hilbert spaces

In a Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded net has a weakly convergent subnet (Hilbert spaces are reflexive). As norm-closed, convex sets are weakly closed (Hahn–Banach theorem), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact. are precompact with respect to the weak operator topology (the weak operator topology is weaker than the ultraweak topology the trace class has the Heine–Borel property, if equipped with either the weak operator or the ultraweak topology.

Relation to the axiom of choice and other statements

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. However, the theorem does rely upon the axiom of choice in the separable case (see above): in this case there actually exists a constructive proof. 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 spaces, suffices for the proof of the Banach–Alaoglu theorem, and is in fact equivalent to it.

The Banach–Alaoglu theorem is equivalent to the ultrafilter lemma, which implies the Hahn–Banach theorem for real vector spaces (HB) but is not equivalent to it (said differently, Banach–Alaoglu is also strictly stronger than HB). However, the Hahn–Banach theorem is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space 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 (also sometimes called );

Compactness implies convex compactness because a topological space is compact if and only if every family of closed subsets having the finite intersection property (FIP) has non-empty intersection. The definition of convex compactness is similar to this characterization of compact spaces in terms of the FIP, except that it only involves those closed subsets that are also convex (rather than all closed subsets).

See also

Notes

Proofs

Citations

References

• See Theorem 3.15, p.68.

Further reading