Banach-Alaoglu theorem
Encyclopedia
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.
A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach
, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu
.
Since the Banach–Alaoglu theorem is proven via Tychonoff's theorem
, it relies on the ZFC
axiomatic framework, in particular the axiom of choice. Most mainstream functional analysis also relies on ZFC.
Specifically, let X be a separable normed space and B the closed unit ball in X∗. Since X is separable, let {xn} be a countable dense subset. Then the following defines a metric for x, y ∈ B
in which
denotes the duality pairing of X∗ with X. Sequential compactness of B in this metric can be shown by a diagonalization argument similar to the one employed in the proof of the Arzelà–Ascoli theorem.
The sequential Banach–Alaoglu theorem is often used in the field of partial differential equations to construct solutions to PDE or variational problems
. For instance, if one wants to minimize a functional
on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence 
which approaches the infimum of F, use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit x, and then establish that x is a minimizer of F. The last step often requires F to obey a (sequential) lower semi-continuity
property in the weak* topology.
When X∗ is the space of finite Radon measures on the real line (so that
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.
.
Given a separated locally convex space X with continuous dual X ' then the polar U0 of any neighbourhood U in X is compact in the weak topology
σ(X ',X) on X '.
In the case of a normed vector space, the polar of a neighbourhood is closed and norm-bounded in the dual space. For example the polar of the unit ball is the closed unit ball in the dual. Consequently, for normed vector space (and hence Banach spaces) the Bourbaki–Alaoglu theorem is equivalent to the Banach–Alaoglu theorem.
and
Since each Dx is a compact subset
of the complex plane, D is also compact in the product topology
by Tychonoff theorem.
We can identify the closed unit ball in X*, B1(X*), as a subset of D in a natural way:
This map is injective and continuous, with B1(X*) having the weak-* topology and D the product topology. Its inverse, defined on its range, is also continuous.
The theorem will be proved if the range of the above map is closed. But this is also clear. If one has a net
in D, then the functional defined by
lies in B1(X*).
As a consequence, B(H) has the Heine–Borel property
, if equipped with either the weak operator or the ultraweak topology.
for all g ∈ Lq(μ) = X* (where 1/p+1/q=1). The corresponding result for p=1 is not true, as L1(μ) is not reflexive.
It should be cautioned that despite appearances, the Banach-Alaoglu theorem does *not* 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 neighbourhood of the origin in the weak-* topology. In fact, it is a result of Weil
that all locally compact Hausdorff
topological vector spaces must be finite dimensional.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
unit ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
of the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
in the weak* topology
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
. A common proof identifies the unit ball with the weak* topology as a closed subset of a product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of compact sets with the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
. As a consequence of Tychonoff's theorem
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
, this product, and hence the unit ball within, is compact.
A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu
Leonidas Alaoglu
Leonidas Alaoglu was a Canadian-American mathematician, most famous for his widely-cited result called Alaoglu's theorem on the weak-star compactness of the closed unit ball in the dual of a normed space, also known as the Banach–Alaoglu theorem.- Life and work :Alaoglu was born in Red Deer,...
.
Since the Banach–Alaoglu theorem is proven via Tychonoff's theorem
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
, it relies on the ZFC
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
axiomatic framework, in particular the axiom of choice. Most mainstream functional analysis also relies on ZFC.
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.Specifically, let X be a separable normed space and B the closed unit ball in X∗. Since X is separable, let {xn} be a countable dense subset. Then the following defines a metric for x, y ∈ B
in which
denotes the duality pairing of X∗ with X. Sequential compactness of B in this metric can be shown by a diagonalization argument similar to the one employed in the proof of the Arzelà–Ascoli theorem.The sequential Banach–Alaoglu theorem is often used in the field of partial differential equations to construct solutions to PDE or variational problems
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. For instance, if one wants to minimize a functional
on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence which approaches the infimum of F, use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit x, and then establish that x is a minimizer of F. The last step often requires F to obey a (sequential) lower semi-continuitySemi-continuity
In mathematical analysis, semi-continuity is a property of extended real-valued functions that is weaker than continuity...
property in the weak* topology.
When X∗ is the space of finite Radon measures on the real line (so that
is the space of continuous functions vanishing at infinity, by the Riesz representation theoremRiesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...
), the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.
Generalization: Bourbaki–Alaoglu theorem
The Bourbaki–Alaoglu theorem is a generalization by Bourbaki to dual topologiesDual topology
In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space....
.
Given a separated locally convex space X with continuous dual X ' then the polar U0 of any neighbourhood U in X is compact in the weak topology
Weak topology (polar topology)
In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair...
σ(X ',X) on X '.
In the case of a normed vector space, the polar of a neighbourhood is closed and norm-bounded in the dual space. For example the polar of the unit ball is the closed unit ball in the dual. Consequently, for normed vector space (and hence Banach spaces) the Bourbaki–Alaoglu theorem is equivalent to the Banach–Alaoglu theorem.
Proof
For any x in X, letand
Since each Dx is a compact subset
of the complex plane, D is also compact in the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
by Tychonoff theorem.
We can identify the closed unit ball in X*, B1(X*), as a subset of D in a natural way:
This map is injective and continuous, with B1(X*) having the weak-* topology and D the product topology. Its inverse, defined on its range, is also continuous.
The theorem will be proved if the range of the above map is closed. But this is also clear. If one has a net
in D, then the functional defined by
lies in B1(X*).
Consequences
- In a Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded sequence has a weakly convergent subsequence (Hilbert spaces are reflexive Reflexive spaceIn functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
) - As norm-closed, convex sets are weakly closed (Hahn–Banach theoremHahn–Banach theoremIn mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact. - Closed and bounded sets in B(H) are precompact with respect to the weak operator topologyWeak operator topologyIn functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
(the WOT is weaker than the ultraweak topologyUltraweak topologyIn functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B* of B, the trace class operators on H...
which is in turn the weak-*-topology with respect to the predual of B(H), the trace classTrace classIn mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....
operators.) Hence bounded sequences of operators have a weak accumulation point.
As a consequence, B(H) has the Heine–Borel property
Heine–Borel theorem
In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:*S is closed and bounded...
, if equipped with either the weak operator or the ultraweak topology.
- If X is a reflexive Banach space, then every bounded sequence in X has a weakly convergent subsequence. (This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of X; or, more succinctly, by applying the Eberlein–Šmulian theoremEberlein–Šmulian theoremIn the mathematical field of functional analysis, the Eberlein–Šmulian theorem is a result relating three different kinds of weak compactness in a Banach space...
.) For example, suppose that X=Lp(μ)Lp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
, 1<p<∞. Let fn be a bounded sequence of functions in X. Then there exists a subsequence fnk and an f ∈ X such that
for all g ∈ Lq(μ) = X* (where 1/p+1/q=1). The corresponding result for p=1 is not true, as L1(μ) is not reflexive.
It should be cautioned that despite appearances, the Banach-Alaoglu theorem does *not* 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
Strong topology
In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:* the final topology on the disjoint union...
, but is usually not a neighbourhood of the origin in the weak-* topology. In fact, it is a result of Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
that all locally compact Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological vector spaces must be finite dimensional.