Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu 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 the most important fact about the weak-* topology - echos throughout functional analysis."In 1912, Helly proved that the unit ball of the continuous dual space of C 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.
The proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
According to Pietsch , there are at least 12 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
If X is a real or complex vector space then we will let X# denote the algebraic dual space of X.If X is a topological vector space, then we will denote the continuous dual space of X by, where note that ⊆ X#.
We denote the weak-* topology on X# by .
Note that the subspace topology that inherits from ) is just.
If X is a normed vector space, then the polar of a neighborhood is closed and norm-bounded in the dual space.
In particular, if U is the open unit ball in X then the polar of U is the closed unit ball in the continuous dual space of X.
Thus this theorem can thus be specialized to:
Note that when the continuous dual space of X is an infinite dimensional normed space then it is impossible for the closed unit ball in to be a compact subset when 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.
This theorem is one motivation for having different topologies on a same space.
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 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.
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 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.
Due to the constructive nature of its proof, 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 lower semi-continuity property in the weak* topology.
When is the space of finite Radon measures on the real line, the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.
Consequences
;Consequences for normed spacesAssume that X is a normed space and endow its continuous dual space with the usual dual norm.
- The closed unit ball in is weak-* compact.
- Note that if is infinite dimensional then its closed unit ball is necessarily not compact in the norm topology by the F. Riesz theorem.
- A Banach space is reflexive if and only if its closed unit ball is -compact.
- If X is a reflexive Banach space, then every bounded sequence in X has a weakly convergent subsequence.
For example, suppose that X = Lp, 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 = X*.
The corresponding result for p=1 is not true, as L1 is not reflexive.
- In a Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded net has a weakly convergent subnet.
- As norm-closed, convex sets are weakly closed, norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact.
- Closed and bounded sets in B are precompact with respect to the weak operator topology Hence bounded sequences of operators have a weak accumulation point.
As a consequence, B has the Heine–Borel property, if equipped with either the weak operator or the ultraweak topology.
Relation to the axiom of choice
Since the Banach–Alaoglu theorem is usually proven via Tychonoff's theorem, it relies on the ZFC axiomatic framework, and in particular the axiom of choice.Most mainstream functional analysis also relies on ZFC.
However, the theorem does not rely upon the axiom of choice in the separable case : in this case one actually has a constructive proof.
In the non-separable case, the Ultrafilter Lemma, which is strictly weaker than the axiom of choice, suffices for the proof of the Banach-Alaoglu theorem, and is in fact equivalent to it.