Banach–Alaoglu theorem

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.[1] 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 - [that] echos throughout functional analysis."[2] In 1912, Helly proved that the unit ball of the continuous dual space of C([a, b]) is countably weak-* compact.[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 12 mathematicians who can lay claim to this theorem or an important predecessor to it.[2]

The Bourbaki–Alaoglu theorem is a generalization[4][5] 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[2]

Statement

See also: Topological vector space § Dual space, Dual system, and Polar set

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 (TVS), then we will denote the continuous dual space of X by ${\displaystyle X^{\prime }}$, where note that ${\displaystyle X^{\prime }}$ ⊆ X^#. We denote the weak-* topology on X^# (resp. on ${\displaystyle X^{\prime }}$) by ${\displaystyle \sigma \left(X^{\#},X\right)}$ (resp. ${\displaystyle \sigma \left(X^{\prime },X\right)}$). Note that the subspace topology that ${\displaystyle X^{\prime }}$ inherits from ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$) is just ${\displaystyle \sigma \left(X^{\prime },X\right)}$.

Alaoglu theorem:[3] For any TVS X (not necessarily Hausdorff or locally convex), the polar U^∘ := ${\displaystyle \left\{x^{\prime }\in X^{\prime }:\sup _{x\in U}\left|x^{\prime }\left(x\right)\right|\leq 1\right\}}$ of any neighborhood U of 0 in X is compact in the weak-* topology[6] σ${\displaystyle \left(X^{\prime },X\right)}$ on ${\displaystyle X^{\prime }}$. Moreover, U^∘ is equal to the polar of U with respect to the canonical system ${\displaystyle \left(X,X^{\#}\right)}$ and it is also a compact subset of ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$.

Proof —

For this proof, we will use the basic properties that are listed in the articles: polar set, dual system, and continuous linear operator.

Recall that when X^# is endowed with the weak-* topology ${\displaystyle \sigma \left(X^{\#},X\right)}$ then ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$ is a complete space (however, ${\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right)}$ may fail to be a complete space). Throughout, unless stated otherwise, all polar sets will be taken with respect to the canonical pairing ${\displaystyle \left(X,X^{\prime }\right)}$ where ${\displaystyle X^{\prime }}$ is the continuous dual space of X.

Let U be a neighborhood of 0 in X and let:

• U^∘ := ${\displaystyle \left\{f\in X^{\prime }:\sup _{u\in U}\left|f\left(u\right)\right|\leq 1\right\}}$ be the polar of U with respect to the canonical pairing ${\displaystyle \left(X,X^{\prime }\right)}$,
• U^∘∘ be the bipolar of U with respect to ${\displaystyle \left(X,X^{\prime }\right)}$
• U^# := ${\displaystyle \left\{f\in X^{\#}:\sup _{u\in U}\left|f(u)\right|\leq 1\right\}}$ be the polar of U with respect to the canonical dual system ${\displaystyle \left(X,X^{\#}\right)}$

Recall that U^∘∘∘ = U^∘.

(1) We first show that U^# = U^∘ and deduce that U^∘ is a ${\displaystyle \sigma \left(X^{\#},X\right)}$-closed subset of ${\displaystyle X^{\#}}$: It is a basic result that the polar of a set is weakly closed so that in particular, U^# is a ${\displaystyle \sigma \left(X^{\#},X\right)}$-closed subset of X^#. Clearly, U^∘ ⊆ U^#. For the reverse inclusion, note that if f ∈ U^# then since ${\displaystyle \sup _{u\in U}\left|f(u)\right|\leq 1}$ and U is a neighborhood of 0 in X, it follows that f is a continuous linear functional (i.e. f ∈ ${\displaystyle X^{\prime }}$) from which it readily follows that U^# ⊆ U^∘.

(2) We now show that U^∘ is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-totally bounded subset of ${\displaystyle X^{\prime }}$: By the bipolar theorem, U ⊆ U^∘∘ so since U is absorbing in X, it follows that U^∘∘ is also an absorbing subset of X, which one can show implies that U^∘ is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-bounded. Since X distinguishes points of ${\displaystyle X^{\prime }}$, it can be shown that a subset of ${\displaystyle X^{\prime }}$ is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-bounded if and only if it is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-totally bounded. From this it follows that U^∘ is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-totally bounded.

(3) We now show that U^∘ is ${\displaystyle \sigma \left(X^{\#},X\right)}$-totally bounded subset of ${\displaystyle X^{\#}}$: Recall that the ${\displaystyle \sigma \left(X^{\prime },X\right)}$ topology on ${\displaystyle X^{\prime }}$ is identical to the subspace topology that ${\displaystyle X^{\prime }}$ inherits from ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$. This fact, together with (2), immediately implies that U^∘ is a ${\displaystyle \sigma \left(X^{\#},X\right)}$-totally bounded subset of X^#.

(4) Deduce that U^∘ is a ${\displaystyle \sigma \left(X^{\prime },X\right)}$-compact subset of ${\displaystyle X^{\prime }}$: Since ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$ is a complete space and U^∘ is a closed (by (1)) and totally bounded (by (3)) subset of ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$, it follows that U^∘ is compact.

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 (or closed) unit ball in X then the polar of U is the closed unit ball in the continuous dual space ${\displaystyle X^{\prime }}$ of X (with the usual dual norm). Thus this theorem can thus be specialized to:

Banach-Alaoglu theorem: If X is a normed space then the closed unit ball in the continuous dual space ${\displaystyle X^{\prime }}$ (endowed with its usual operator norm) is compact with respect to the weak-* topology.

Note that when the continuous dual space ${\displaystyle X^{\prime }}$ of X is an infinite dimensional normed space then it is impossible for the closed unit ball in ${\displaystyle X^{\prime }}$ to be a compact subset when ${\displaystyle X^{\prime }}$ 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 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 {x_n} be a countable dense subset. Then the following defines a metric for x, y ∈ B

${\displaystyle \rho (x,y)=\sum _{n=1}^{\infty }\,2^{-n}\,{\frac {\left|\langle x-y,x_{n}\rangle \right|}{1+\left|\langle x-y,x_{n}\rangle \right|}}}$

in which ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 (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. For instance, if one wants to minimize a functional  ${\displaystyle F:X^{\prime }\to \mathbb {R} }$  on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence  ${\displaystyle x_{1},x_{2},\ldots \in X^{\prime }}$  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 ${\displaystyle X^{\prime }}$ is the space of finite Radon measures on the real line (so that  ${\displaystyle X=C_{0}\left(\mathbb {R} \right)}$  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.

Proof —

For any x in X, let

${\displaystyle D_{x}=\left\{z\in \mathbb {C}$ :\left|z\right|\leq \|x\|\right\}} ,

and

${\displaystyle D=\prod _{x\in X}D_{x}}$.

Since each D_x is a compact subset of the complex plane, D is also compact in the product topology by Tychonoff's theorem.

We can identify the closed unit ball in ${\displaystyle X^{\prime }}$, B₁(X*), as a subset of D in a natural way:

${\displaystyle f\in B_{1}\left(X^{\prime }\right)\mapsto \left(f(x)\right)_{x\in X}\in D}$.

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

${\displaystyle \left(f_{\alpha }(x)\right)_{x\in X}\to \left(\lambda _{x}\right)_{x\in X}}$

in D, then the functional defined by

${\displaystyle g(x)=\lambda _{x}}$

lies in B₁(${\displaystyle X^{\prime }}$).

Consequences

Consequences for normed spaces

Assume that X is a normed space and endow its continuous dual space ${\displaystyle X^{\prime }}$ with the usual dual norm.

• The closed unit ball in ${\displaystyle X^{\prime }}$ is weak-* compact.[3]
  • Note that if ${\displaystyle X^{\prime }}$ is infinite dimensional then its closed unit ball is necessarily not compact in the norm topology by the F. Riesz theorem (despite it being weak-* compact).
• A Banach space is reflexive if and only if its closed unit ball is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-compact.[3]
• 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 theorem.) For example, suppose that X = L^p(μ), 1<p<∞. Let f_n be a bounded sequence of functions in X. Then there exists a subsequence f_nk and an f ∈ X such that

  ${\displaystyle \int f_{n_{k}}g\,d\mu \to \int fg\,d\mu }$

  for all g ∈ L^q(μ) = X* (where 1/p+1/q=1).

  The corresponding result for p=1 is not true, as L¹(μ) 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.
• Closed and bounded sets in B(H) are precompact with respect to the weak operator topology (the weak operator topology is weaker than the ultraweak topology which is in turn the weak-* topology with respect to the predual of B(H), the trace class operators.) Hence bounded sequences of operators have a weak accumulation point. As a consequence, B(H) 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 (see below): 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.

See also

• Bishop–Phelps theorem
• Banach–Mazur theorem
• Delta-compactness theorem
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• Goldstine theorem
• James' theorem
• Krein-Milman theorem
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space
• Topological vector space – Vector space with a notion of continuity

References

1. Rudin 1991, Theorem 3.15.
2. Narici 2011, pp. 235-240. sfn error: no target: CITEREFNarici2011 (help)
3. Narici 2011, pp. 225-273. sfn error: no target: CITEREFNarici2011 (help)
4. Köthe 1969, Theorem (4) in §20.9.
5. Meise & Vogt 1997, Theorem 23.5.
6. Explicitly, a subset ${\displaystyle B^{\prime }\subseteq X^{\prime }}$ is said to be "compact (resp. totally bounded, etc.) in the weak-* topology" if when ${\displaystyle X^{\prime }}$ is given the weak-* topology and the subset ${\displaystyle B^{\prime }}$ is given the subspace topology inherited from ${\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right)}$, then ${\displaystyle B^{\prime }}$ is a compact (resp. totally bounded, etc.) space.

• Köthe, Gottfried (1969). Topological Vector Spaces I. New York: Springer-Verlag.CS1 maint: ref=harv (link) See §20.9.
• Meise, Reinhold; Vogt, Dietmar (1997). Introduction to Functional Analysis. Oxford: Clarendon Press. ISBN 0-19-851485-9.CS1 maint: ref=harv (link) See Theorem 23.5, p. 264.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, W. (1991). Functional Analysis (2nd ed.). Boston, MA: McGraw-Hill. ISBN 0-07-054236-8.CS1 maint: ref=harv (link) See Theorem 3.15, p. 68.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press.
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)

Further reading

• John B. Conway (1994). A course in functional analysis (2nd ed.). Berlin: Springer-Verlag. ISBN 0-387-97245-5. See Chapter 5, section 3.
• Peter B. Lax (2002). Functional Analysis. Wiley-Interscience. pp. 120–121. ISBN 0-471-55604-1.