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

If X is a real or complex vector space then we will let ${\displaystyle X^{\#}}$ denote the algebraic dual space of X. If X is a topological vector space (TVS), then denote the continuous dual space of X by ${\displaystyle X^{\prime }}$, where ${\displaystyle X^{\prime }\subseteq X^{\#}}$ necessarily holds. Denote the weak-* topology on ${\displaystyle X^{\#}}$ (resp. on ${\displaystyle X^{\prime }}$) by ${\displaystyle \sigma \left(X^{\#},X\right)}$ (resp. ${\displaystyle \sigma \left(X^{\prime },X\right)}$). Importantly, 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

${\displaystyle U^{\circ }=\left\{x^{\prime }\in X^{\prime }:\sup _{x\in U}\left|x^{\prime }\left(x\right)\right|\leq 1\right\}}$

of any neighborhood U of 0 in X is compact in the weak-* topology[6] ${\displaystyle \sigma \left(X^{\prime },X\right)}$ on ${\displaystyle X^{\prime }.}$ Moreover, ${\displaystyle U^{\circ }}$ 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 the origin in X and let:

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

A well known fact about polars of sets is that U^∘∘∘ = U^∘.

(1) First show that U^# = U^∘ and then deduce that U^∘ is a ${\displaystyle \sigma \left(X^{\#},X\right)}$-closed subset of ${\displaystyle X^{\#}:}$ It is a well-known result that the polar of a set is weakly closed, which implies that ${\displaystyle U^{\#}}$ is a ${\displaystyle \sigma \left(X^{\#},X\right)}$-closed subset of ${\displaystyle X^{\#}.}$ Because every continuous linear functional is a linear functional, U^∘ ⊆ U^# holds. For the reverse inclusion, 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 (that is, ${\displaystyle f\in X^{\prime }}$) from which it follows that U^# ⊆ U^∘).

(2) 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 ${\displaystyle U^{\circ \circ }}$ is also an absorbing subset of X, which one can show implies that ${\displaystyle U^{\circ }}$ 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 ${\displaystyle U^{\circ }}$ is ${\displaystyle \sigma \left(X^{\prime },X\right)}$-totally bounded.

(3) Now show that ${\displaystyle U^{\circ }}$ 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), implies that ${\displaystyle U^{\circ }}$ is a ${\displaystyle \sigma \left(X^{\#},X\right)}$-totally bounded subset of ${\displaystyle X^{\#}.}$

(4) Finally, deduce that ${\displaystyle U^{\circ }}$ is a ${\displaystyle \sigma \left(X^{\prime },X\right)}$-compact subset of ${\displaystyle X^{\prime }:}$ Because ${\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}$ is a complete space and ${\displaystyle U^{\circ }}$ 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). Consequently, this theorem can 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.

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 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 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)^∞
_n=1 be a countable dense subset. Then the following defines a metric, where for any 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 every x ∈ 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.

The closed unit ball in ${\displaystyle X^{\prime }}$, B₁(X*) can be identified 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. This map's inverse, defined on its range, is also continuous.

To finish proving this theorem, it will now be shown that the range of the above map is closed. Given a net

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

in D, the functional defined by

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

lies in ${\displaystyle B_{1}(X^{\prime }).}$

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 nearness

References

1. Rudin 1991, Theorem 3.15.
2. Narici & Beckenstein 2011, pp. 235-240.
3. Narici & Beckenstein 2011, pp. 225-273.
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. See §20.9.
• Meise, Reinhold; Vogt, Dietmar (1997). Introduction to Functional Analysis. Oxford: Clarendon Press. ISBN 0-19-851485-9. 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.
• Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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.