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| In [[functional analysis]] and related branches of [[mathematics]], the '''Banach–Alaoglu theorem''' (also known as '''Alaoglu's theorem''') states that the [[closed set|closed]] [[ball (mathematics)|unit ball]] of the [[dual space]] of a [[normed vector space]] is [[compact space|compact]] in the [[weak topology|weak* topology]].<ref>Rudin, section 3.15.</ref> A common proof identifies the unit ball with the weak* topology as a closed subset of a [[cartesian product|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.
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| A proof of this theorem for [[separable space|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]].
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| Since the Banach–Alaoglu theorem is proven via [[Tychonoff's theorem]], it relies on the [[Zermelo–Fraenkel set theory|ZFC]] axiomatic framework, 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.
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| This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any states can be written as a convex linear combination of so-called pure states.
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| ==The theorem==
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| Let X be a normed space, the dual X* is hence also a normed space (with the operator norm).
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| The closed unit ball of X* is compact with respect to the weak* topology. (cf. also section "dual" in the article "topological vector space")
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| This is a motivation for having different topologies on a same space since in contrast the unit ball in the norm topology is compact if and only if the space is finite dimensional, cf. [[Riesz lemma]]
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| ==Sequential Banach–Alaoglu theorem==
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| 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 metric space|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.
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| Specifically, let ''X'' be a separable normed space and ''B'' the closed unit ball in ''X''<sup>∗</sup>. Since ''X'' is separable, let {''x''<sub>''n''</sub>} be a countable dense subset. Then the following defines a metric for ''x'', ''y'' ∈ ''B''
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| :<math>\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|}</math>
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| in which <math>\langle\cdot,\cdot\rangle</math> denotes the duality pairing of ''X''<sup>∗</sup> 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]].
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| 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 [[calculus of variations|variational problems]]. For instance, if one wants to minimize a functional  <math>F: X^* \to {\Bbb R}</math>  on the dual of a separable normed vector space ''X'', one common strategy is to first construct a minimizing sequence  <math>x_1, x_2, \ldots \in X^*</math>  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) [[Semi-continuity|lower semi-continuity]] property in the weak* topology.
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| When ''X''<sup>∗</sup> is the space of finite Radon measures on the real line (so that  <math>X = C_0({\Bbb R})</math>  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]].
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| ==Generalization: Bourbaki–Alaoglu theorem==
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| The '''Bourbaki–Alaoglu theorem''' is a generalization by [[Bourbaki]] to [[dual topology|dual topologies]].
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| Given a separated [[locally convex space]] ''X'' with [[continuous dual]] ''X'' ' then the [[polar set|polar]] ''U''<sup>0</sup> of any [[neighbourhood (topology)|neighbourhood]] ''U'' in ''X'' is compact in the [[weak topology (polar topology)|weak topology]] σ(''X'' ',''X'') on ''X'' '.
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| 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.
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| ==Proof==
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| For any ''x'' in ''X'', let
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| :<math>D_x=\{z\in\mathbb{C}: \left|z\right|\leq \|x\|\},</math>
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| :<math>D=\Pi_{x\in X} D_x .</math>
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| Since each ''D<sub>x</sub>'' is a compact subset
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| of the complex plane, ''D'' is also compact in the [[product topology]] by [[Tychonoff theorem]].
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| We can identify the closed unit ball in ''X*'', ''B''<sub>1</sub>(''X*''), as a subset of ''D'' in a natural way:
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| :<math> f \in B_1\left(X^*\right) \mapsto (f(x))_{x \in X} \in D.</math> | |
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| This map is injective and continuous, with ''B''<sub>1</sub>(''X*'') having the weak-* topology and ''D'' the product topology. Its inverse, defined on its range, is also continuous.
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| The theorem will be proved if the range of the above map is closed. But this is also clear. If one has a net
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| :<math>(f_{\alpha}(x))_{x \in X} \rightarrow (\lambda_x)_{x \in X}</math>
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| in ''D'', then the functional defined by
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| :<math>g(x) = \lambda_x \,</math>
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| lies in ''B''<sub>1</sub>(''X*'').
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| ==Consequences==
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| * 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 space|reflexive]].)
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| * 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.
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| * Closed and bounded sets in ''B(H)'' are precompact with respect to the [[weak operator topology]] (the WOT 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.
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| As a consequence, ''B(H)'' has the [[Heine–Borel theorem|Heine–Borel property]], if equipped with either the weak operator or the ultraweak topology.
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| * 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''=[[Lp space|''L''<sup>p</sup>(μ)]], 1<''p''<∞. Let ''f''<sub>n</sub> be a bounded sequence of functions in ''X''. Then there exists a subsequence ''f''<sub>n<sub>k</sub></sub> and an ''f'' ∈ ''X'' such that
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| :<math>\int f_{n_k} g\,d\mu \to \int f g\,d\mu</math> | |
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| for all ''g'' ∈ ''L''<sup>q</sup>(μ) = ''X''* (where 1/''p''+1/''q''=1). The corresponding result for ''p''=1 is not true, as ''L''<sup>1</sup>(μ) is not reflexive.
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| 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 [[André Weil|Weil]] that all [[locally compact]] [[Hausdorff space|Hausdorff]] topological vector spaces must be finite dimensional.
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| ==See also==
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| *[[Bishop–Phelps theorem]]
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| *[[Eberlein–Šmulian theorem]]
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| *[[Goldstine theorem]]
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| *[[James' theorem]]
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| *[[Mazur's lemma]]
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| ==Notes==
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| {{reflist}}
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| ==Further reading==
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| * {{cite book | author = John B. Conway | title = A course in functional analysis | publisher = Springer-Verlag | location = Berlin | year = 1994 | edition = 2nd | isbn = 0-387-97245-5}} See Chapter 5, section 3.
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| * {{Cite book
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| | last = Rudin
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| | first = W.
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| | title = Functional Analysis
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| | place = Boston, MA
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| | publisher = McGraw-Hill
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| | isbn = 0-07-054236-8
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| | edition = 2nd
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| | year = 1991
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| | postscript = <!--None--> }}. See section 3.15, p. 68.
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Banach-Alaoglu theorem}}
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| [[Category:Functional analysis]]
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| [[Category:Articles containing proofs]]
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| [[Category:Compactness theorems]]
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