Banach-Alaoglu theorem

The Banach-Alaoglu Theorem (also Alaoglu Theorem or Alaoglu-Bourbaki Theorem or in a more general version Banach-Alaoglu-Bourbaki Theorem ) is a compactness theorem and is generally assigned to the field of functional analysis , although it is a purely topological proposition and essentially follows from Tychonoff's theorem.

It is named after Stefan Banach and Leonidas Alaoglu .

The sentence

Let it be a normalized space and its topological dual space . Then the crowd ${\displaystyle E}$ ${\displaystyle E^{\,\prime }}$

${\displaystyle M:=\{\varphi \in E^{\,\prime }\,|\,\|\varphi \|_{E^{\,\prime }}\leq 1\}}$

compact in terms of the weak - * topology in . ${\displaystyle E^{\,\prime }}$

discussion

The meaning of this statement results above all from the comparison with Riesz's lemma , according to which the norm-closed unit sphere of a normalized space is compact with respect to the norm topology if and only if the space has finite dimensions . The topological dual space , i.e. the space of all continuous linear functionals in a normalized space , is itself able to normalize again ${\displaystyle E^{\,\prime }}$ ${\displaystyle E}$

${\displaystyle \|\varphi \|_{E^{\,\prime }}:=\sup\{|\varphi (x)|\,|\,x\in E,\,\|x\|\leq 1\},\quad \varphi \in E'.}$

The standardized unit sphere in is precisely the set . Mit is also of infinite vector space dimension. Applied to, it follows from Riesz's lemma that the case is not norm-compact. But it is compact in the weaker weak - * topology. ${\displaystyle E^{\,\prime }}$ ${\displaystyle M}$ ${\displaystyle E}$ ${\displaystyle E^{\,\prime }}$ ${\displaystyle E^{\,\prime }}$ ${\displaystyle M}$ ${\displaystyle \dim E=\infty }$ ${\displaystyle M}$

Please note at this point that the norm of is used for the construction of , but the compactness does not apply in the norm topology, but in the weak - * - topology. ${\displaystyle M}$ ${\displaystyle E^{\,\prime }}$

In connection with the above comparison, the classification of the Banach-Alaoglu theorem in the field of functional analysis can be justified, because the statement is nontrivial only when the underlying normalized space is infinite ( and with the above norm, finite-dimensional areas are topologically isomorphic , and the weak - * - topology is the same as the standard topology). ${\displaystyle E}$ ${\displaystyle E^{\,\prime }}$

Note that the Banach-Alaoglu theorem does not imply the local compactness of the weak - * - topology, because this is coarser than the norm topology and the closed unit sphere is not a null neighborhood. Every locally compact topological vector space is finitely dimensional.

application

Compact sets are always of great importance in (functional) analysis. Since they are rather rare in infinite-dimensional normalized spaces (according to Riesz's lemma mentioned above and, more generally, non-local compactness), the change to the weaker * topology does not mean any major restriction in many situations or this topology naturally comes into play comes, one of these sentences is a wealth of “new” compact quantities at hand. A prominent example is the proof of Gelfand-Neumark's theorem from the theory of C * -algebras , which produces an isometric isomorphism between any commutative C * -algebra and the continuous functions on a compact set . The compactness of the set follows from an application of the Banach-Alaoglu theorem. ${\displaystyle A}$ ${\displaystyle C(\Gamma _{A})}$ ${\displaystyle \Gamma _{A}}$ ${\displaystyle \Gamma _{A}}$

In addition, the Banach-Alaoglu theorem is a central element of the proof of the fundamental theorem of Young measures . It allows a weak - * - convergent subsequence to be selected from a sequence of atomic measures.

Generalizations and other formulations

Generalization: Alaoglu-Bourbaki theorem

The Banach-Alaoglu theorem can be formulated for more general topological vector spaces.

Let be a locally convex space . For a neighborhood in is ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle X}$

${\displaystyle U^{\circ }:=\{\varphi \in X^{\,\prime }\,|\,\mathrm {Re} (\varphi (x))\leq 1\ \forall \,x\in U\}}$

(the so-called polar of ) a weakly - * - compact set. ${\displaystyle U}$

For Banach rooms

The unit sphere in the dual space of a Banach space is weak - * - compact. ${\displaystyle B^{*}:=\{x^{*}\in X^{*}\,|\,\|x^{*}\|\leq 1\}}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$

For separable Banach rooms

The unit sphere in the dual space of a separable Banach space is compact with the weak - * - topology and also weak - * - metrizable , which is why it is also weak - * - sequence-compact. Ie a sequence has a weakly * convergent subsequence with a limit value in . ${\displaystyle B^{*}:=\{x^{*}\in X^{*}\,|\,\|x^{*}\|\leq 1\}}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }\subseteq B^{*}}$ ${\displaystyle B^{*}}$

literature

• Dirk Werner : Functional Analysis. Springer, Berlin 1995, ISBN 3-540-59168-0 , p. 335 f.
• Herbert Schröder: Functional Analysis. 2nd Edition. German, Frankfurt am Main 2000, ISBN 3-8171-1623-3 , pp. 93 f.
• Klaus Jänich: Topology. 4th edition. Springer, Berlin 1994, ISBN 3-540-57471-9 , p. 201 f.

Web links

• László Erdős: Banach-Alaoglu theorems. (PDF; 94 kB) Accessed August 16, 2013 . 

Individual evidence

1. ^ Nicolas Bourbaki : V. Topological Vector Spaces (=  Elements of Mathematics ). Springer , Berlin 2003, ISBN 3-540-42338-9 , I, p. 15 (Original title: Éspaces vectoriels topologiques . Paris 1981. Translated by HG Eggleston and S. Madan).
2. ↑ Stefan Müller: Variational models for microstructure and phase transitions . In: Calculus of Variations and Geometric Evolution Problems: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Cetraro, Italy, June 15-22, 1996 (=  Lecture Notes in Mathematics ). Springer Berlin Heidelberg, Berlin, Heidelberg 1999, ISBN 978-3-540-48813-2 , pp. 85-210 , doi : 10.1007 / bfb0092670 .
3. ^ Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5