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- Banach–Alaoglu_theorem abstract "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. 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.A proof of this theorem for 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.Since the Banach–Alaoglu theorem is proven via Tychonoff's theorem, it relies on the 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.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.".
- Banach–Alaoglu_theorem wikiPageID "577366".
- Banach–Alaoglu_theorem wikiPageLength "10049".
- Banach–Alaoglu_theorem wikiPageOutDegree "61".
- Banach–Alaoglu_theorem wikiPageRevisionID "704578056".
- Banach–Alaoglu_theorem wikiPageWikiLink André_Weil.
- Banach–Alaoglu_theorem wikiPageWikiLink Arzelà–Ascoli_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Axiom_of_choice.
- Banach–Alaoglu_theorem wikiPageWikiLink Ball_(mathematics).
- Banach–Alaoglu_theorem wikiPageWikiLink Bishop–Phelps_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Calculus_of_variations.
- Banach–Alaoglu_theorem wikiPageWikiLink Cantors_diagonal_argument.
- Banach–Alaoglu_theorem wikiPageWikiLink Cartesian_product.
- Banach–Alaoglu_theorem wikiPageWikiLink Category:Articles_containing_proofs.
- Banach–Alaoglu_theorem wikiPageWikiLink Category:Compactness_theorems.
- Banach–Alaoglu_theorem wikiPageWikiLink Category:Functional_analysis.
- Banach–Alaoglu_theorem wikiPageWikiLink Closed_set.
- Banach–Alaoglu_theorem wikiPageWikiLink Compact_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Delta-convergence.
- Banach–Alaoglu_theorem wikiPageWikiLink Dual_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Dual_topology.
- Banach–Alaoglu_theorem wikiPageWikiLink Eberlein–Šmulian_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Functional_analysis.
- Banach–Alaoglu_theorem wikiPageWikiLink Goldstine_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Hahn–Banach_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Hausdorff_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Heine–Borel_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Hellys_selection_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink James_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Krein–Milman_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Leonidas_Alaoglu.
- Banach–Alaoglu_theorem wikiPageWikiLink Locally_compact_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Locally_convex_topological_vector_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Lp_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Mathematics.
- Banach–Alaoglu_theorem wikiPageWikiLink Mazurs_lemma.
- Banach–Alaoglu_theorem wikiPageWikiLink Metrization_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Neighbourhood_(mathematics).
- Banach–Alaoglu_theorem wikiPageWikiLink Nicolas_Bourbaki.
- Banach–Alaoglu_theorem wikiPageWikiLink Normed_vector_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Partial_differential_equation.
- Banach–Alaoglu_theorem wikiPageWikiLink Polar_set.
- Banach–Alaoglu_theorem wikiPageWikiLink Product_topology.
- Banach–Alaoglu_theorem wikiPageWikiLink Reflexive_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Riesz_representation_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Rieszs_lemma.
- Banach–Alaoglu_theorem wikiPageWikiLink Semi-continuity.
- Banach–Alaoglu_theorem wikiPageWikiLink Separable_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Sequentially_compact_space.
- Banach–Alaoglu_theorem wikiPageWikiLink Stefan_Banach.
- Banach–Alaoglu_theorem wikiPageWikiLink Strong_topology.
- Banach–Alaoglu_theorem wikiPageWikiLink Trace_class.
- Banach–Alaoglu_theorem wikiPageWikiLink Tychonoffs_theorem.
- Banach–Alaoglu_theorem wikiPageWikiLink Ultraweak_topology.
- Banach–Alaoglu_theorem wikiPageWikiLink Weak_operator_topology.
- Banach–Alaoglu_theorem wikiPageWikiLink Weak_topology.
- Banach–Alaoglu_theorem wikiPageWikiLink Weak_topology_(polar_topology).
- Banach–Alaoglu_theorem wikiPageWikiLink Zermelo–Fraenkel_set_theory.
- Banach–Alaoglu_theorem wikiPageWikiLinkText "Banach–Alaoglu theorem".
- Banach–Alaoglu_theorem wikiPageWikiLinkText "sequential Banach–Alaoglu theorem".
- Banach–Alaoglu_theorem wikiPageUsesTemplate Template:Cite_book.
- Banach–Alaoglu_theorem wikiPageUsesTemplate Template:Functional_Analysis.
- Banach–Alaoglu_theorem wikiPageUsesTemplate Template:Reflist.
- Banach–Alaoglu_theorem subject Category:Articles_containing_proofs.
- Banach–Alaoglu_theorem subject Category:Compactness_theorems.
- Banach–Alaoglu_theorem subject Category:Functional_analysis.
- Banach–Alaoglu_theorem type Function.
- Banach–Alaoglu_theorem type Proof.
- Banach–Alaoglu_theorem type Redirect.
- Banach–Alaoglu_theorem type Theorem.
- Banach–Alaoglu_theorem comment "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. 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.".
- Banach–Alaoglu_theorem label "Banach–Alaoglu theorem".
- Banach–Alaoglu_theorem sameAs Q537618.
- Banach–Alaoglu_theorem sameAs Satz_von_Banach-Alaoglu.
- Banach–Alaoglu_theorem sameAs Alaoglun_lause.
- Banach–Alaoglu_theorem sameAs Théorème_de_Banach-Alaoglu-Bourbaki.
- Banach–Alaoglu_theorem sameAs משפט_בנך-אלאוגלו.
- Banach–Alaoglu_theorem sameAs Teorema_di_Banach-Alaoglu.
- Banach–Alaoglu_theorem sameAs バナッハ=アラオグルの定理.
- Banach–Alaoglu_theorem sameAs Stelling_van_Banach-Alaoglu.
- Banach–Alaoglu_theorem sameAs Twierdzenie_Banacha-Alaoglu.
- Banach–Alaoglu_theorem sameAs m.02rv4g.
- Banach–Alaoglu_theorem sameAs Теорема_Банаха-Алаоглу.
- Banach–Alaoglu_theorem sameAs Banach–Alaoglus_sats.
- Banach–Alaoglu_theorem sameAs Q537618.
- Banach–Alaoglu_theorem wasDerivedFrom Banach–Alaoglu_theorem?oldid=704578056.
- Banach–Alaoglu_theorem isPrimaryTopicOf Banach–Alaoglu_theorem.