About: Weak topology

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In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

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rdf:type	Thing yago:WikicatTopologicalVectorSpaces yago:Abstraction100002137 yago:Attribute100024264 yago:Space100028651
rdfs:label	Schwache Topologie (de) Topologie faible (fr) 약한 위상 (ko) 弱位相 (ja) Słaba topologia (pl) Topologia fraca (pt) Weak topology (en)
rdfs:comment	En mathématiques, la topologie faible d'un espace vectoriel topologique E est une topologie définie sur E au moyen de son dual topologique E'. On définit également sur E' une topologie dite faible-* au moyen de E. Dans tout cet article, sauf mention contraire, on notera ⟨φ, x⟩ := φ(x) pour x ∈ E et φ forme linéaire sur E. (fr) 弱位相（じゃくいそう、英: weak topology）とは、ノルム空間X上に定義される位相の一つである。体K上のノルム空間にはノルムから定まる位相（ノルム位相。弱位相と区別するため強位相とも呼ばれる）があるが、弱位相はこれよりも弱い（強くない）位相であり、X上のK値有界線形写像（すなわちXの共役空間X*の元）が全て連続になる最弱な位相である。なお弱位相は位相空間論におけるの特別な場合に当たる。強位相に関するものと区別するため、弱位相に関する連続性、収束性、コンパクト性はそれぞれ弱連続性、弱収束性、弱コンパクト性と呼ばれる。本項では弱位相の関連概念である＊弱位相についても述べる。 (ja) Em análise funcional, a topologia fraca em espaços vetoriais topológicos (como espaços de Banach) é a menor topologia que faz com que os funcionais lineares contínuos na topologia original permaneçam contínuos nessa nova topologia. Tendo menos abertos, terá mais compactos, o que pode simplificar algumas operações (como aquelas que envolvam maximizações de funções). (pt) Die schwache Topologie ist eine spezielle Topologie und im Grenzgebiet der beiden mathematischen Teilgebiete der Topologie und Funktionalanalysis anzusiedeln. Sie wird auf normierten Räumen oder allgemeiner auf lokalkonvexen Hausdorff-Räumen definiert. (de) In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. (en) Słaba topologia – alternatywna (w stosunku do wyjściowej) topologia na danej przestrzeni liniowo-topologicznej, będąca uogólnieniem idei zbieżności po współrzędnych (w przypadku przestrzeni skończenie wymiarowych słaba topologia pokrywa się z wyjściową topologią). jeśli jest (mocną) topologią w to słabą topologię oznacza się zwykle symbolem Innym sposobem wprowadzenia tej topologii jest podanie bazy otoczeń zera. (pl)
rdfs:seeAlso	Polar topology
dcterms:subject	Topology General topology Topology of function spaces
Wikipage page ID	33662 (xsd:integer)
Wikipage revision ID	1108332887 (xsd:integer)
Link from a Wikipage to another Wikipage	Scalar multiplication Rigged Hilbert space Topological complement Topologies on the set of operators on a Hilbert space David Hilbert Derivative Vector space Initial topology Compact space Complex number Complex numbers Continuous dual space Continuous function Continuous linear operator Holomorphic function Mathematics Convergence of measures Locally convex space Locally convex topological vector space Lp space Stefan Banach Closed set Comparison of topologies Functional analysis Orthonormal basis Polar set Banach space Banach–Alaoglu theorem Topological field Topological vector space Topologies on spaces of linear maps Weak-* topology Locally compact space Absolute value Topology Dual system Field (mathematics) Double dual Continuity (topology) Pointwise convergence Uniform convergence Linear functional Heine-Borel theorem Hilbert space Surjective General topology Bilinear map Eberlein compactum Weak convergence (Hilbert space) Distribution (mathematics) Topology of function spaces Marcel Riesz Polar topology Continuous function (topology) Metrizable TVS Metrizable Net (mathematics) Operator topology Real number Reflexive space Seminorm Sequence (mathematics) Strong operator topology Weak operator topology Riemann–Lebesgue lemma Locally convex Subbase Injective Normed vector space Space (mathematics) Vague topology Strong topology Test function Algebraic dual space Algebraic dual Compact set

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