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Statements

Subject Item: dbr:Ptak_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Robert_Phelps
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Schwartz_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Envelope_(category_theory)
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:List_of_functional_analysis_topics
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Montel_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Smith_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Barrelled_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Bornological_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Bounded_set_(topological_vector_space)
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Holomorphic_function
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:John_von_Neumann
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:DF-space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Uniform_boundedness_principle
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Ursescu_theorem
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Vector-valued_Hahn–Banach_theorems
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Inductive_tensor_product
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:LB-space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Nuclear_operators_between_Banach_spaces
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Complemented_subspace
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Complete_topological_vector_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Continuous_linear_operator
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

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dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Convex_set
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Measure_(mathematics)
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Gelfand_pair
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

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dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

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dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Transpose_of_a_linear_map
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Quasi-complete_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Quasi-interior_point
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Fréchet_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Fréchet–Urysohn_space
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Gateaux_derivative
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Bounded_operator
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Minkowski_functional
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Convex_hull
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Orlicz–Pettis_theorem
dbo:wikiPageWikiLink: dbr:Locally_convex_topological_vector_space

Subject Item: dbr:Locally_convex_space
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Subject Item: dbr:Locally_convex_topological_vector_space
rdf:type: yago:Space100028651 owl:Thing yago:Property113244109 yago:WikicatPropertiesOfTopologicalSpaces yago:Abstraction100002137 yago:Relation100031921 yago:WikicatTopologicalVectorSpaces yago:Attribute100024264 yago:Possession100032613 dbo:Building
rdfs:label: Espaço localmente convexo Przestrzeń liniowo-topologiczna lokalnie wypukła Lokaal convexe topologische vectorruimte 국소 볼록 공간 Espace localement convexe Locally convex topological vector space Spazio localmente convesso 局所凸位相ベクトル空間 Lokalkonvexer Raum Локально опуклий простір Espacio localmente convexo Локально выпуклое пространство
rdfs:comment: In matematica, uno spazio localmente convesso è uno spazio vettoriale topologico che generalizza il concetto di spazio normato. La topologia localmente convessa su uno spazio vettoriale topologico (reale o complesso) è una topologia formata da una base di insiemi convessi tale per cui le operazioni lineari sullo spazio sono continue. Non si tratta necessariamente di una topologia di Hausdorff. In de functionaalanalyse, een deelgebied van de wiskunde, is een lokaal convexe topologische vectorruimte een topologische vectorruimte waarin lokaal, dus in ieder punt willekeurig veel convexe omgevingen zijn. Equivalent daarmee is dat de topologie wordt voortgebracht door een familie van seminormen. 関数解析学および関連する数学の分野において、局所凸位相ベクトル空間（きょくしょとついそうベクトルくうかん、英: locally convex topological vector space）あるいは局所凸空間（locally convex space）は、ノルム空間を一般化する位相ベクトル空間（TVS）の例である。それらは、均衡かつ併呑な凸集合の平行移動によって位相が生成されるような位相ベクトル空間として定義される。または代わりに、それらは半ノルムの族を伴うベクトル空間として定義され、その族に関して位相を定義することが出来る。一般にこのような空間は必ずしもノルム化可能ではないが、零ベクトルに対する凸局所基の存在はハーン＝バナッハの定理の成立を保証する上で十分に強く、その結果として連続線型汎函数に関する豊富な理論がもたらされた。フレシェ空間は、距離化可能かつその距離に関して完備であるような局所凸空間である。それらは、ノルムに関する完備ベクトル空間であるようなバナッハ空間の一般化である。 En análisis funcional y en áreas relativas a las matemáticas, espacios vectoriales topológicos localmente convexos o espacios localmente convexos son ejemplos de espacios vectoriales topológicos los cuales generalizan los espacios normados. Pueden ser definidos como espacios vectoriales topológicos cuya topología es generada por transformaciones de equilibrio, absorbentes, conjuntos convexos. Paralelamente, pueden ser definidos como un espacio vectorial con una familia de seminormas y una topología puede ser definida en términos de esa familia. Aunque en general tales espacios no son necesariamente normables, la existencia de una base localmente convexa para el vector cero es lo suficientemente fuerte para sustentar el teorema de Hahn-Banach, produciendo así una teoría lo suficientemente r ( 비슷한 이름의 국소 볼록 집합에 관해서는 해당 문서를 참조하십시오.) 함수해석학에서 국소 볼록 공간(局所볼록空間, 영어: locally convex space)은 그 위상이 일련의 반노름들에 대한 시작 위상으로 유도되는 위상 벡터 공간이다.:§5, 38–49 함수해석학에서 다루는 가장 일반적인 공간 가운데 하나이다. Przestrzeń liniowo-topologiczna lokalnie wypukła – przestrzeń liniowo-topologiczna mająca złożoną ze zbiorów wypukłych. Ze względu na dobre własności jest to ważna klasa przestrzeni liniowo-topologicznych rozważanych w analizie funkcjonalnej. Em análise funcional e áreas da matemática relativas a ela, espaço localmente convexo ou espaço vetorial topológico localmente convexo é um espaço vectorial topológico que admite uma formada por conjuntos convexos. A importância do estudo destes espaço provém do fato que, embora não sejam necessariamente espaços normáveis, sua estrutura permite que se estabeleça o teorema da categoria de Baire e o teorema de Hahn-Banach. Lokalkonvexe Räume (genauer: lokalkonvexe topologische Vektorräume) sind im mathematischen Teilgebiet der Funktionalanalysis untersuchte topologische Vektorräume mit zusätzlichen Eigenschaften. Es handelt sich dabei um topologische Vektorräume, in denen jeder Punkt über „beliebig kleine“ konvexe Umgebungen verfügt. Alternativ können lokalkonvexe Räume auch als Vektorräume definiert werden, deren Topologie durch eine Familie von Halbnormen erzeugt wird. En mathématiques, un espace localement convexe est un espace vectoriel topologique dont la topologie peut être définie à l'aide d'une famille de semi-normes. C'est une généralisation de la notion d'espace normé. Локально выпуклое пространство — линейное топологическое пространство с системой полунорм, удовлетворяющей некоторым условиям. Локально опуклий простір — лінійний топологічний простір з системою напівнорм, що задовольняє деяким умовам. In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
rdfs:seeAlso: dbr:Topological_vector_space
dbp:name: Theorem
dcterms:subject: dbc:Convex_analysis dbc:Functional_analysis dbc:Topological_vector_spaces
dbo:wikiPageID: 518844
dbo:wikiPageRevisionID: 1124729345
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dbo:wikiPageExternalLink: n28:books%3Fid=7n9sAAAAMAAJ
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dbp:note: Kalton
dbo:abstract: Przestrzeń liniowo-topologiczna lokalnie wypukła – przestrzeń liniowo-topologiczna mająca złożoną ze zbiorów wypukłych. Ze względu na dobre własności jest to ważna klasa przestrzeni liniowo-topologicznych rozważanych w analizie funkcjonalnej. In matematica, uno spazio localmente convesso è uno spazio vettoriale topologico che generalizza il concetto di spazio normato. La topologia localmente convessa su uno spazio vettoriale topologico (reale o complesso) è una topologia formata da una base di insiemi convessi tale per cui le operazioni lineari sullo spazio sono continue. Non si tratta necessariamente di una topologia di Hausdorff. Da un punto di vista analitico uno spazio localmente convesso può essere caratterizzato considerando uno spazio vettoriale topologico nel quale è definita una famiglia di seminorme. Lo spazio viene detto localmente convesso se: La topologia naturale che caratterizza uno spazio localmente convesso è dunque la topologia più debole tale per cui le seminorme della famiglia sono funzioni continue, e continua è l'operazione di addizione. ( 비슷한 이름의 국소 볼록 집합에 관해서는 해당 문서를 참조하십시오.) 함수해석학에서 국소 볼록 공간(局所볼록空間, 영어: locally convex space)은 그 위상이 일련의 반노름들에 대한 시작 위상으로 유도되는 위상 벡터 공간이다.:§5, 38–49 함수해석학에서 다루는 가장 일반적인 공간 가운데 하나이다. 関数解析学および関連する数学の分野において、局所凸位相ベクトル空間（きょくしょとついそうベクトルくうかん、英: locally convex topological vector space）あるいは局所凸空間（locally convex space）は、ノルム空間を一般化する位相ベクトル空間（TVS）の例である。それらは、均衡かつ併呑な凸集合の平行移動によって位相が生成されるような位相ベクトル空間として定義される。または代わりに、それらは半ノルムの族を伴うベクトル空間として定義され、その族に関して位相を定義することが出来る。一般にこのような空間は必ずしもノルム化可能ではないが、零ベクトルに対する凸局所基の存在はハーン＝バナッハの定理の成立を保証する上で十分に強く、その結果として連続線型汎函数に関する豊富な理論がもたらされた。フレシェ空間は、距離化可能かつその距離に関して完備であるような局所凸空間である。それらは、ノルムに関する完備ベクトル空間であるようなバナッハ空間の一般化である。 En análisis funcional y en áreas relativas a las matemáticas, espacios vectoriales topológicos localmente convexos o espacios localmente convexos son ejemplos de espacios vectoriales topológicos los cuales generalizan los espacios normados. Pueden ser definidos como espacios vectoriales topológicos cuya topología es generada por transformaciones de equilibrio, absorbentes, conjuntos convexos. Paralelamente, pueden ser definidos como un espacio vectorial con una familia de seminormas y una topología puede ser definida en términos de esa familia. Aunque en general tales espacios no son necesariamente normables, la existencia de una base localmente convexa para el vector cero es lo suficientemente fuerte para sustentar el teorema de Hahn-Banach, produciendo así una teoría lo suficientemente rica de funcionales lineales continuos. Los espacios de Fréchet son espacios localmente convexos que están dotados de una métrica y son completos respecto a esta métrica. Son generalizaciones de los espacios de Banach, que a su vez son espacios vectoriales completos con respecto a una norma. Em análise funcional e áreas da matemática relativas a ela, espaço localmente convexo ou espaço vetorial topológico localmente convexo é um espaço vectorial topológico que admite uma formada por conjuntos convexos. A importância do estudo destes espaço provém do fato que, embora não sejam necessariamente espaços normáveis, sua estrutura permite que se estabeleça o teorema da categoria de Baire e o teorema de Hahn-Banach. In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. Локально опуклий простір — лінійний топологічний простір з системою напівнорм, що задовольняє деяким умовам. Lokalkonvexe Räume (genauer: lokalkonvexe topologische Vektorräume) sind im mathematischen Teilgebiet der Funktionalanalysis untersuchte topologische Vektorräume mit zusätzlichen Eigenschaften. Es handelt sich dabei um topologische Vektorräume, in denen jeder Punkt über „beliebig kleine“ konvexe Umgebungen verfügt. Alternativ können lokalkonvexe Räume auch als Vektorräume definiert werden, deren Topologie durch eine Familie von Halbnormen erzeugt wird. Ein lokalkonvexer Raum kann als eine Verallgemeinerung eines normierten Vektorraumes bzw. eines normierbaren Vektorraumes betrachtet werden, denn die Normkugeln um 0 sind konvexe Umgebungen des Nullpunktes. En mathématiques, un espace localement convexe est un espace vectoriel topologique dont la topologie peut être définie à l'aide d'une famille de semi-normes. C'est une généralisation de la notion d'espace normé. Локально выпуклое пространство — линейное топологическое пространство с системой полунорм, удовлетворяющей некоторым условиям. In de functionaalanalyse, een deelgebied van de wiskunde, is een lokaal convexe topologische vectorruimte een topologische vectorruimte waarin lokaal, dus in ieder punt willekeurig veel convexe omgevingen zijn. Equivalent daarmee is dat de topologie wordt voortgebracht door een familie van seminormen.
dbp:mathStatement: If is a TVS and if is a linear functional on , then is continuous if and only if there exists a continuous seminorm on such that Let be a Fréchet space over the field Then the following are equivalent: # does admit a continuous norm . # contains a vector subspace that is TVS-isomorphic to # contains a complemented vector subspace that is TVS-isomorphic to Suppose that is a vector space and let be a filter base of subsets of such that: # Every is convex, balanced, and absorbing; # For every there exists some real satisfying such that Then is a neighborhood base at 0 for a locally convex TVS topology on Let be a linear operator between TVSs where is locally convex . Then is continuous if and only if for every continuous seminorm on , there exists a continuous seminorm on such that Suppose that is a vector space and let be a non-empty collection of convex, balanced, and absorbing subsets of Then the set of all of all positive scalar multiples of finite intersections of sets in forms a neighborhood base at the origin for a locally convex TVS topology on Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.
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