| dbo:abstract |
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after borné, the French word for "bounded". (en) Bornologische Räume sind in dem mathematischen Teilgebiet Funktionalanalysis spezielle lokalkonvexe Räume, für deren lineare Operatoren die aus der Theorie der normierten Räume bekannte Äquivalenz von Stetigkeit und Beschränktheit gilt. Diese Räume lassen sich durch ihre Nullumgebungsbasen charakterisieren und haben weitere Eigenschaften mit normierten Räumen gemeinsam. (de) 수학에서 유계형 집합(有界型集合, 영어: bornological set)은 유계 부분 집합들의 집합족이 명시된 집합이다. (ko) 数学、特に函数解析学における有界型空間（ゆうかいけいくうかん、ゆうかいがたくうかん界相空間（かいそうくうかん、英: bornological space; ボルノロジー空間）は、集合や函数の有界性の問題をある意味で考えるのに最低限必要な構造というものを抽出した空間のクラスである（これは位相空間が連続性の問題を考えるのに最低限必要な構造を抽出したものであったことと同様の考え方である）。界相空間を初めて考えたのはマッキーで、命名はブルバキによる（フランス語で有界を意味する borné (と位相 topology) に由来）。 (ja) |
| dbo:wikiPageID |
1804457 (xsd:integer) |
| dbo:wikiPageLength |
26252 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID |
1124682504 (xsd:integer) |
| dbo:wikiPageWikiLink |
dbr:Quasi-complete dbr:Montel_space dbr:Valued_field dbr:Barrelled_space dbr:Bounded_set_(topological_vector_space) dbc:Topological_vector_spaces dbr:LB-space dbr:Complete_topological_vector_space dbr:Continuous_dual dbr:Continuous_linear_operator dbr:Convex_set dbr:Mathematics dbr:Separable_space dbr:Fréchet_space dbr:George_Mackey dbr:Bounded_set dbr:Minkowski_functional dbr:Convex_hull dbr:Locally_convex_space dbr:Locally_convex_topological_vector_space dbr:Functional_analysis dbr:Mackey_space dbr:Mackey_topology dbr:Balanced_hull dbr:Balanced_set dbr:Banach_space dbr:Topological_vector_space dbr:Topologies_on_spaces_of_linear_maps dbr:Distinguished_space dbr:Hausdorff_space dbr:Absolutely_convex_set dbr:American_Mathematical_Society dbr:F-space dbr:Field_(mathematics) dbr:Nicolas_Bourbaki dbr:Normed_space dbr:Absorbing_set dbr:LF-space dbr:Bornivorous_set dbr:Ultrabornological_space dbr:Infrabarreled_space dbr:Metrizable_topological_vector_space dbr:Neighborhood_basis dbr:Reflexive_space dbr:Seminorm dbr:Sequential_space dbr:Locally_convex dbr:Topological_space dbr:Strong_dual_space dbr:Bornivorous dbr:Quasi-barrelled dbr:Continuous_map_(topology) dbr:Banach_disk dbr:Bounded_linear_operator dbr:Seminormed_space dbr:Ulam_measure |
| dbp:mathStatement |
The product of a collection locally convex bornological spaces is bornological if and only if does admit an Ulam measure. (en) Let and be locally convex TVS and let denote endowed with the topology induced by von Neumann bornology of Define similarly. Then a linear map is a bounded linear operator if and only if is continuous. Moreover, if is bornological, is Hausdorff, and is continuous linear map then so is If in addition is also ultrabornological, then the continuity of implies the continuity of where is the ultrabornological space associated with (en) |
| dbp:name |
Theorem (en) Mackey-Ulam theorem (en) |
| dbp:wikiPageUsesTemplate |
dbt:Annotated_link dbt:Cite_book dbt:Em dbt:Main dbt:Reflist dbt:Sfn dbt:Short_description dbt:Wikt-lang dbt:Bourbaki_Topological_Vector_Spaces_Part_1_Chapters_1–5 dbt:Berberian_Lectures_in_Functional_Analysis_and_Operator_Theory dbt:Conway_A_Course_in_Functional_Analysis dbt:Edwards_Functional_Analysis_Theory_and_Applications dbt:Functional_analysis dbt:Grothendieck_Topological_Vector_Spaces dbt:Hogbe-Nlend_Bornologies_and_Functional_Analysis dbt:Jarchow_Locally_Convex_Spaces dbt:Khaleelulla_Counterexamples_in_Topological_Vector_Spaces dbt:Köthe_Topological_Vector_Spaces_I dbt:Math_theorem dbt:Narici_Beckenstein_Topological_Vector_Spaces dbt:Schaefer_Wolff_Topological_Vector_Spaces dbt:Schechter_Handbook_of_Analysis_and_Its_Foundations dbt:Swartz_An_Introduction_to_Functional_Analysis dbt:Topological_vector_spaces dbt:Wilansky_Modern_Methods_in_Topological_Vector_Spaces dbt:Adasch_Topological_Vector_Spaces dbt:Boundedness_and_bornology |
| dct:subject |
dbc:Topological_vector_spaces |
| gold:hypernym |
dbr:Space |
| rdf:type |
yago:WikicatTopologicalVectorSpaces yago:Abstraction100002137 yago:Attribute100024264 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 yago:Space100028651 yago:WikicatPropertiesOfTopologicalSpaces |
| rdfs:comment |
Bornologische Räume sind in dem mathematischen Teilgebiet Funktionalanalysis spezielle lokalkonvexe Räume, für deren lineare Operatoren die aus der Theorie der normierten Räume bekannte Äquivalenz von Stetigkeit und Beschränktheit gilt. Diese Räume lassen sich durch ihre Nullumgebungsbasen charakterisieren und haben weitere Eigenschaften mit normierten Räumen gemeinsam. (de) 수학에서 유계형 집합(有界型集合, 영어: bornological set)은 유계 부분 집합들의 집합족이 명시된 집합이다. (ko) 数学、特に函数解析学における有界型空間（ゆうかいけいくうかん、ゆうかいがたくうかん界相空間（かいそうくうかん、英: bornological space; ボルノロジー空間）は、集合や函数の有界性の問題をある意味で考えるのに最低限必要な構造というものを抽出した空間のクラスである（これは位相空間が連続性の問題を考えるのに最低限必要な構造を抽出したものであったことと同様の考え方である）。界相空間を初めて考えたのはマッキーで、命名はブルバキによる（フランス語で有界を意味する borné (と位相 topology) に由来）。 (ja) In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. (en) |
| rdfs:label |
Bornologischer Raum (de) Bornological space (en) 有界型空間 (ja) 유계형 집합 (ko) |
| owl:sameAs |
freebase:Bornological space yago-res:Bornological space wikidata:Bornological space dbpedia-de:Bornological space dbpedia-ja:Bornological space dbpedia-ko:Bornological space https://global.dbpedia.org/id/5385u |
| prov:wasDerivedFrom |
wikipedia-en:Bornological_space?oldid=1124682504&ns=0 |
| foaf:isPrimaryTopicOf |
wikipedia-en:Bornological_space |
| is dbo:wikiPageRedirects of |
dbr:Bornological dbr:Bornological_convergence dbr:Mackey_complete_topological_vector_space dbr:Mackey_convergence dbr:Mackey_convergence_condition dbr:Mackey_convergent_sequence dbr:Von-Neumann_bornology dbr:Weak_Mackey_convergence |
| is dbo:wikiPageWikiLink of |
dbr:List_of_functional_analysis_topics dbr:Montel_space dbr:Bornological dbr:Bounded_set_(topological_vector_space) dbr:John_von_Neumann dbr:LB-space dbr:Pre-abelian_category dbr:Positive_linear_functional dbr:Complete_topological_vector_space dbr:Continuous_linear_operator dbr:Convenient_vector_space dbr:Quasibarrelled_space dbr:Fréchet_space dbr:George_Mackey dbr:Bounded_operator dbr:Locally_convex_topological_vector_space dbr:Locally_convex_vector_lattice dbr:Mackey_space dbr:Topological_vector_space dbr:Topologies_on_spaces_of_linear_maps dbr:Webbed_space dbr:Distinguished_space dbr:Family_of_sets dbr:LF-space dbr:Distribution_(mathematics) dbr:Bornivorous_set dbr:Bornology dbr:Polar_topology dbr:Spaces_of_test_functions_and_distributions dbr:Ultrabornological_space dbr:Metrizable_topological_vector_space dbr:Semi-abelian_category dbr:Sequentially_complete dbr:Vector_bornology dbr:Strong_dual_space dbr:Bornological_convergence dbr:Mackey_complete_topological_vector_space dbr:Mackey_convergence dbr:Mackey_convergence_condition dbr:Mackey_convergent_sequence dbr:Von-Neumann_bornology dbr:Weak_Mackey_convergence |
| is foaf:primaryTopic of |
wikipedia-en:Bornological_space |