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In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers or complex numbers such that is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space. The space of all sequences such that the series is convergent (possibly conditionally) is denoted by cs. This is a closed vector subspace of bs, and so is also a Banach space with the same norm.

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rdf:type	yago:WikicatBanachSpaces yago:Abstraction100002137 yago:Attribute100024264 yago:Space100028651
rdfs:label	Bs space (en) 有界級数空間 (ja)
rdfs:comment	数学の函数解析学の分野における有界級数（ゆうかいきゅうすう、英: bounded series）の空間 bs は、その部分和（series; 有限級数）の列が有界 (bounded) となるような実または複素無限数列全体の成す数列空間としてで与えられる。この空間 bs は項ごとの和とスカラー倍に関してベクトル空間を成し、ノルム ‖ • ‖bs を与えてノルム空間の構造を持つ。さらに bs はこのノルムの誘導する距離に関して完備、従ってバナッハ空間となる。 bs の部分空間として、収斂級数 (convergent series) の空間 csは、その和（無限級数）が収斂（でもよい）する無限数列全体の成す数列空間を言う。cs は、バナッハ空間 bs の（ノルム ‖ • ‖bs に関する）閉部分空間となるから、それ自身バナッハ空間を成す。空間 bs は有界数列の空間 ℓ∞ に、写像を通じて等距同型であり、さらに同じ写像 T によって cs は収斂数列の空間 c に等距同型となる。 (ja) In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers or complex numbers such that is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space. The space of all sequences such that the series is convergent (possibly conditionally) is denoted by cs. This is a closed vector subspace of bs, and so is also a Banach space with the same norm. (en)
dcterms:subject	Banach spaces Functional analysis
Wikipage page ID	17555165 (xsd:integer)
Wikipage revision ID	1034623230 (xsd:integer)
Link from a Wikipage to another Wikipage	Banach spaces Vector space Complex numbers Mathematics Conditional convergence Convergent series Closed set Complete metric space Functional analysis Banach space Normed space Isomorphism Isometry Functional analysis C space Real numbers Sequence (mathematics) Series (mathematics) Metric (mathematics) Vector subspace Componentwise operation Space of bounded sequences Space of convergent sequences
sameAs	Bs space Bs space Bs space Bs space Bs space
dbp:wikiPageUsesTemplate	dbt:Banach_spaces dbt:Annotated_link dbt:Citation dbt:Reflist dbt:Functional_analysis dbt:Mathanalysis-stub
has abstract	In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers or complex numbers such that is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space. The space of all sequences such that the series is convergent (possibly conditionally) is denoted by cs. This is a closed vector subspace of bs, and so is also a Banach space with the same norm. The space bs is isometrically isomorphic to the Space of bounded sequences via the mapping Furthermore, the space of convergent sequences c is the image of cs under (en) 数学の函数解析学の分野における有界級数（ゆうかいきゅうすう、英: bounded series）の空間 bs は、その部分和（series; 有限級数）の列が有界 (bounded) となるような実または複素無限数列全体の成す数列空間としてで与えられる。この空間 bs は項ごとの和とスカラー倍に関してベクトル空間を成し、ノルム ‖ • ‖bs を与えてノルム空間の構造を持つ。さらに bs はこのノルムの誘導する距離に関して完備、従ってバナッハ空間となる。 bs の部分空間として、収斂級数 (convergent series) の空間 csは、その和（無限級数）が収斂（でもよい）する無限数列全体の成す数列空間を言う。cs は、バナッハ空間 bs の（ノルム ‖ • ‖bs に関する）閉部分空間となるから、それ自身バナッハ空間を成す。空間 bs は有界数列の空間 ℓ∞ に、写像を通じて等距同型であり、さらに同じ写像 T によって cs は収斂数列の空間 c に等距同型となる。 (ja)
prov:wasDerivedFrom	wikipedia-en:Bs_space?oldid=1034623230&ns=0
page length (characters) of wiki page	1860 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Bs_space
is Link from a Wikipage to another Wikipage of	Sequence space Cs space
is Wikipage redirect of	Cs space
is foaf:primaryTopic of	wikipedia-en:Bs_space

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