About: Complemented subspace

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In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (topological) complement in , such that is the direct sum in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result preserves many nice properties from the operation of direct sum in finite-dimensional vector spaces.

Attributes	Values
rdf:type	Thing
rdfs:label	Complemented subspace (en) Subespaço complementado (pt)
rdfs:comment	Em análise funcional, um subespaço fechado de um espaço vetorial normado é dito complementado em se existe um subespaço fechado tal que . Uma motivação para o estudo de espaços complementados é o seguinte resultado: se é espaço de Banach e é complementado em , com complemento , então é homeomorfo a com a topologia produto. (pt) In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (topological) complement in , such that is the direct sum in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result preserves many nice properties from the operation of direct sum in finite-dimensional vector spaces. (en)
rdfs:seeAlso	Coproduct
dcterms:subject	Functional analysis
Wikipage page ID	63033543 (xsd:integer)
Wikipage revision ID	1100860620 (xsd:integer)
Link from a Wikipage to another Wikipage	Cantor-Schroder-Bernstein theorem Complement (set theory) Complete topological vector space Continuous function Continuous linear map Mathematics Closure (topology) Fréchet space Morphism Context (language use) Coproduct Locally convex topological vector space Bijective Closed set Functional analysis Banach space Additive group Topological group Topological vector space Linear map List of Banach spaces Normed space Direct sum Direct sum of modules Direct sum of topological groups Product (category theory) Projection (mathematics) Hilbert space Hamel bases Functional analysis Codimension Homeomorphism Open map Orthogonal complement Category of vector spaces Up to Image (mathematics) Separating set Restriction map Free vector space Vector space isomorphism Vector subspace Direct sum of Banach spaces Direct sum of topological vector spaces Indiscrete topology Quotient map Quotient vector space Identity map Lebesgue decomposition Bounded linear map Surjection TVS-isomorphism Tim Gowers dbr:Absolutely_summing_operator
sameAs	Complemented subspace Complemented subspace Complemented subspace
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group	proof (en)
has abstract	In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (topological) complement in , such that is the direct sum in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result preserves many nice properties from the operation of direct sum in finite-dimensional vector spaces. Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not. In general, classifying all complemented subspaces is a difficult problem, which has been solved only for some well-known Banach spaces. The concept of a complemented subspace is analogous to, but distinct from, that of a set complement. The set-theoretic complement of a vector subspace is never a complementary subspace. (en) Em análise funcional, um subespaço fechado de um espaço vetorial normado é dito complementado em se existe um subespaço fechado tal que . Uma motivação para o estudo de espaços complementados é o seguinte resultado: se é espaço de Banach e é complementado em , com complemento , então é homeomorfo a com a topologia produto. (pt)
prov:wasDerivedFrom	wikipedia-en:Complemented_subspace?oldid=1100860620&ns=0
page length (characters) of wiki page	20745 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Complemented_subspace
is Link from a Wikipage to another Wikipage of	Topological complement Topologically complemented Complete topological vector space Fréchet space Locally convex topological vector space Banach space Topological direct sum Topological vector space Direct sum

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