About: Kuiper's theorem

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In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators. A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible, ie. all its homotopy groups are trivial. This result has important uses in topological K-theory.

Attributes	Values
rdf:type	yago:WikicatTheoremsInTopology yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Satz von Kuiper (de) Kuiper's theorem (en)
rdfs:comment	Der Satz von Kuiper ist ein mathematischer Lehrsatz, der im Übergangsfeld zwischen dem Gebiet der Funktionalanalysis und dem Gebiet der Topologie angesiedelt ist und der auf eine Arbeit des niederländischen Mathematikers Nicolaas Hendrik Kuiper aus dem Jahre 1965 zurückgeht. Kuiper behandelt hier Homotopieeigenschaften der Gruppe der invertierbaren beschränkten linearen Operatoren eines unendlich-dimensionalen separablen Hilbertraums und bestätigt mit seinem Satz eine von Michael Atiyah, Albert Solomonowitsch Schwarz und Richard Sheldon Palais aufgestellte Vermutung. (de) In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators. A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible, ie. all its homotopy groups are trivial. This result has important uses in topological K-theory. (en)
dcterms:subject	Hilbert space Operator theory Theorems in topology Topology of Lie groups K-theory
Wikipage page ID	2993692 (xsd:integer)
Wikipage revision ID	1032234181 (xsd:integer)
Link from a Wikipage to another Wikipage	Andrey Nikolayevich Tychonoff Norm topology Homogeneous space Hilbert space Operator theory Theorems in topology Mathematics General linear group Classifying space Gram-Schmidt process Bott periodicity theorem Bounded operator Topology of Lie groups Connected space Contractible space Homotopic Linear operator Shizuo Kakutani Unit sphere Maximal compact subgroup Banach space K-theory Adrien Douady Fibre bundle Brouwer fixed-point theorem Nicolaas Kuiper Mathematical folklore Hilbert space Invertible Jacques Dixmier Topological K-theory Homeomorphism Homotopy group Homotopy theory Topology (journal) Diffeomorphic CW complex Circle group Classifying space for U(n) Fredholm operator Michael Atiyah Strong operator topology Unitary group Vector bundle Space (mathematics) Weakly contractible Upper triangular matrices Matrix polar decomposition Separable Hilbert space Hilbert bundle Homotopy long exact sequence Homotopy type Trivial bundle Weak homotopy equivalence dbr:Klaus_Jänich
sameAs	Kuiper's theorem Kuiper's theorem Kuiper's theorem Kuiper's theorem Kuiper's theorem
dbp:wikiPageUsesTemplate	dbt:Cite_journal dbt:Reflist dbt:Short_description
has abstract	Der Satz von Kuiper ist ein mathematischer Lehrsatz, der im Übergangsfeld zwischen dem Gebiet der Funktionalanalysis und dem Gebiet der Topologie angesiedelt ist und der auf eine Arbeit des niederländischen Mathematikers Nicolaas Hendrik Kuiper aus dem Jahre 1965 zurückgeht. Kuiper behandelt hier Homotopieeigenschaften der Gruppe der invertierbaren beschränkten linearen Operatoren eines unendlich-dimensionalen separablen Hilbertraums und bestätigt mit seinem Satz eine von Michael Atiyah, Albert Solomonowitsch Schwarz und Richard Sheldon Palais aufgestellte Vermutung. (de) In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators. A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible, ie. all its homotopy groups are trivial. This result has important uses in topological K-theory. (en)
gold:hypernym	Result
prov:wasDerivedFrom	wikipedia-en:Kuiper's_theorem?oldid=1032234181&ns=0
page length (characters) of wiki page	7980 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Kuiper's_theorem
is Link from a Wikipage to another Wikipage of	Lie group General linear group Kuiper Adrien Douady Nicolaas Kuiper Hilbert manifold Contractibility of unit sphere in hilbert space List of theorems Kuiper theorem Atiyah-Jänich theorem Atiyah–Jänich theorem Contractibility of unit sphere in Hilbert space
is Wikipage redirect of	Contractibility of unit sphere in hilbert space

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