About: Quotient of subspace theorem

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In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: * The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant. * The induced norm || · || on E, defined by is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that for with K > 1 a universal constant.

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rdfs:label	Quotient of subspace theorem (en)
rdfs:comment	In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. Let (X, \|\|·\|\|) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: * The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant. * The induced norm \|\| · \|\| on E, defined by is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that for with K > 1 a universal constant. (en)
dcterms:subject	Banach spaces Theorems in functional analysis Asymptotic geometric analysis
Wikipage page ID	9176798 (xsd:integer)
Wikipage revision ID	987112011 (xsd:integer)
Link from a Wikipage to another Wikipage	Banach spaces Vitali Milman Theorems in functional analysis Vertical bar Asymptotic geometric analysis Normed space Isomorphism Quadratic form Quotient space (linear algebra)
sameAs	Quotient of subspace theorem Quotient of subspace theorem Quotient of subspace theorem Quotient of subspace theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Reflist dbt:Functional_analysis
has abstract	In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. Let (X, \|\|·\|\|) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: * The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant. * The induced norm \|\| · \|\| on E, defined by is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that for with K > 1 a universal constant. The statement is relative easy to prove by induction on the dimension of Z (even for Y=Z, X=0, c=1) with a K that depends only on N; the point of the theorem is that K is independent of N. In fact, the constant c can be made arbitrarily close to 1, at the expense of theconstant K becoming large. The original proof allowed (en)
prov:wasDerivedFrom	wikipedia-en:Quotient_of_subspace_theorem?oldid=987112011&ns=0
page length (characters) of wiki page	2597 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Quotient_of_subspace_theorem
is Link from a Wikipage to another Wikipage of	Vitali Milman List of theorems Quotient of Subspace theorem
is Wikipage redirect of	Quotient of Subspace theorem
is foaf:primaryTopic of	wikipedia-en:Quotient_of_subspace_theorem

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