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2012-06-11T11:16:46Z

See also: remove redlink

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In mathematics, the '''quotient of subspace theorem''' is an important property of finite dimensional [[normed space]]s, discovered by [[Vitali Milman]].<ref>The original proof appeared in {{harvtxt|Milman|1984}}. See also {{harvtxt|Pisier|1989}}.</ref>

Let (''X'', ||·||) be an ''N''-dimensional normed space. There exist subspaces ''Z'' ⊂ ''Y'' ⊂ ''X'' such that the following holds:
* The [[quotient space (linear algebra)|quotient space]] ''E'' = ''Y'' / ''Z'' is of dimension dim E ≥ ''c'' ''N'', where ''c'' > 0 is a universal constant.
* The induced [[norm (mathematics)|norm]] || · || on ''E'', defined by

:: <math>\| e \| =\min_{y \in e} \| y \|, \quad e \in E, </math>

is [[isomorphism|isomorphic]] to Euclidean. That is, there exists a positive [[quadratic form]] ("Euclidean structure") ''Q'' on ''E'', such that

:: <math>\frac{\sqrt{Q(e)}}{K} \leq \| e \| \leq K \sqrt{Q(e)}</math> for <math>e \in E,</math>

:with ''K'' > 1 a universal constant.

In fact, the constant ''c'' can be made arbitrarily close to 1, at the expense of the
constant ''K'' becoming large. The original proof allowed

:<math> c(K) \approx 1 - \text{const} / \log \log K. </math><ref>See references for improved estimates.</ref>

==Notes==
{{Reflist}}

== References ==
* {{citation|last=Milman|first=V.D.|author-link=Vitali Milman|title=Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space|journal=Israel seminar on geometrical aspects of functional analysis|volume=X|publisher=Tel Aviv Univ.|location=Tel Aviv|year=1984}}
* {{citation|first=Y.|last= Gordon|title=On Milman's inequality and random subspaces which escape through a mesh in '''R'''<sup>''n''</sup>|journal=Geometric aspects of functional analysis|pages=84–106|series=Lecture Notes in Math.|volume=1317|publisher=Springer|location=Berlin|year=1988|doi=10.1007/BFb0081737|isbn=978-3-540-19353-1}}
* {{citation|first=G.|last=Pisier|author-link=Gilles Pisier|title=The volume of convex bodies and Banach space geometry|series=Cambridge Tracts in Mathematics|volume=94|publisher=Cambridge University Press|location=Cambridge|year=1989}}

[[Category:Functional analysis]]
[[Category:Banach spaces]]
[[Category:Asymptotic geometric analysis]]
[[Category:Theorems in functional analysis]]

Watterson estimator - Revision history

en>Melcombe: /* See also */ remove redlink