en>Scetoaux: was missing a space between words

2015-01-11T05:11:59Z

was missing a space between words

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	~~In [[mathematics]], in the theory of [[Banach space]]s, '''Dvoretzky's theorem'''~~ is ~~an important structural theorem proved by [[Aryeh Dvoretzky]] in~~ the ~~early 1960s.<ref>{{Cite book \|first=A. \|last=Dvoretzky \|chapter=Some results~~ on ~~convex bodies and Banach spaces \|year=1961 \|title=Proc~~. ~~Internat~~. ~~Sympos~~. ~~Linear Spaces (Jerusalem, 1960) \|pages=123–160 \|publisher=Jerusalem Academic Press \|location=Jerusalem \|isbn= }}<~~/~~ref> It answered a question of [[Alexander Grothendieck~~]~~]. A new proof found by~~ [~~[Vitali Milman]] in the 1970s<ref>{{Cite journal \|first=V. D~~. ~~\|last=Milman \|title=A new proof of A~~. ~~Dvoretzky's theorem on cross-sections of convex bodies \|language=Russian \|journal=Funkcional. Anal. i Prilozhen. \|volume=5 \|year=1971 \|issue=4 \|pages=28–37 \|doi= }}<~~/~~ref> was one of the starting points for the development of [[asymptotic geometric analysis]] (also called ''asymptotic functional analysis'' or the ''local theory of [[Banach spaces]]'')~~.~~<ref>{{Cite book \|quote=The full significance of measure concentration was first realized by Vitali Milman in his revolutionary proof [Mil1971~~] ~~of the theorem of Dvoretzky ... Dvoretzky~~'s ~~theorem, especially as proved by Milman,~~ is ~~a milestone in the local (that~~ is, finite-dimensional) theory of Banach spaces. While I feel sorry for a mathematician who cannot see its intrinsic appeal, this appeal on its own does not explain the enormous influence that the proof has had, well beyond Banach space theory, as a result of planting the idea of measure concentration in the minds of many mathematicians. Huge numbers of papers have now been published exploiting this idea or giving new techniques for ~~showing that~~ it ~~holds. \|first=W. T~~. \|last=Gowers \|chapter=The two cultures of mathematics \|title=Mathematics: frontiers and perspectives \|pages=65–78 \|publisher=Amer. Math. Soc. \|location=Providence, RI \|year=2000 \|isbn=0-8218-2070-2 }}</ref>		Hi there. My name is Garland although it is not the title on my birth car warranty certification. [http://www.Volvocars.com/us/sales-services/sales/volvo_certified_preowned/Pages/default.aspx Managing] car [http://www.toyotacertified.com/warranty.html warranty people] is how I make cash and it's something I truly appreciate. What he really enjoys performing is to perform handball but he is struggling to discover time for it. Delaware is the only place I've been residing in.<br><br>Also visit [http://www.true-life.org/Activity-Feed/My-Profile/UserId/2850 extended auto warranty] car warranty my [http://Automobiles.honda.com/certified-pre-owned/warranty.aspx webpage extended] [http://www.amazinghostingsolutions.com/ActivityFeed/MyProfile/tabid/57/UserId/20310/Default.aspx auto warranty] ([http://kdfw.de.vu/index/users/view/id/17092 please click the up coming article])

	~~==Original formulation==~~

	For every natural number ''k'' ∈ '''N''' and every ''ε'' > 0 there exists ''N''(''k'', ''ε'') ∈ '''N''' such that if (''X'',  ‖.‖) is ~~a Banach space of dimension ''N''(''k'', ''ε''),there exist a subspace~~
	~~''E'' ⊂ ''X'' of dimension ''k'' and a positive [[quadratic form]]~~
	~~''Q'' on ''E'' such that~~ the ~~corresponding Euclidean norm~~

	~~:<math>\| \cdot \| = \sqrt{Q(\cdot)} </math>~~

	on '~~'E'' satisfies:~~

	:<~~math~~> ~~\|x\| \leq \\|x\\| \leq (1+\epsilon)\|x\| \quad \text{for every} \quad x \in E.~~<~~/math~~>

	~~==Further development==~~

	~~In 1971, [[Vitali Milman]] gave a new proof of Dvoretzky's theorem, making use of the~~ [[concentration of measure]] on the sphere to show that a random ''k''-dimensional subspace satisfies the above inequality with probability very close to 1. The proof gives the sharp dependence on ''k'':

	~~:<math>N(k,\epsilon)\leq\exp(C(\epsilon)k).<~~/~~math>~~

	~~Equivalently, for every Banach space (''X'',  ‖~~.~~‖) of dimension ''N'', there exists a subspace ''E'' ⊂ ''X'' of dimension~~
	~~''k'' ≥ ''c''(''ε'') log ''N'' and a Euclidean norm \|.\| on ''E'' such that the inequality above holds~~.

	~~More precisely, let ''S''<sup>''n'' − 1<~~/~~sup> be the unit sphere with respect to some Euclidean structure ''Q'', and let ''σ'' be the invariant probability measure on ''S''<sup>''n'' − 1<~~/~~sup>. Then:~~
	* There exists such a subspace ''E'' with

	~~:: <math>k = \dim E \geq c(\epsilon) \, \left(\frac{\int_{S^{n~~-~~1}} \\| \xi \\| \, d\sigma(\xi)}{\max_{\xi \in S^{n-1}} \\| \xi \\|}\right)^2 \, N. <~~/~~math>~~

	* For any ''X'' one may choose ''Q'' so that the term in the brackets will be at most

	:~~: <math> c_1 \sqrt{\frac{\log N}{N}}~~.</~~math>~~

	~~Here ''c''<sub>1<~~/~~sub> is a universal constant~~. ~~The best possible ''k'' is denoted ''k''<sub>*</sub>(''X'') and called the [[Dvoretzky dimension]~~] ~~of ''X''.~~

	~~The dependence on ''ε'' was studied by~~ [~~[Yehoram Gordon]],<ref>{{Cite journal \|first=Y~~. ~~\|last=Gordon \|title=Some inequalities for Gaussian processes and applications \|journal=Israel J~~. ~~Math. \|volume=50 \|year=1985 \|issue=4 \|pages=265–289 \|doi= }}<~~/ref><ref>{{Cite journal \|first=Y. \|last=Gordon \|title=Gaussian processes and almost spherical sections of convex bodies \|journal=Annals of Probability \|volume=16 \|year=1988 \|issue=1 \|pages=180–188 \|doi= }}</~~ref> who showed that ''k''<sub>*<~~/~~sub>(''X'') ≥ ''c''<sub>2<~~/~~sub> ''ε''<sup>2<~~/~~sup> log ''N''~~. ~~Another proof of this result was given by [[Gideon Schechtman]~~]~~.<ref>{{Cite book \|first=G. \|last=Schechtman \|chapter=A remark concerning the dependence on ε in Dvoretzky's theorem \|title=Geometric aspects of functional analysis~~ (~~1987–88) \|pages=274–277 \|series=Lecture Notes in Math. \|volume=1376 \|publisher=Springer \|location=Berlin \|year=1989 \|isbn=0-387-51303-5 }}<~~/~~ref>~~

	[[Noga Alon]] and [[Vitali Milman]] showed that the logarithmic bound on the dimension of the subspace in Dvoretzky's theorem can be significantly improved, if one is willing to accept a subspace that is close either to a Euclidean space or to a [[Chebyshev distance\|Chebyshev space]]. Specifically, for some constant ''c'', every ''n''-dimensional space has a subspace of dimension ''k'' ≥ exp(c√(log ''N'')) that is close either to ''ℓ''{{su\|b=2\|p=''k''}} or to ''ℓ''{{su\|b=∞\|p=''k''}}.~~<ref>{{citation~~
	~~\| last1 = Alon \| first1 = N~~. ~~\| author1-link = Noga Alon~~
	~~\| last2 = Milman \| first2 = V. D. \| author2-link = Vitali Milman~~
	~~\| doi = 10.1007~~/~~BF02804012~~
	~~\| mr = 720303~~
	~~\| issue = 4~~
	~~\| journal = Israel Journal of Mathematics~~
	~~\| pages = 265–280~~
	~~\| title = Embedding of <math>\scriptstyle l^{k}_{\infty}<~~/~~math> in finite-dimensional Banach spaces~~
	~~\| volume = 45~~
	~~\| year = 1983}}.<~~/~~ref>~~

	~~Important related results were proved by [[Tadeusz Figiel]~~], [[Joram Lindenstrauss]] and Milman.<ref>{{Cite journal \|first=T. \|last=Figiel \|first2=J. \|last2=Lindenstrauss \|first3=V. D. \|last3=Milman \|title=The dimension of almost spherical sections of convex bodies \|journal=Bull. Amer. Math. Soc. \|volume=82 \|year=1976 \|issue=4 \|pages=575–578 \|doi= }}, expanded in
	~~"The dimension of almost spherical sections of convex bodies",~~
	~~Acta Math. '''139''' (1977~~)~~, 53–94.</ref>~~

	~~==References==~~
	~~{{Reflist}}~~

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

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