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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>
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| ==Original formulation==
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| 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
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| ''E'' ⊂ ''X'' of dimension ''k'' and a positive [[quadratic form]]
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| ''Q'' on ''E'' such that the corresponding Euclidean norm
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| :<math>| \cdot | = \sqrt{Q(\cdot)} </math>
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| on ''E'' satisfies:
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| :<math> |x| \leq \|x\| \leq (1+\epsilon)|x| \quad \text{for every} \quad x \in E.</math>
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| ==Further development==
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| 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'':
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| :<math>N(k,\epsilon)\leq\exp(C(\epsilon)k).</math>
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| Equivalently, for every Banach space (''X'', ‖.‖) of dimension ''N'', there exists a subspace ''E'' ⊂ ''X'' of dimension
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| ''k'' ≥ ''c''(''ε'') log ''N'' and a Euclidean norm |.| on ''E'' such that the inequality above holds.
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| 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:
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| * There exists such a subspace ''E'' with
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| :: <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>
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| * For any ''X'' one may choose ''Q'' so that the term in the brackets will be at most
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| :: <math> c_1 \sqrt{\frac{\log N}{N}}.</math> | |
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| 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''.
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| 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>
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| [[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
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| | last1 = Alon | first1 = N. | author1-link = Noga Alon
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| | last2 = Milman | first2 = V. D. | author2-link = Vitali Milman
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| | doi = 10.1007/BF02804012
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| | mr = 720303
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| | issue = 4
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| | journal = Israel Journal of Mathematics
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| | pages = 265–280
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| | title = Embedding of <math>\scriptstyle l^{k}_{\infty}</math> in finite-dimensional Banach spaces
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| | volume = 45
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| | year = 1983}}.</ref>
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| 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
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| "The dimension of almost spherical sections of convex bodies",
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| Acta Math. '''139''' (1977), 53–94.</ref>
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| ==References==
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| {{Reflist}}
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| [[Category:Functional analysis]]
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| [[Category:Banach spaces]]
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| [[Category:Asymptotic geometric analysis]]
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| [[Category:Theorems in functional analysis]]
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