Dvoretzkys sats

Inom matematiken, i teorin av Banachrum, är Dvoretzkys sats ett viktigt strukturellt resultat bevisat av Aryeh Dvoretzky under tidiga 1960-talet.^[1] Den besvarar en fråga av Alexander Grothendieck. Ett nytt bevis upptäckt av Vitali Milman på 1970-talet^[2] var startpunkten för utvecklingen av asymptotisk geometrisk analys (även känt under namnen asymptotisk funktionalanalys och lokala teorin av Banachrum).^[3]

Ursprunglig formulering[redigera | redigera wikitext]

För varje naturligt tal k ∈ N och varje ε > 0 finns det N(k, ε) ∈ N så att om (X, ‖.‖) är ett Banachrum av dimension N(k, ε) finns det ett delrum E ⊂ X med dimension k och en positiv kvadratisk form Q över E så att den korresponderande euklidiska normen

|\cdot |={\sqrt {Q(\cdot )}}

över E satisfierar

|x|\leq \|x\|\leq (1+\epsilon )|x|\quad {\text{for every}}\quad x\in E.

Källor[redigera | redigera wikitext]

Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Dvoretzky's theorem, 7 juli 2014.

^ Dvoretzky, A. (1961). ”Some results on convex bodies and Banach spaces”. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960). Jerusalem: Jerusalem Academic Press. sid. 123–160
^ Milman, V. D. (1971). ”A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies” (på ryska). Funkcional. Anal. i Prilozhen. 5 (4): sid. 28–37.
^ Gowers, W. T. (2000). ”The two cultures of mathematics”. Mathematics: frontiers and perspectives. Providence, RI: Amer. Math. Soc. sid. 65–78. ISBN 0-8218-2070-2. ”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.”

[1] Dvoretzky, A. (1961). ”Some results on convex bodies and Banach spaces”. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960). Jerusalem: Jerusalem Academic Press. sid. 123–160

[2] Milman, V. D. (1971). ”A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies” (på ryska). Funkcional. Anal. i Prilozhen. 5 (4): sid. 28–37.

[3] Gowers, W. T. (2000). ”The two cultures of mathematics”. Mathematics: frontiers and perspectives. Providence, RI: Amer. Math. Soc. sid. 65–78. ISBN 0-8218-2070-2. ”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.”

[1]

[2]

[3]