On the dimension of the $l^{n}_{p}$-subspaces of Banach spaces, for $1\leq p<2$
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Abstract:
We give an estimate relating the stable type $p$ constant of a Banach space $X$ with the dimension of the $l_p^n$-subspaces of $X$. Precisely, let $C$ be this constant and assume $1 < p < 2$. We show that, for each $\varepsilon > 0,X$ must contain a subspace $(1 + \varepsilon )$-isomorphic to $l_p^k$, for every $k$ less than $\delta (\varepsilon ){C^{p’}}$ where $\delta (\varepsilon ) > 0$ is a number depending only on $p$ and $\varepsilon$.References
- D. Amir and V. D. Milman, Unconditional and symmetric sets in $n$-dimensional normed spaces, Israel J. Math. 37 (1980), no. 1-2, 3–20. MR 599298, DOI 10.1007/BF02762864
- Alejandro de Acosta, Inequalities for $B$-valued random vectors with applications to the strong law of large numbers, Ann. Probab. 9 (1981), no. 1, 157–161. MR 606806
- William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
- T. Figiel, J. Lindenstrauss, and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), no. 1-2, 53–94. MR 445274, DOI 10.1007/BF02392234
- Jørgen Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159–186. MR 356155, DOI 10.4064/sm-52-2-159-186
- William B. Johnson and Gideon Schechtman, Embedding $l^{m}_{p}$ into $l^{n}_{1}$, Acta Math. 149 (1982), no. 1-2, 71–85. MR 674167, DOI 10.1007/BF02392350
- J. L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), no. 1, 1–29. MR 407568, DOI 10.2307/1971054
- H. Lemberg, Nouvelle démonstration d’un théorème de J.-L. Krivine sur la finie représentation de $l_{p}$ dans un espace de Banach, Israel J. Math. 39 (1981), no. 4, 341–348 (French, with English summary). MR 636901, DOI 10.1007/BF02761678
- Raoul LePage, Michael Woodroofe, and Joel Zinn, Convergence to a stable distribution via order statistics, Ann. Probab. 9 (1981), no. 4, 624–632. MR 624688
- M. B. Marcus and G. Pisier, Characterizations of almost surely continuous $p$-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), no. 3-4, 245–301. MR 741056, DOI 10.1007/BF02392199
- Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90
- Haskell P. Rosenthal, On a theorem of J. L. Krivine concerning block finite representability of $l^{p}$ in general Banach spaces, J. Functional Analysis 28 (1978), no. 2, 197–225. MR 493384, DOI 10.1016/0022-1236(78)90086-1 V. V. Yurinski, Exponential bounds for large deviations, Theor. Probability Appl. 19 (1974), 154-155.
- V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37 (Russian). MR 0293374
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 276 (1983), 201-211
- MSC: Primary 46B20; Secondary 60B11
- DOI: https://doi.org/10.1090/S0002-9947-1983-0684503-5
- MathSciNet review: 684503