On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝&lt;2

Abstract.Every m-dimensional subspace of Lp, 1 < p < 2, (1 + t)-embeds into I" as long as n > rjm1 +<1//'"(log m)"1. where r¡ = t¡( p. e) < oo. For subspaces of L¡ we get a somewhat weaker result.In [1 and 8] good estimates were obtained for the dimension of the largest l™, 1 ^ p < 2, space which embeds nicely into /" (in [1]) or into more general finitedimensional spaces (in [8]).In this note we are interested in obtaining a similar lower bound for the largest m such that every m-dimensional subspace of L 1 < p < 2, embeds nicely into /" (or into an arbitrary fixed Banach space).We obtain the following two theorems. STp(U)-the stable type p constant of U -is defined below. then any m-dimensional subspace X of L (0,1) (1 + a)-embeds into U (i.e. there exists a 1-1 operator Tfrom X onto a subspace of U with ||7,||||7'"1|| < 1 + a).Since STp(l")q = n we get that if n > tj • ml + l/p(]og m)~l then every mdimensional subspace of L (0,1) embeds into /". For subspaces of LX(Q, 1) we get a worse result.Theorem 2. For each a > 0 there exists a K = K(a) such that if n > mK then every m-dimensional subspace ofXj(0,1)(1 + a)-embeds into I".The proof of the two theorems is based on the proof of the main theorem in [8].We begin by reviewing some of the ideas and results of [8]. Let (Aj)j>l be an independent and identically distributed (i.i.d.) sequence of exponential random variables, i.e., P(Aj > t) = e'', t > 0, and let r. = H'n = lAn. Let (yj)j>i be any sequence of symmetric i.i.d. random variable in Lp(¡x), ¡x a probability measure, 1 < p < 2. Then X = YZy-1(TJ)~l/p ■ y¡ is a symmetric /7-stable random

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On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2

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References 13 publications

Probabilistic Limit Theorems in the Setting of Banach Spaces

Probabilistic Limit Theorems in the Setting of Banach Spaces

Fine embeddings of finite-dimensional subspaces of 𝐿_{𝑝},1≤𝑝<2, into 𝑙^{𝑚}₁

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