Sign-embeddings of $l^{n}_{1}$
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- by John Elton PDF
- Trans. Amer. Math. Soc. 279 (1983), 113-124 Request permission
Abstract:
If $({e_i})_{i = 1}^n$ are vectors in a real Banach space with $\parallel {e_i}\parallel \leqslant 1$ and $\text {Average}_{{\varepsilon _1} = \pm 1}\parallel \sum \nolimits _{i = 1}^n {{\varepsilon _i}{e_i}\parallel \geqslant \delta n}$, where $\delta > 0$, then there is a subset $A \subseteq \{ 1,\ldots ,n\}$ of cardinality $m \geqslant cn$ such that ${({e_i})_{i \in A}}$ is $K$-equivalent to the standard $l_1^m$ basis, where $c > 0$ and $K < \infty$ depend only on $\delta$. As a corollary, if $1 < p < \infty$ and $l_1^n$ is $K$-isomorphic to a subspace of ${L_p}(X)$, then $l_1^m(m \geqslant cn)$ is $K’$-isomorphic to a subspace of $X$, where $c > 0$ and $K’ < \infty$ depend only on $K$ and $p$.References
- J. Bourgain and H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), no. 2, 149–188. MR 707202, DOI 10.1016/0022-1236(83)90080-0
- Herman Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statistics 23 (1952), 493–507. MR 57518, DOI 10.1214/aoms/1177729330 W. B. Johnson and G. Schechtman, Embedding $l_p^m$ into $l_1^n$ (to appear).
- M. G. Karpovsky and V. D. Milman, Coordinate density of sets of vectors, Discrete Math. 24 (1978), no. 2, 177–184. MR 522926, DOI 10.1016/0012-365X(78)90197-8
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367 M. Loève, Probability theory. I, 4th ed., Springer-Verlag, Berlin and New York, 1977. V. D. Milman, Some remarks about embeddings of $l_1^k$ in finite dimensional spaces (to appear).
- Gilles Pisier, De nouvelles caractérisations des ensembles de Sidon, Mathematical analysis and applications, Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 685–726 (French, with English summary). MR 634264
- Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411 —, Sign-embeddings of ${L^1}$ (to appear).
- N. Sauer, On the density of families of sets, J. Combinatorial Theory Ser. A 13 (1972), 145–147. MR 307902, DOI 10.1016/0097-3165(72)90019-2
- Gideon Schechtman, Random embeddings of Euclidean spaces in sequence spaces, Israel J. Math. 40 (1981), no. 2, 187–192. MR 634905, DOI 10.1007/BF02761909
- Saharon Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247–261. MR 307903
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 279 (1983), 113-124
- MSC: Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9947-1983-0704605-4
- MathSciNet review: 704605