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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Quotients of Banach spaces of cotype $ q$


Author: Gilles Pisier
Journal: Proc. Amer. Math. Soc. 85 (1982), 32-36
MSC: Primary 46B20; Secondary 28C20, 60B11
MathSciNet review: 647892
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Abstract: Let $ Z$ be a Banach space and let $ X \subset Z$ be a $ B$-convex subspace (equivalently, assume that $ X$ does not contain $ l_1^n$'s uniformly). Then every Bernoulli series $ \Sigma _{n = 1}^\infty {\varepsilon _n}{z_n}$ which converges almost surely in the quotient $ Z/X$ can be lifted to a Bernoulli series a.s. convergent in $ Z$. As a corollary, if $ Z$ is of cotype $ q$, then $ Z/X$ is also of cotype $ q$. This extends a result of [4] concerning the particular case $ Z = {L_1}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0647892-8
PII: S 0002-9939(1982)0647892-8
Keywords: Banach space of cotype $ q$, $ B$-convexity, Bernoulli series, sum of independent, Banach space valued random variables
Article copyright: © Copyright 1982 American Mathematical Society