Quotients of Banach spaces of cotype 𝑞

Pisier, Gilles

doi:10.1090/S0002-9939-1982-0647892-8

Quotients of Banach spaces of cotype $q$
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by Gilles Pisier

Proc. Amer. Math. Soc. 85 (1982), 32-36

DOI: https://doi.org/10.1090/S0002-9939-1982-0647892-8

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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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