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A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
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Pisier's property $(\alpha)$
I suggest a paper of N. Kalton from 2006 on the Rademacher Decoupling property. Every space with Pisier's property Alpha has the decoupling property so it is more general and gets to the heart of such …