Absolutely representing systems, uniform smoothness, and type

Abstract

Absolutely representing system (ARS) in a Banach space $X$ is a set $D \subset X$ such that every vector $x$ in $X$ admits a representation by an absolutely convergent series $x = \sum_i a_i x_i$ with $(a_i)$ reals and $(x_i) \subset D$. We investigate some general properties of ARS. In particular, ARS in uniformly smooth and in B-convex Banach spaces are characterized via $\epsilon$-nets of the unit balls. Every ARS in a B-convex Banach space is quick, i.e. in the representation above one can achieve $\|a_i x_i\| < cq^i\|x\|$, $i=1,2,...$ for some constants $c>0$ and $q \in (0,1)$.