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 $\Vert a_i{x}_i\Vert <c{q}^i\Vert x\Vert$, $i=1,2,...$ for some constants $c>0$ and $q\in (0,1)$.

15 pages