Absolutely representing systems,uniform smoothness and type

An absolutely representing system (ARS) in a Banach space X is a set D ? X such that every vector x in X admits a representation by an absolutely convergent series x = Σ _i a _i x _i with (a _i ) ? R and (x _i ) ? D. We investigate some general properties of absolutely representing systems. In particular, absolutely representing systems in uniformly smooth and in B-convex Banach spaces are characterized via ?-nets of the unit balls. Every absolutely representing system 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 ? (0,1).