Absolutely Representing Systems, Uniform Smoothness and Type
نویسنده
چکیده
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 = ∑ i aixi with (ai) ⊂ R and (xi) ⊂ D. We investigate some general properties of ARS. In particular, ARS in uniformly smooth and in B-convex Banach spaces are characterized via ε-nets of the unit balls. Every ARS in a B-convex Banach space is quick, i.e. in the representation above one can achieve ‖aixi‖ < cqi‖x‖, i = 1, 2, . . . for some constants c > 0 and q ∈ (0, 1).
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