Absolutely λ-summable sequences lying in the range of a vector measure I

Abstract

Let X be a real and infinite dimensional Banach space. By λ_X we denote the vector space of all sequences (α_n ) of real numbers such that (α_nx_n ) lies inside the range of some X-valued measure with bounded variation for every null sequence (x_n ) in X. Among other results we prove: (i) λ_X is the largest normal sequence space μ satisfying that every sequence (x_n ) ∈ μ {X} lies inside the range of some X**-valued measure with bounded variation, and (ii) λ_X is a perfect space for which l ₁ ⊂ λ_X ⊂ l ₂. We also determinate the sequence space λ_X when X is an L _p ?space (1 ≤ p ≤ +∞).