Ideal convergent subseries in banach spaces

Abstract

Assume that I is an ideal on N, and Σ_nx_n is a divergent series in a Banach space X. We study the Baire category, and the measure of the set A(I) := {t ∈ {0,1}^N : Σ_n ^t(n)xn is I-convergent. In the category case, we assume that I has the Baire property and Σ_nx_n is not unconditionally convergent, and we deduce that A(I) is meager. We also study the smallness of A(I) in the measure case when the Haar probability measure λ on {0,1}^N is considered. If I is analytic or coanalytic, and Σ_n x_n is I-divergent, then λ(A(I)) = 0 which extends the theorem of Dindoš, Ŝalát and Toma. Generalizing one of their examples, we show that, for every  ideal I on N, with the property of long intervals, there is a divergent series of reals such that λ(A(Fin)) = 0 and λ(A(I)) = 1.

Mathematics Subject Classification (2010): 40A05, 46B25, 54E52, 28A05.

Keywords: Series in Banach spaces, ideal convergence, Baire category, measure on the Cantor space, subseries