Ideal convergent subseries in Banach spaces

Abstract

Assume that I is an ideal on N, and n xn 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 : n t(n)xn is I-convergent . In the category case, we assume that I has the Baire property and n xn 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 n xn is I-divergent, then λ(A(I)) = 0 which extends the theorem of Dindoš, Šalát 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.