The Baire category of ideal convergent subseries and rearrangements

Abstract Let I be a 1-shift-invariant ideal on N with the Baire property. Assume that a series ∑ n x n with terms in a real Banach space X is not unconditionally convergent. We show that the sets of I -convergent subseries and of I -convergent rearrangements of a given series are meager in the respective Polish spaces. A stronger result, dealing with I -bounded partial sums of a series, is obtained if X is finite-dimensional. We apply the main theorem to series of functions with the Baire property, from a Polish space to a separable Banach space over R , under the assumption that the ideal I is analytic or coanalytic.