On Series with Alternating Signs in the Euclidean Metric

Abstract

This paper presents one of many interesting aspects of relatively convergent series. Namely, given a sequence of elements of a Hilbert space we consider all possible ways of placing plus or minus signs in front of these elements to create an alternating sequence. In a convenient metric it makes sense to ask what is the `size' of the set of those choices of $+$ or $-$ for which the resulting series converges. The term `size' here refers to either Baire category or Lebesgue measure of this set. It turns out that especially the question of the Lebesgue measure of this set is quite intriguing and leads to interesting results generalizing known results for real-valued sequence.