Linear structure of sets of divergent sequences and series

Abstract

We show that there exist infinite dimensional spaces of series, every non-zero element of which, enjoys certain pathological property. Some of these properties consist on being (i) conditional convergent, (ii) divergent, or (iii) being a subspace of l(infinity) of divergent series. We also show that the space 1(1)(omega)(X) of all weakly unconditionally Cauchy series in X has an infinite dimensional vector space of non-weakly convergent series, and that the set of unconditionally convergent series on X contains a vector space E, of infinite dimension, so that if x is an element of E \ {0} then Sigma(i) parallel to x(i)parallel to = infinity.