Ideals of subseries convergence and F-spaces

Abstract

Let X be an F-space and \({\boldsymbol x=(x_n)}\) be a sequence of vectors in X. Ideals \({\mathcal{C}(\boldsymbol x)}\) of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c0 obtained by using the ideals \({\mathcal{C}(\boldsymbol x)}\) breaks down in every Frechet space not isomorphic to a Banach space. On the other hand, the result can be extended to some F-spaces via the definition of a new class of F-spaces satisfying a stronger version of the condition (O) of Orlicz. A theorem discriminating between the finite and infinite dimensional case is obtained about the family \({\mathcal{C}(X)}\) of all ideals associated with the F-space X.