Ideals of subseries convergence and F-spaces

Let X be an F-space and be a sequence of vectors in X. Ideals of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c (0) obtained by using the ideals breaks down in every Fr,chet 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 of all ideals associated with the F-space X.