General methods of convergence and summability

Abstract

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of N and study the space of convergence associated with the filter. We notice that c(X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then l infinity (X) is a space of convergence associated with any free ultrafilter of N; and that if X is not complete, then l infinity (X) is never the space of convergence associated with any free filter of N. Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that l infinity (X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c(X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c(X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set HB(lim):={T is an element of B(l infinity (X),X):T|c(X)=lim and parallel to T parallel to =1} and prove that HB(lim) is a face of BLX0</mml:msubsup> if X has the Bade property, where LX0</mml:msubsup>:={T is an element of B(<mml:msub>l infinity (X),X):<mml:msub>c0(X)subset of ker(T)}. Finally, we study the multipliers associated with series for the above methods of convergence.