Generalized Sectional Convergence and Multipliers

We define a generalized sectional convergence scheme (gscs) as a sequence (Tn) of finitely non-zero matrices which converges coordinatewise to the identity matrix. If Tnx converges to x for each x in a topological sequence space S, then we say that S has AK (Tn), a generalization of sectional convergence (AK). We prove that a generalization of the Dieudonné Weak Basis Theorem is valid in this new context. A gscs (Tn) together with a dense subspace Λ of l1 determines a matrix space M = Λ(Tn). If Λ is barrelled and each member of M is a matrix representation of a linear operator T which maps the locally convex K space S into itself (denote this by MS ⊂ S), then we can draw conclusions about topological and approximation properties of S. For an appropriate type of (Tn) we will have the following: If S is an FK space with AD and MS ⊂ S, then S has the generalized sectional convergence associated with (Tn). If S is a sequence space, which has AD in the βφ topology and MS ⊂ S, then S is barrelled in the βφ topology. The many dense barrelled subspaces of l1 are examples of dense βφ subspaces Λ of l1, and the latter still support our conclusions even though Λ and the corresponding multiplier space M may be very small. This reduction of M is novel even in the context of ordinary sectional convergence.