Suppose X is a locally convex space, Y is a topological vector space and λ(X)βY is the β-dual of some X valued sequence space λ(X). When λ(X) is c0(X) or l∞(X), we have found the largest M ⊂ 2λ(X) for which (Aj) ∈ λ(X)βY if and only if Σ
j=1∞Aj(xj) converges uniformly with respect to (xj) in any M ∈ M. Also, a remark is given when λ(X) is lp(X) for 0 < p < + ∞.
The concept of absolute convergence for series is generalized to locally convex spaces and an invariant theorem for absolutely convergent series in duality is established: when a locally convex space
is weakly sequentially complete, an admissible topology which is strictly stronger than the weak topology on
in the dual pair (
) is given such that it has the same absolutely convergent series as the weak topology in