A study on absolutely convergent series in locally convex spaces

The concept of absolute convergence for series,a previously simple concept in normed spaces,was generalized to locally convex spaces.In addition,the equivalent characterization for an absolutely convergent series in locally convex spaces was obtained,in which an equicontinuous set in the duality of the locally convex spaces was employed to characterize the absolutely convergent series.Then this result was generalized to an equicontinuous set in the space of all continuous linear operators from the locally convex space to another locally convex space.Meanwhile,absolutely convergent series were researched in paranormed locally convex spaces and an interrelationship between absolutely convergent series and the paranorm in the space was established.