Continuous operators on paranormed spaces and matrix transformations

The concept of a paranormed /3-space is defined and some theorems of Banach-Steinhaus type are proved for sequences of continuous linear f unctionals on such a space. For example, necessary and sufficient conditions are given for a sequence (A n (x)) of continuous linear functionals to be in the space of generalized entire sequences, for each x belonging to a paranormed /3-space. The general theorems are then used to characterize matrix transformations between generalized l p spaces and generalized entire sequences.