is paper, the concept of a paranormed fi.-space is definede where ? X> 1 is aninteger, and two Banach-Steinhaus type theorems are proved for the sequences of continuous linear ûınctionals on such space. For example, the 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 fi-speace. These results are then used to characterize the matrix transformations between the strong Cesaro summble sequence space w (p) and the space of generalized entire sequences.