On the convergence of sequences of linear operations
Rick Bailey · Bulletin of the American Mathematical Society · 1943
1. Introduction.Let 5 be a sequence {U n (x)}(n = l, 2, ... ) of linear operations defined over the elements {x} of a Banach space 1 E with values lying in a second B space E\.The fundamental convergence theorem associated with such sequences may be stated as follows :BANACH-STEINHAUS THEOREM.For the convergence of S over E it is necessary and sufficient that S converge over a set H dense in a sphere K of E and that the set of norms {| U n \ } be bounded.