A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded." Symbolically, if is finite for each in the unit ball, then is finite. The theorem is a corollary of the Banach-Steinhaus theorem.
Stated another way, let be a Banach space and be a normed space. If is a collection of bounded linear mappings of into such that for each , then .