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 
.
