| Uniform
Boundedness Principle |
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 also called 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
 .
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