The Banach-Steinhaus theorem is a result in the field of functional analysis which relates the "size" of a certain subset of points defined relative to a family of linear mappings between topological vector spaces to a certain continuity property of the maps involved.
More precisely, suppose that 
and 
are topological vector
spaces, that 
is a collection of continuous
linear maps from 
into 
, and that 
denotes the set of all points 
whose orbits 
are bounded in
.
The Banach-Steinhaus theorem says that if 
is of second category in
,
then it necessarily follows that 
and that the collection 
is equicontinuous.
The statement of the Banach-Steinhaus theorem is often given in various forms, some apparently differing from the above. As a result, various corollaries thereof are
sometimes considered part of the actual theorem. One such example is the identification
of the Banach-Steinhaus theorem with the so-called uniform
boundedness principle, which states that any family of continuous linear operators
between Banach spaces is uniformly bounded provided
that it is bounded pointwise. This result is actually a corollary
of the above-stated version of the Banach-Steinhaus theorem along with the observation
that in the above-described framework, an equicontinuous family 
necessarily satisfies a uniform boundedness property in
which every bounded subset 
of 
implies the existence of a bounded subset 
of 
satisfying 
for every 
in 
.
