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 .