# Banach-Steinhaus theorem

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A general appellation for several results concerning the linear-topological properties of the space of continuous linear mappings of one linear topological space into another. Let and be locally convex linear topological spaces, where is a barrelled space, or let and be linear topological spaces, where is a Baire space. The following propositions are then valid. 1) Any subset of the set of continuous linear mappings of into which is bounded in the topology of simple convergence is equicontinuous (the uniform boundedness principle); 2) If a filter in contains a set bounded in the topology of simple convergence, and converges in the topology of simple convergence to some mapping of into , then is a continuous linear mapping of into , and converges uniformly to on each compact subset of [2], [3].

These general results make it possible to render the classical results of S. Banach and H. Steinhaus [1] more precise: Let and be Banach spaces and let be a subset of the second category in . Then, 1) if and is finite for all , then ; 2) if is a sequence of continuous linear mappings of into , and if the sequence converges in for all , then converges uniformly on any compact subset of to a continuous linear mapping of into .

#### References

 [1] S. Banach, H. Steinhaus, "Sur le principe de la condensation de singularités" Fund. Math. , 9 (1927) pp. 50–61 [2] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French) [3] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)