Abstract
While the content of the classical Banach-Steinhaus theorem varies somewhat in the literature, one very common variation is the following: if E and F are locally convex topological vector spaces and E is barrelled, then every pointwise bounded subset of 𝓛(E, F) is equicontinuous. This powerful theorem is used, for example to show that the pointwise limit of a sequence of continuous linear mappings is a continuous linear mapping. It is used as well to derive the continuity of separately continuous bilinear mappings.
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