Corollary

— The set H is equicontinuous if, and only if, for every semi-norm $q \in \Gamma$, there exists a neighbourhood of 0 in $\prod_{i=1}^n E_i$, in which the functions $q \circ u$, for $u \in \mathbf{H}$, are uniformly bounded.

The condition is evidently necessary, and the demonstration of prop. 4 shows that it implies an inequality of the form (5) for all $u \in \mathbf{H}$, and therefore the equicontinuity of H.

We state explicitly the particular case of prop. 4 for linear mappings.