Exercise 14, § 5

Extend the definitions and results of § 5 to arbitrary multilinear mappings. Let E, F, G be three topological vector spaces, S (resp. T) a family of bounded subsets of E (resp. F), and U a family of bounded subsets of the space L_{S,T}(E, F ; G) of bilinear (S, T)-hypocontinuous mappings from E × F into G, endowed with the (S, T)-topology (exerc. 13). Show that the trilinear map (x, y, u) ↦ u(x, y) from E × F × L_{S,T}(E, F ; G) into G is (S, T)-hypocontinuous; in order that it is (S, U)-hypocontinuous, it is necessary and sufficient that every set L ∈ U is S-equihypocontinuous (III, p. 47, exerc. 7).