Proposition 18

— Let G_1, G_2, H_1, H_2 be four Hausdorff locally convex spaces, with H_1 and H_2 assumed to be quasi-complete. Let A : G_1 → G_2 and B : H_1 → H_2 be two continuous linear mappings. Let U_1 : T → L(G_1; H_1), U_2 : T → L(G_2; H_2) be two mappings satisfying the conditions of Prop. 16 (resp. Prop. 17), and suppose that for every t ∈ T, B ◦ U_1(t) = U_2(t) ◦ A. Then, for every measure with compact support (resp. every bounded measure) λ on T, B ◦ U_1(λ) = U_2(λ) ◦ A.

Indeed, for every x ∈ G_1, one has (No. 1, Prop. 1)