Proposition 17

— Let G, H be two Hausdorff locally convex spaces, with H assumed to be quasi-complete. Let U be a mapping of T into L(G; H) such that (t, x) ↦ U(t) · x is continuous on T × G, and U(T) is equicontinuous. Then, if M^1(T) is equipped with its Banach space topology, the bilinear mapping (λ, x) ↦ U(λ) · x of M^1(T) × G into H is continuous (which implies, in particular, that the linear mapping λ ↦ U(λ) of M^1(T) into L(G; H) is continuous when L(G; H) is equipped with the topology of bounded convergence).