Mappings between function spaces

In the following we shall let U = [u] and Q3 = [v] denote arbitrary Banach spaces, and Z = [t] denote a Hausdorff space. X and 2) are to denote the spaces of all continuous functions mapping Z into U and into Q3 respectively. We shall let Q denote the space of all linear continuous mappings of U into Z3. A function K on Z to Q which is bounded on Z and continuous in the strong topology of Q induces a linear continuous operator K on I to 2) by the formula(2)