Mappings between function spaces

In the following we shall let U= u and 33 = v denote arbitrary Banach spaces, and £= t denote a Hausdorff space.ï and g) are to denote the spaces of all continuous functions mapping X into U and into 33 respectively.We shall let ß denote the space of all linear continuous mappings of U into 33.A function K on Ï to fi which is bounded on X and continuous in the strong topology of ß induces a linear continuous operator k on 3£ to g) by the formula( 2)The main result of this paper, contained in §3, gives sufficient conditions that k map H onto the whole space g).Applications to nonlinear equations, integral equations, and to underdetermined systems of differential equations, including certain types of partial differential equations, are given in § §4, 5, and 7.In §6 we consider the case when X is a compact interval of the real axis, and let 7L' and §)' denote the subspaces of H and g) consisting of those functions which are of class C. Sufficient conditions are given that k map ï' onto g)', and this result is applied to the case of nonlinear equations.In the case when, for each /, K(t) gives a one-to-one correspondence between the spaces U and 33, the results of this paper have been long known and are easily derived.The methods used here enable us to prove the existence of solutions with certain useful properties in cases when the solution is not unique.1. Topological preliminaries.J. A. Dieudonné has discussed 3 a generalization of the notion of compactness of a topological space which will be of considerable use in §3.We therefore summarize a few definitions and results in the form that we shall need.In the following all coverings of topological spaces are supposed to consist of open sets.Definitions.A covering 33 is said to be a refinement of the covering 3Í if every set in 33 is contained in some set in 2(.A covering is said to be neighborhood-finite if every point has a neighborhood which intersects only a finite