Generalized Interpolation Spaces

Abstract

In this paper we introduce the notion of “generalized” interpolation space, and state and prove a “generalized” interpolation theorem. This apparently provides a foundation for an axiomatic treatment of interpolation space theory, for subsequently we show that the “mean” interpolation spaces of Lions-Peetre, the “complex” interpolation spaces of A. P. Calderón, and the “complex” interpolation spaces of M. Schechter are all generalized interpolation spaces. Furthermore, we prove that each of the interpolation theorems established respectively for the above-mentioned interpolation spaces is indeed a special case of our generalized interpolation theorem. In §III of this paper we use the generalized interpolation space concept to state and prove a “generalized” duality theorem. The very elegant duality theorems proved by Calderón, Lions-Peetre and Schechter, respectively, are shown to be special cases of our generalized duality theorem. Of special interest here is the isolation by the general theorem of the need in each of the separate theorems for certain “base” spaces to be duals of others. At the close of §II of this paper we employ our generalized interpolation theorem “structure” to construct new interpolation spaces which are neither complex nor mean spaces.