A note on the Riesz representation theorem

Abstract

the norm on the space is defined f If | = max I f(x) 0; O < x < 1. An extensive bibliography for representation theorems is given in [1]. In 1961, S. E. James [2] generalized this result by considering continuous functions whose range of values was a subset of a Banach space S and considered bounded linear transformations T from this space into S. James' result required that the transforination T be such that there exist a functional T from the real valued continuous functions on [0, 1 ] into the reals such that for each real valued continuous function g on [0, 1] and for each h in S, T[g(x)h]= T [g] *h. The purpose of this note is to extend James' result in the following way: suppose S, is a linear normed space, S2 is a Banach space, C is the space of continuous functions from [0, 1] into S, with norm defined flgjIc=fo|g(x)jIsjdx and B[S1, S2] is the space of continuous linear transformations from S1 into S2.