RIESZ TYPE THEOREM IN LOCALLY CONVEX SPACES

Abstract

The present paper is concerned with some representatons of linear mappings of continuous functions into locally convex vector spaces, namely Theorem. If X is a complete Hausdorff locally convex vector space, then a general form of weakly compact mapping T : C(a,b) → X is of the form Tg = R b a g(t)dx(t), where the function x(·) : (a,b) → X has a weakly compact semivariation on (a,b). This theorem is a generalization of the result from Banach spaces to locally convex vector spaces.