Riesz type theorem in locally convex vector spaces

Abstract: The present paper is concerned with some representatons of linear mappings of continuous functions into locally convex vector spaces, namely: If X is a complete Hausdorff locally convex vector space, then a general form of weakly compact mapping T:C{[a,b]}\to X is of the form Tg=\int_a^bg(t)dx(t), where the function $x(\cdot):[a,b] \to X$ has a weakly compact semivariation on $[a,b]$. This theorem is a generalization of the result from Banach spaces to locally convex vector spaces.