Functions with bounded variation in locally convex space

Abstract

ABSTRACT The present paper is concerned with some properties of functions with values in locally convex vector space, namely functions having bounded variation and generalizations of some theorems for functions with values in locally convex vector spaces replacing Banach spaces, namely Theorem: If X is a sequentially complete locally convex vector space, then the function x(・) : [a, b] → X having a bounded variation on the interval [a, b] defines a vector-valued measure m on borelian subsets of [a, b] with values in X and with the bounded variation on the borelian subsets of [a, b]; the range of this measure is also a weakly relatively compact set in X. This theorem is an extension of the results from Banach spaces to locally convex spaces.