Functions with bounded variation in locally convex space

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 $X$, then the function $x(.):[a,b] \to X$ 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 which measure is also a weakly relatively compact subset in $X$. This theorem is an extension of the  results (\cite {di}) from Banach spaces to locally convex spaces.