On variation topology

Abstract

Let $I$ be a real interval and $X$ be a Banachspace. It is observed that spaces $\mathrm{\Lambda }B{V}^{(p)}([a,b],R)$,$LBV(I,X)$ (locally bounded variation), $B{V}_0(I,X)$ and $LB{V}_0(I,X)$share many properties of the space $BV([a,b],R)$. Here we haveproved that the space $\mathrm{\Lambda }B{V}_0^{(p)}(I,X)$ is a Banach spacewith respect to the variation norm and the variation topologymakes $L\mathrm{\Lambda }B{V}_0^{(p)}(I,X)$ a complete metrizablelocally convex vector space (i.e\. a Fréchet space).