On variation topology

R. G. Vyas

Let $I$ be a real interval and $X$ be a Banach space. It is observed that spaces $\Lambda BV^{(p)}([a, b],R)$, $LBV(I,X)$ (locally bounded variation), $BV_0(I,X)$ and $LBV_0(I,X)$ share many properties of the space $BV([a,b],R)$. Here we have proved that the space $\Lambda BV^{(p)}_0(I,X)$ is a Banach space with respect to the variation norm and the variation topology makes $L\Lambda BV^{(p)}_0(I,X)$ a complete metrizable locally convex vector space (i.e\.a Fr├ęchet space).