# Banach algebra of BV functions

I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.

-
The norm inequality $\|f\|_{bv} \cdot \|g\|_{bv} \ge \|f\cdot g \|_{bv}$ is Proposition 13.12 in Carother's Real Analysis book. –  Bill Johnson Dec 10 '13 at 3:42

For the second part: are you considering functions of bounded variation on some interval? If so, then such a function can be written as the difference of two non decreasing functions, and hence is in $L^\infty$. So, this is answered by the first case.