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

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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 first part: wikipedia has a proof. So, I am just posting the link here.

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.

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Since you ask for references, there are two readily traceable articles on algebras of functions of bounded variation in Manuacripta Mathematica by Blümlinger and Tichy.

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