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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4$\begingroup$ The norm inequality $\|f\|_{bv} \cdot \|g\|_{bv} \ge \|f\cdot g \|_{bv}$ is Proposition 13.12 in Carother's Real Analysis book. $\endgroup$– Bill JohnsonDec 10, 2013 at 3:42
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2 Answers
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For the first part: wikipedia has a proof. So, I am just posting the link here.
http://en.wikipedia.org/wiki/Bounded_variation#BV.28.CE.A9.29_is_a_Banach_algebra
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.
