I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$. Given any $x,y \in M$, $$d(x,y) = ||x-y||_{L^2} + |V_0^1(x) - V_0^1(y)|$$

Fourier series is not convergent in this metric! But if I use this (not proven yet!) concept, which is a slightly modified form of Fourier expansion, the resulting series converges in the above metric.

My question is, is this metric been studied anywhere in the literature, if so, what are the interesting properties/aspects of it. My hope is that, if this metric is interesting, then it would serve as a good motivation for validating this concept.

The space of functions of bounded variation and certain general spacesby Clarence Raymond Adams [Trans. AMS 40 (1936), 421-438]. See also Abstract #1 on p. 19 here (1937) and Abstract #1 on p. 27 here (1940). $\endgroup$ – Dave L Renfro Nov 17 '15 at 22:01