Space of functions of bounded variation

Summary

The space $BV([a, b])$ collects the real functions on $[a, b]$ that have bounded variation. With a norm built from the total variation, it is a Banach space. Its elements are exactly the differences of two monotone functions, and they serve as integrators in the Riemann–Stieltjes integral. In several variables, $BV(\Omega)$ is the space of integrable functions whose distributional gradient is a finite measure. It plays a role in the calculus of variations and in image processing.