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  "body": "\n## Summary\n\nThe space $BV([a, b])$ collects the real functions on $[a, b]$ that have [[bounded variation|A8251BHCMP]].\nWith a norm built from the [[total variation|8QA5DEWNYD]], it is a [[Banach space|06GZ956657]].\nIts elements are exactly the differences of two monotone functions, and they serve as integrators in the Riemann–Stieltjes integral.\nIn several variables, $BV(\\Omega)$ is the space of integrable functions whose distributional gradient is a finite measure.\nIt plays a role in the calculus of variations and in image processing.\n\n## Explanations\n\n### L1 human\n\nA function has bounded variation when the total distance its values travel up and down is finite.\nThese functions form a vector space, because the variation of a sum is at most the sum of the variations.\nThe size of a function in this space is its starting value plus the total distance travelled.\nFor example, $x \\mapsto x^2$ on $[-1, 1]$ goes down by $1$ and then up by $1$, so its total variation is $2$.\n",
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