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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers.\nThe *space of functions of bounded variation* $BV([a, b])$ is the set of functions $f \\colon [a, b] \\to \\R$ of [[bounded variation|A8251BHCMP]], with pointwise operations.\nIts norm is\n$$\\norm{f}_{BV} = \\abs{f(a)} + V_a^b(f),$$\nwhere $V_a^b(f)$ is the [[total variation|8QA5DEWNYD]] of $f$ on $[a, b]$.\n\n## Remarks\n\nThis is a [[normed space|03J5RAKB30]].\nThe variation is subadditive and $V_a^b(t f) = \\abs{t} V_a^b(f)$, which gives the triangle inequality and homogeneity.\nIf $\\norm{f}_{BV} = 0$, then $V_a^b(f) = 0$, so $f$ is constant, and $f(a) = 0$ gives $f = 0$.\nThe space is complete, so it is a Banach space.\n\nExample: $f(x) = x$ on $[0, 1]$.\nIt is increasing, so $V_0^1(f) = f(1) - f(0) = 1$ and $\\norm{f}_{BV} = 0 + 1 = 1$.\n\nNon-example: $f(x) = x \\sin(1/x)$ for $0 \u003c x \\le 1$, with $f(0) = 0$.\nIt is continuous on $[0, 1]$, but its total variation is infinite, so it is not in $BV([0, 1])$.\n\nAn equivalent norm is $\\sup_{x} \\abs{f(x)} + V_a^b(f)$.\n",
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