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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]] with [[dual space|07JBSZE0V4]] $X^*$, and write $\\ip{\\ell}{u} = \\ell(u)$ for $\\ell \\in X^*$ and $u \\in X$.\nLet $D \\subseteq X$ and let $T \\colon D \\to X^*$ be a map, not necessarily linear.\nThe map $T$ is a *monotone operator* if\n$$\\ip{T(u) - T(v)}{u - v} \\ge 0 \\quad \\text{for all } u, v \\in D.$$\nIt is *strictly monotone* if equality holds only for $u = v$.\n\n## Remarks\n\nA set-valued map $T$ from $X$ to subsets of $X^*$ is monotone if $\\ip{u^* - v^*}{u - v} \\ge 0$ whenever $u^* \\in T(u)$ and $v^* \\in T(v)$.\nIt is *maximal monotone* if its graph is not properly contained in the graph of another monotone map.\nFor a real Hilbert space one identifies $X^*$ with $X$.\nA linear operator $T$ is then monotone if and only if $\\ip{Tu}{u} \\ge 0$ for all $u$, that is, if and only if $-T$ is [[dissipative|3S7HJ5GQRF]].\nThe subdifferential of a proper convex lower semicontinuous function is maximal monotone.\n\nExample: let $f \\colon \\R^n \\to \\R$ be a differentiable [[convex function|07Q7CNM5WF]].\nAdding the inequalities $f(v) \\ge f(u) + \\nabla f(u) \\cdot (v - u)$ and $f(u) \\ge f(v) + \\nabla f(v) \\cdot (u - v)$ gives $(\\nabla f(u) - \\nabla f(v)) \\cdot (u - v) \\ge 0$, so $\\nabla f$ is monotone.\n\nNon-example: $T(u) = -u$ on $\\R$ gives $(T(1) - T(0))(1 - 0) = -1 \u003c 0$.\n\nOn $X = \\R$, a map $T \\colon \\R \\to \\R$ is monotone exactly when it is a nondecreasing [[monotone function|0A7KV2WB7A]].\n",
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