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  "body": "\n## Summary\n\nA monotone operator is a map $T$ from a Banach space $X$ to its dual $X^*$ with $\\ip{T(u) - T(v)}{u - v} \\ge 0$ for all $u, v$.\nIt generalises the increasing functions of one real variable, the [[monotone functions|0A7KV2WB7A]], to infinite dimensions.\nGradients of convex functionals and many nonlinear elliptic operators are monotone.\nMonotone operator theory, with results such as the [[Browder–Minty theorem|NWK9Z80BT2]], solves nonlinear equations $T(u) = f$ without compactness.\n\n## Explanations\n\n### L1 human\n\nA function $f$ on the real line is increasing when $(f(u) - f(v))(u - v) \\ge 0$: larger inputs never give smaller outputs.\nA monotone operator asks the same thing in many dimensions, with the product replaced by a pairing.\nFor example, the gradient of a convex function on $\\R^n$ is monotone, because a convex function bends upwards in every direction.\nThe map $T(u) = -u$ is not monotone, since it reverses order.\n",
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