Monotone operator

Summary

A 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 $u, v$. It generalises the increasing functions of one real variable, the monotone functions, to infinite dimensions. Gradients of convex functionals and many nonlinear elliptic operators are monotone. Monotone operator theory, with results such as the Browder–Minty theorem, solves nonlinear equations $T(u) = f$ without compactness.