Editing Monotonic function (section)

== In functional analysis ==

In [[functional analysis]] on a [[topological vector space]] <math>X</math>, a (possibly non-linear) operator <math>T: X \rightarrow X^*</math> is said to be a ''monotone operator'' if

<math display="block">(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.</math>
[[Kachurovskii's theorem]] shows that [[convex function]]s on [[Banach space]]s have monotonic operators as their derivatives.

A subset <math>G</math> of <math>X \times X^*</math> is said to be a ''monotone set'' if for every pair <math>[u_1, w_1]</math> and <math>[u_2, w_2]</math> in <math>G</math>,

<math display="block">(w_1 - w_2, u_1 - u_2) \geq 0.</math>
<math>G</math> is said to be ''maximal monotone'' if it is maximal among all monotone sets in the sense of set inclusion.  The graph of a monotone operator <math>G(T)</math> is a monotone set. A monotone operator is said to be ''maximal monotone'' if its graph is a ''maximal monotone set''.