# The intuition behind the definition of a monotone operator

I started reading about monotone operators in Zeidlers' book on Nonlinear functional analysis:

The operator $A:X \to X^*$ is monotone on the reflexive Banach space $X$ if: $\langle Ax - Ay, x-y\rangle \geq 0$ for all $u,v \in X$.

This definition (if I get it right) is supposed to be a generalization of a monotone function $f: \mathbb{R} \to \mathbb{R}$. He proceeds with an example:

Set $X=\mathbb{R}$ and $F(u)=Au$. Then $X^*= \mathbb{R}$ and: $\langle Ax - Ay, x-y\rangle =\big(F(u) -F(v) \big)(u-v)$.

But I'm thinking if $F$ is strictly decreasing and we have $v<u, u,v\in \mathbb{R}$ then $u-v >0$ while $F(u) -F(v)<0$, so $\big(F(u) -F(v) \big)(u-v)<0$. What am I missing here? If this were a generalization of a monotone function of $\mathbb{R}$ it should be positive for all $u,v \in \mathbb{R}$ I think? If someone could help me and explain where the definition comes from I'd also be very grateful.

• Yes, increasing monotone. They are Frechet derivative of convex functions – user553213 Apr 28 '18 at 11:20
• I'm sorry - I didn't really get this. Could you please elaborate? – user202542 Apr 28 '18 at 11:26
• A definition is motivated when it is a hypothesis of a theorem. There is your theorem. The other thing is that, as you observed, if generalizes 'increasing monotone' functions. For decreasing monotone you would need to change the inequality. That can also be seen in the theorem, the derivative of a real convex function on the reals is an increasing monotone function. – user553213 Apr 28 '18 at 11:30
• Ok! So it's rather a generalization of an increasing function? – user202542 Apr 28 '18 at 12:13