In functional analysis, an operator $T: V \to V$ is monotone if $(Tu - Tv, u - v) \geq 0$ for all $u,v \in V$. The function $f: \mathbb{R} \to \mathbb{R}$ defined $f(x) = -x$ seems not to fit this definition, for $(f(u) - f(v), u - v) = (f(u) - f(v)) \cdot (u - v) = - (u - v)^2 < 0$ for any $u \neq v$.
This seems counterintuitive to me because the function $f(x) = -x$ is clearly monotone (albeit monotone decreasing) in the usual analysis sense. So it seems that "monotone" in the functional analysis sense is more restrictive than in the usual analysis sense.
What I'm trying to understand is (1) If I've made some mistake in my description above, and (2) if not, why the functional analysis definition would classify functions like $f(x) = -x$ as "not monotone". Why doesn't functional analysis agree with classical analysis? What is gained by narrowing the definition of monotone?
"Monotone" here is just a generalization of "(weakly) monotonically increasing." If you want a generalization of monotonically decreasing then just reverse the sign and ask for $\langle T(u - v), u - v \rangle \le 0$.
I don't know exactly what theorems and results use this definition but in general there are lots of reasons to distinguish positive and negative things as well as increasing and decreasing things, e.g. we talk about positive definite matrices and not matrices which are either positive or negative definite, we talk about convex or concave functions and not functions which are either convex or concave. This seems harmless to me.