While studying some books on Semigroups theory, I always get that monotone operators are defined on Hilbert spaces and my professor always warn us that we can't define such an operator in a banach space since we don't have the inner product. However, I came across this definition quite often and it really throws me off:
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 $x,y \in X$.
How is it possible to define inner product in a Banach space?