I am reading Lectures on Amenability, and I am a little unsure about the definition of a mean:
Let $G$ be a locally compact group, and let $E$ be a subspace of $L^{\infty}(G)$ containing the constant functions. A mean on E is a functional $m \in E^{*}$ such that $\langle 1,m \rangle=\|m\|=1$
My understanding is that $L^{\infty}(G)$ is the set of functions on $G$ that are almost everywhere bounded, that $E^{*}$ is the dual space of $E$ ie the Banach-space of all bounded linear functionals $E \to \mathbb{C}$ with the operator norm.
My confusion is with the last part of the definition: $\langle 1,m \rangle=\|m\|=1$. Maybe I am missing something here, but what is the inner product of the dual space $E^{*}$? And what is the reasoning behind choosing this requirement?
The inner product notation $\langle v, f\rangle$ for $v\in V, \, f\in V^*$ simply denotes the function value $f(v)$.