Consider the function space $F=\{ f : \mathbb{R}^m \rightarrow \mathbb{R}^n\}$ and the empirical scalarproduct: $$ \langle f,g\rangle:=1/n\sum^n_{i=1}f(x_i)^Tg(x_i), $$ for a a finite dataset $x_1, \ldots, x_n \in \mathbb{R}^m$. The Dualspace is defined as $F^*=\{\langle d,\cdot\rangle:d\in F\}$. The cost functional $C$ (for example: $$C(f)=1/n\sum_{i=1}^n\|f(x_i)-f*(x_i)\|^2$$ ) only depends on the values of $f \in F$ at the data points. As a result it was said that, the (functional) derivative of the $\operatorname{cost} C$ at point $f_0 \in F$ can be viewed as an element of $F^{*}$, which we write $\left.\partial_{f} C\right|_{f_0}$, such that $\left.\partial_f C\right|_{f_0}=\langle d|_{f_0}, \cdot\rangle$ for some $d|_{f_0} \in F$.
Can someone explain why the last equation holds? Also im not familiar with the functional derivative, but according to the definition of wiki:https://en.wikipedia.org/wiki/Functional_derivative shouldn't the derivative at $f_0$ also be a function from $\mathbb{R}^m$ to $\mathbb{R}^n$?