Let $~f: U \subseteq \mathbb R^n \to \mathbb R$ where $U$ is open, then the Fréchet derivative defines the function $f': U \to (\mathbb R^n)'$ that maps vectors to a unique linear functional for which the limit defined by the frechet derivative is true.
Further, given $u\in \mathbb R^n$, $f$ has a directional derivative in direction $h \in \mathbb R^n$ if following limit exists: $$ \frac{\partial f}{\partial h}(u) := \lim_{x \to 0} \frac{f(u+xh) - f(u)}{x} $$ then we have the relationship: $$ \frac{\partial f}{\partial h}(u) = f'(u) h $$
I am confused about the notation here, especially in the last statement, is the term $f'(u)h$ saying that the unique linear functional for which the Fréchet derivative is satisfied at the point $u$ is evaluated at the point $h$? Is there a more common notation for this?
This notation comes up in the following set of lecture notes: http://www.math.ist.utl.pt/~czaja/ISEM/internetseminar200910.pdf although they use $\mathscr{E}$ where I have used $f$
The notation $f^\prime(u).h$ or $f^\prime(u)h$ is classical.
As $f': U \to (\mathbb R^n)'$, $f^\prime(u)$ is a linear form from $\mathbb R^n$ to $\mathbb R$. $f^\prime(u).h$ is the real value of $f^\prime(u)$ at vector $h$ which can also be denoted by $[f^\prime(u)](h)$ or $f^\prime(u)h$.
This is also the standard notation for a function defined between two Banach spaces $f : E \to F$. The Fréchet derivative of $f$ is a function from $E$ to $\mathcal L (E,F)$, the space of continuous linear maps between $E$ and $F$. So the value of $f^\prime$ at a point $u \in E$ is a continuous linear map from $E$ to $F$. And $f^\prime(u).h$ is the value of the linear map at vector $h$.
Please note the "accent aigu - é" for Fréchet.