Computing derivative with respect to a function Consider the functional
$$
\begin{align*}
J\colon (X:= C^{1}( [ 0, 1]))\to \mathbb{R}, \quad J[ f] = \int_{ 0}^{1} f'( x)^{2}\mathrm{~d}x 
.\end{align*}
$$
The task asks to compute $\frac{\partial }{\partial f} J $. So far I haven't
encounted functionals, so I tried to endow $X$ with the infinity norm
and consider some pertubation function $h \in X$. One has
$$
\begin{align*}
J[ f + h] - J[f]
= \int_{ 0}^{1} f'( x)h'( x)\mathrm{~d}x 
+ \int_{ 0}^{1} h'( x)^{2}\mathrm{~d}x 
.\end{align*}
$$
With the above computation wanted to try to exploit the linearity of the
differentiation/integral operator, but didn't succeed to bring it into the desired form
$$
\begin{align*}
J[ f + h] = J[ f] + \mathrm{D}J\!\left( f\right)( h) + o(h )
.\end{align*}
$$
I would suspsect
$$
\begin{align*}
\mathrm{D}J\!\left( f\right)( h) = \int_{ 0}^{1} f'( x) h'( x)\mathrm{~d}x , 
\quad \int_{ 0}^{1} h'( x)^{2}\mathrm{~d}x \in o( h )
\end{align*}
$$
but I don't even know whether this is a fruitful approach to solve this task.
 A: A way to generalize the notion of derivative defined on $\mathbb R^n$ to normed vector space is by introducing the Fréchet derivative.
Let $V, W$ be two normed vector spaces and $J: V \to W$. We say that $J$ is Fréchet differentiable at $x \in V$ if there exists a bounded linear map $A: V \to W$ such that
$$J(x + h) = J(x) + Ah + o(h),$$
for all $h \in V$, or equivalently
$$\lim_{h \to 0} \frac{\|J(x + h) - J(h) - Ah\|_W}{\|h\|_V} = 0.$$
If $A$ exists, it is unique a is denoted by $DJ(x)$ (it depends on $x$).
Here, the norm on $C^1([0,1])$ is defined by
$$\|f\|_{C^1} = \sup_{[0,1]}|f(x)| + \sup_{[0,1]}|f'(x)|,$$
and we take the usual norm on $\mathbb R$. As you guessed, in this case we have
$$DJ(f): C^{1}([0,1]) \to \mathbb R: h \mapsto 2\int_0^1 f' h'$$
(be careful, you forgot the $2$ before the integral). It is a bounded linear map as
$$|DJ(f)(h)| \le 2\|f'\|_{\infty} \|h'\|_{\infty} \le 2\|f\|_{C^1} \|h\|_{C^1}, $$
so that
$$\|DJ(f)\|_{L(C^1, \mathbb R)} \le 2 \|f\|_{C^1},$$
where $L(C^1, \mathbb R)$ stands for the space of bounded linear map between $C^1$ and $\mathbb R$. Now you just have to check that
$$\int_0^1h'^2$$
is a small $o$ of $h$, i.e.
$$\lim_{h \to 0} \frac{\left|\int_0^1 h'^2\right|}{\|h\|_{C^1}} = 0,$$
which is quite obvious.
