Consider the space $C\bigl([a,b];\mathbb{R}\bigr)$ equipped with the $\sup$ norm. Define the operator $\mathfrak{f} : C\bigl([a,b];\mathbb{R}\bigr) \to C\bigl([a,b];\mathbb{R}\bigr) \: \text{by} \ \ \mathfrak{f}(\varphi)(t) = \int_{a}^{b}(\varphi(s))^{3} ds \cdot \varphi(t), \ \ \text{for} \ \varphi\in C\bigl([a,b];\mathbb{R}\bigr)$
Now for a given $\chi\in C\bigl([a,b];\mathbb{R}\bigr)$,I want to find a linear operator $\mathscr{L} : C\bigl([a,b];\mathbb{R}\bigr) \to C\bigl([a,b];\mathbb{R}\bigr)$ satisfying $\lim_{||\varphi||_{\infty}\to 0} \: \frac{\mathfrak{f}(\chi+\psi)-\mathfrak{f}(\chi)-\mathscr{L}\varphi}{||\varphi||_{\infty}}=0.$
I also want to show $\mathscr{L}$ is continuous. I know that it suffices to show $\mathscr{L}$ is bounded.
Also I want to calculate the derivative of $D\mathfrak{f}(\chi)$ of $\mathfrak{f}$ at $\chi$ $?$
A solution would be of great help.