I'm given a function $F:C([0,1])\rightarrow C([0,1])$, $F(f)=f^2$ (where $C([0,1])$ is given the supremum norm) and I want to find $D_g F(f)$ for any $f,g\in C([0,1])$. I find that $D_g F(f)= 2fg$, but I did not use anywhere the supremum norm. In another question, I'm asked to find the differential using the $L^1$ norm so surely the process uses the norm somewhere?
Edit: To clarify what I want to know, since $f$ under the $L^1$ norm is not even continuous, then surely I can't find $DF(f)$ (the Frechet derivative). I want to find the Gateaux derivative when $C([0,1])$ is endowed the $L^1$ norm, which is defined to be $$D_g F(f)=\lim_{t\rightarrow 0} \frac{F(f+tg)-F(f)}{t}$$ which has no mention of the norm. Am I misunderstanding something here?
You have $$F(f+h)=(f+h)^2= f^2+2fh +h^2=F(f)+2fh +h^2$$ Hence $$dF_f(h)=2fh$$ is a good candidate for the Fréchet derivative.
Now you have to verify that $$\lim\limits_{h \to 0} \frac{\Vert F(f+h)-F(h)-d_f(h)\Vert}{\Vert h \Vert}= \lim\limits_{h \to 0} \frac{\Vert h^2 \Vert}{\Vert h\Vert} =0$$ to conclude that $dF_f$ is the derivative at point $f$. This is where the norms come into play.
The $\sup $ norm $\Vert \cdot \Vert_\infty$ satisfies the inequality $$\Vert fg \Vert_\infty \le \Vert f\Vert_\infty \Vert g \Vert_\infty$$ so your conclusion is correct in that case.
I let you finish for the case of the norm $\Vert \cdot \Vert_1$. Considering the sequence $(f_n)$ of piecewise linear functions defined by $f_n(0)=n, f_n(\frac{1}{n^2})=f_n(1)=0$ might be useful.