Fréchet derivative: dependency of the choice of norm

Question

Let $ f_1: L^2(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $, $ f_2: H^1(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $ and $f_2: H^1(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{H^1 (\Omega)}^2$.

Now it is possible to calculate the Gateaux-Derivatives:

1. $f_1'(u)h=(2u,h)_{L^2(\Omega)}$
2. $f_2'(u)h=(2u,h)_{L^2(\Omega)}$
3. $f_3'(u)h=(2u,h)_{H^1(\Omega)}$

Is the following statement true?

$f_2$ is not Fréchet differentiable with respect to $\Vert \cdot \Vert _{H^1 (\Omega)}$.

My current calculations are:

$\frac{{\vert f_2(u+h)-f_2(u)-(2u,h)_{L^2} \vert}}{{\Vert h\Vert_{H^1}}}=\frac{{\vert (h,h)_{L^2} \vert}}{{\Vert h\Vert_{H^1}}}$