Find the length of the vector f(x) = x in L[0, 1] Find the length of the vector $f(x) = x$ in $L_2[0, 1]$
Is the solution $L=\int\sqrt{(1+(f'(x))^2dx)}$ hence $L=\sqrt{2}$ ?
Or am I completly wrong?

The equation $L = \int_I \sqrt{1+{f'}^2}$ is the euclidean length of the parametrised curve $(x,f(x))_{x\in I}$. Here, it seems that the question is not about the length of the curve as a geometric object in the plane, but would rather be the norm of the vector $\mathrm{id}$ in the abstract normed linear space $\left(L^2([0,1]), \|.\|_2\right)$ where
$$
\|f\|_2 = \sqrt{\int_0^1 |f|^2}
$$
Hence, in this case
$$
\|\mathrm{id}\|_2 = \sqrt{\int_0^1 x^2 \mathrm{d}x} 
$$
This will lead to another value than $\sqrt{2}$.
