Study the equivalence of these norms I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique.
So I define the following norm
$$\Vert f\Vert=(\Vert g\Vert_{H_1}^2+\Vert h\Vert_{H_2}^2)^{\frac{1}{2}}$$
Is this equivalent to
$$|||f|||=\Vert g\Vert_{H_1}+\Vert h\Vert_{H_2}$$?
I've followed this reasoning:
from sublinearity of square root I have
 $$\Vert f\Vert\leq|||f|||$$;
for the other direction I observe that
$$(\Vert g\Vert_{H_1}+\Vert h\Vert_{H_2})^2\leq 2(\Vert g\Vert_{H_1}^2+\Vert h\Vert_{H_2}^2)=2\Vert f\Vert^2$$
And so
 $$\Vert f\Vert\leq|||f|||\leq\sqrt{2}\Vert f\Vert$$
Is it right?

Your calculation should be right - it is just the equivalence of the $1$-norm and the $2$-norm on $\mathbb R^2$.
