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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?

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

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