This is a really basic question sorry, I just need to make sure I have my understanding correct.
Given an infinite dimensional Hilbert space $\mathcal{H}$ and two subspaces, also Hilbert spaces, $\mathcal{H}_1$ and $\mathcal{H}_2$. If one can write:
$\mathcal{H}=\mathcal{H}_1 \bigoplus \mathcal{H}_2$
This means that $\mathcal{H}$ is isomorphic to the direct sum, not 'equal', right?
I'm just a little confused as the definiton of the direct sum of two Hilbert spaces (see http://en.wikipedia.org/wiki/Hilbert_space#Direct_sums) does not look to have the same 'form' as $\mathcal{H}$, so it seems a little strange to say they're equal.
I'm fairly sure, if they are isomorphic, the definition used for the direct sum isn't all that important, but I feel like I may be missing something.