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I have read in proofwiki that a direct sum of Hilbert spaces is a Hilbert space. However, Wikipedia Page about direct sum says it is not necessarily true, that is, the direct sum of Hilbert spaces is not always a complete space. Which of them is right? In what Conditions the direct sum of Hilbert spaces is a Hilbert space?

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Note that proofwiki uses what they call the "Hilbert space direct sum" which is what Wikipedia describes in the relevant section a few lines later, starting with "Alternatively and equivalently ..." – t.b. Sep 16 '12 at 16:22
But Wikipédia presents a condition that is not present on proofwiki: the sum of all norms for each function on the direct sum must converge. – user38397 Sep 16 '12 at 16:32
No, as usual proofwiki is terribly written, but they impose that condition, too: – t.b. Sep 16 '12 at 16:34

The answer is not in general, in fact you need to add some conditions, please see this proof 1, I think it can be of help.

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