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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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    No, as usual proofwiki is terribly written, but they impose that condition, too: http://www.proofwiki.org/wiki/Definition:Hilbert_Space_Direct_Sum2012-09-16

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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.