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?
is a direct sum of Hilbert spaces a Hilbert space.?
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3Note 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 ..." – 2012-09-16
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0But Wikipédia presents a condition that is not present on proofwiki: the sum of all norms for each function on the direct sum must converge. – 2012-09-16
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2No, as usual proofwiki is terribly written, but they impose that condition, too: http://www.proofwiki.org/wiki/Definition:Hilbert_Space_Direct_Sum – 2012-09-16