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In wiki's page of Orthonormal basis, there's such a sentence.

Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis.

Seems that it's problematic. If the space is not separable- no countable orthonormal basis is admitted, then the basis could be uncountable, how can you apply the Gram–Schmidt process?

Michael
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    You can apply Gram-Schmidt by transfinite recursion on an uncountable set the same way you apply it by ordinary recursion on a countable set. – Eric Wofsey Nov 26 '22 at 05:00
  • @EricWofsey If you are at step $\alpha$ which is a limit ordinal, you need to subtract an infinite series (actually an $\alpha$-sequence) from a vector, how do you know that's always viable? – Michael Nov 26 '22 at 11:01
  • @EricWofsey Ok, I get it now. Only at most countably many inner products of a vector with another from an orthonormal set could be nonzero. – Michael Nov 26 '22 at 13:09

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