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Let $H$ be a Hilbert space and $S$ a vector subspace of $H$ which is dense in H. Does there exist an orthonormal basis $(u_{\alpha})_{\alpha \in A}$ of $H$ such that $\{ u_{\alpha} : \alpha \in A \} \subseteq S$?

I think the answer is generally negative, but I can't find a counterexample. Any help is welcome.

Maurizio Barbato
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1 Answers1

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There is a post here that can be useful.

Community
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Marko
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  • Thank you very much, Marko, for having drawn my attention to this post, which contains the answer to my question. As I suspected, the answer is generally negative. – Maurizio Barbato Jan 20 '16 at 22:49
  • No problem Maury. It is an intersting question. At first I thought the answer is true but the true beauty of mathematics is that there are some answers that are not obvious or even intuitive. – Marko Jan 20 '16 at 22:51