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Let $H$ be a Hilbert space and $V$ is a dense linear subspace of $H$. Can we find an orthonormal basis of $H$ such that all elements are in $V$?

Summer
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  • Can we always carry on Gram-Schmidt on a given basis? – Timbuc Dec 01 '14 at 18:53
  • If $H$ is separable, you simply have to apply Gram-Schimdt orthonormalization in a dense countable set contained in $V$. I would guess this is not true in general, but I'm not sure. – Luiz Cordeiro Dec 01 '14 at 18:54
  • This post http://math.stackexchange.com/questions/201119/a-complete-orthonormal-system-contained-in-a-dense-sub-space answers the separable case and provides a counterexample for the non-separable case. – PhoemueX Dec 01 '14 at 21:22

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