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Let $H$ be a separable, infinite dimensional, complex Hilbert space.

Let $BH$ be a orthonormal basis of $H$.

Let $V$ be a Hilbert subspace of $H$.

Is it true that exists a subset $BV \subset BH$ such that $BV$ is orthonormal basis of $V$

Thanks.

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No, consider $H = \mathbb C^2$ with basis $\{(0,1),(1,0)\}$ and $V = \mathrm{span} \{(1,1)\}$. Of course this example extends to the infinite dimensional case.

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  • $\begingroup$ ok @André S. thanks $\endgroup$
    – Matey Math
    Jan 16 '17 at 9:42

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