Orthonormal basis of a infinite dimensional Hilbert subspace

Question

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.

Answer 1

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.