Proof of equivalence of two definitions of an orthonormal basis on a Hilbert space

Question

The following are two equivalent definitions of an orthonormal basis $\mathcal{B}$ on a Hilbert space $H$:

1. $\mathcal{B}$ is orthonormal and complete ($\mathcal{B}$ is dense in $H$).
2. if $\langle v, b\rangle = 0$ for all $b\in \mathcal{B}$ and some $v\in H$, then $v=0$.

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I was wondering where I could find a proof of the above equivalence, since it is stated in my notes without any demostration.