# Separable Hilbert space have a countable orthonormal basis

I want to show that every an infinite-dimensional separable (contains countable dense set) Hilbert space has a countable orthonormal basis.

I know that every orthogonal set in a separable Hilbert space is countable,it is help me with the proof?

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Hint: Take a countable dense subset $Q$ and build an orthonormal basis of $\text{span}(Q)$.

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Build by Gram–Schmidt process ? –  ali baba Mar 8 '13 at 9:20
How am I define Q ?: –  ali baba Mar 8 '13 at 9:26
@ali baba: $Q$ is any countable dense subset of $\mathcal{H}$, which exists because you have assumed that $\mathcal{H}$ is separable. –  Haskell Curry Mar 8 '13 at 9:32