# Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal direct sum (finite or countable) of cyclic sub-spaces (i.e. spaces of the form $\operatorname{cls}(\operatorname{span}\{U^nx/n\in\mathbb{Z}\})$ for some vector $x$).

I couldn't find a proof for that, so if someone could give me a reference or a sketch of the proof it would be great.

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 I think you can find the answer in "A Course in Functional Analysis" by Conway. I don't have it with me right now, but I'm almost positive it is in there. – Nonliapunov Sep 25 '12 at 8:36