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* ''U'' is [[surjective]] and preserves the [[inner product]] on the Hilbert space, so that for all [[vector]]s ''x'' and ''y'' in the Hilbert space,
:<math>\langle Ux, Uy \rangle = \langle x, y \rangle. t= p \dot x'</math>
[[Unitary matrix|Unitary matrices]] are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalisation of the notion of a unitary matrix.
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