= Stone theorem for the circle group
{c}
{title2=$U_z=\sum_{n\in\mathbb Z}z^nP_n$}
A strongly continuous <unitary representation> of the <circle group> splits as an orthogonal sum of character subspaces. The <projections> are $P_n=\int_{\mathbb T}z^{-n}U_z\,dz$. Character <orthogonality> makes them mutually orthogonal, and <Fejér sum> shows their ranges span the represented <Hilbert space>. The <self-adjoint> generator has integer <eigenvalues>, with domain given by $\sum_n n^2\|P_n\xi\|^2<\infty$.
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