4. Spectral reduction of compact normal operators

A model for this is provided by Fourier series decomposition. If we denote S ¹ the unit circle, then for θ ∊ S ¹ , the functions $e^{\mathit{in\theta }},\mathrm{\hspace{0.17em}}n\in \mathbb{Z}$ form a self-adjoint subalgebra of C ⁰ (S ¹ ) that separates the points, so it is dense in C ⁰ (S ¹ ) by virtue of Stone-Weierstraβ's theorem. In other words, any continuous periodic function f is the sum of a uniformly convergent series of the...