= Weyl-von Neumann theorem
{c}
Every <self-adjoint operator> on a separable complex <Hilbert space> is diagonal modulo a <compact operator>. For a bounded operator the compact self-adjoint error can have arbitrarily small <Hilbert-Schmidt norm>. In a cyclic spectral representation, partition a bounded real interval into $N$ intervals of width at most $\delta$ and successively bisect. Orthogonal step-function differences give basis vectors supported in the parent intervals. A diagonal value chosen in each support interval gives total squared column error at most $3N\delta^2$. Since $N=O(\delta^{-1})$, this tends to zero. Apply the construction to countably many cyclic summands with summable squared error bounds. For an unbounded operator first split into bounded spectral intervals, obtaining a bounded compact error on the same domain.
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