spectral theorem

The spectral theorem is the grand generalization of 'a symmetric matrix can be diagonalized by an orthonormal basis.' It says that the nicest operators on a Hilbert space — self-adjoint or, more generally, normal — are, after a change of viewpoint, nothing more than multiplication by a function. Diagonalization, which is so clean in finite dimensions, survives into infinite dimensions in the language of integration against a spectral measure.

In its compact form: a compact self-adjoint operator T has an orthonormal basis of eigenvectors, and T x = sum lambda_n <x, e_n> e_n with real eigenvalues lambda_n tending to 0. In its general (bounded) form: every bounded normal operator T is unitarily equivalent to multiplication by a bounded measurable function on some L^2(mu) space; equivalently T = integral lambda dE(lambda) for a unique projection-valued spectral measure E supported on the spectrum.

The catch is that genuine eigenvectors may not exist — there may be no nonzero vector that the operator merely scales. The continuous spectrum is handled not by a sum over eigenvalues but by an integral against the projection-valued measure E, which assigns to each Borel subset of the spectrum an orthogonal projection. This is the precise sense in which 'diagonalization' persists even when no eigenbasis exists, and it is the rigorous foundation of the functional calculus.