spectral theorem for compact self-adjoint operators

This is the theorem where the comfortable finite-dimensional picture survives almost untouched into infinite dimensions. A symmetric matrix has a full orthonormal eigenbasis and real eigenvalues; a compact self-adjoint operator on a Hilbert space has essentially the same thing — a countable orthonormal eigenbasis and real eigenvalues — with one new wrinkle dictated by compactness: the eigenvalues must march toward zero.

Precisely: let T be a compact self-adjoint operator on a Hilbert space H. Then T has an at most countable set of real eigenvalues lambda_1, lambda_2, ... which, if infinite, converge to 0, each nonzero eigenvalue having finite-dimensional eigenspace. The corresponding eigenvectors can be chosen orthonormal, and on the closure of their span T acts as T x = sum lambda_k <x, e_k> e_k. The whole space decomposes as the eigenvectors plus the kernel of T.

Why the accumulation at zero is forced: if infinitely many eigenvalues stayed bounded away from 0, the corresponding orthonormal eigenvectors would map to vectors of bounded length with no convergent subsequence, contradicting compactness. So 0 is the only allowed accumulation point — and 0 itself need not be an eigenvalue, but it always lies in the spectrum of a compact operator on an infinite-dimensional space.

Why it matters: this is the rigorous engine behind separation of variables, Sturm-Liouville theory, the eigenfunction expansions of mathematical physics, and the singular value decomposition extended to operators. The diagonalization x = sum <x,e_k> e_k followed by T x = sum lambda_k <x,e_k> e_k is the exact infinite-dimensional analog of writing a symmetric matrix as Q D Q^T.