Spectral Theorem (Compact Self-Adjoint Operators)

For a compact self-adjoint operator T on a Hilbert space H, the spectral theorem states that H admits an orthonormal basis of eigenvectors of T, the corresponding eigenvalues are real, countable (with possible accumulation at 0) and each nonzero eigenvalue has finite multiplicity; T is representable as a convergent series T = Σ λ_k ⟨·,u_k⟩ u_k.