This textbook derives the spectral theorem for self-adjoint operators from the results of linear algebra regarding the diagonalization of Hermitian matrices. Lebesgue-Stieltjes integrals are utilized, and Helly's selection and convergence theorems for monotone functions are provided. We construct the spectral family through a technically demanding approximation, with the Stieltjes inversion formula at the center of the proof. One result of this is that self-adjoint operators possess not only a discrete spectrum but also a continuous spectrum. The occurring scattering eigenvalues cannot be obtained through variational methods. We then turn to the central question of which elliptic differential operators have a self-adjoint extension and thus fall within the scope of the spectral theorem.