Spectral Theorem
This PDF will give you a clear introduction to the spectral theorem through the concrete example of the Dirichlet Laplacian operator on the Hilbert space. We will look at definitions of self-adjoint and normal operators, verify the self-adjointness of the Laplacian via integration by parts, and use the spectral theorem to explicitly compute its spectral decomposition in terms of eigenvalues and normalized eigenfunctions. The PDF includes 7 exercises with detailed solutions, as well as an appendix. Thank you for supporting our work!