The prototypical problems for the study of eigenvalues and eigenfunctions of elliptic operators are the search for Dirichlet eigenpairs for the negative Laplacian u ∈ H 0 1 ( Ω ) , λ ∈ R , − Δ u = λ u , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath9_1.jpg"/> and their Neumann counterparts, u ∈ H 1 ( Ω ) , λ ∈ R , − Δ u = λ u , ∂ n u = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath9_2.jpg"/> We will next develop formulations for these two problems and place them in the context of the spectral orthogonal decomposition for compact self-adjoint operators, also known as the Hilbert-Schmidt theorem. We will also study how the associated orthogonal (Fourier) series associated to the eigenfunctions can be used to characterize some Sobolev spaces.