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Just finished the third piece in my series on regular Sturm-Liouville theory. This one is about self-adjoint operators. The central idea is the symmetry: \langle Tu,v\rangle=\langle u,Tv\rangle. When an operator satisfies this identity, two fundamental spectral properties follow at once: every eigenvalue is real, and eigenvectors corresponding to distinct eigenvalues are automatically orthogonal. These are exactly the same reasons a Hermitian matrix has a real spectrum and orthogonal eigenvectors; the Sturm–Liouville operator is the differential-operator analogue of that finite-dimensional situation. The article first recalls the matrix representation of a linear operator and its adjoint, then shows how the same symmetry appears for second-order differential operators with the classical boundary conditions, and finally explains why this is the source of the orthogonal eigenfunction expansions that make Sturm–Liouville theory so useful. In the next article we will use the self-adjointness established here to prove that a regular Sturm–Liouville operator has a countably infinite sequence of real eigenvalues tending to +\infty, and that the corresponding eigenfunctions form a complete orthogonal basis of L^2[a,b]. This will finally justify the eigenfunction expansions that make the theory so powerful. Here’s the link: https://t.co/sAVFa2wkeE