DIAGONALISATION OF LINEAR OPERATORS IN INFINITE DIMENSIONS WITH APPLICATION TO PDE

Sammanfattning

We first prove the spectral theorem for symmetric operators on finite-dimensional real vector spaces via the Rayleigh quotient, motivating a variational approach that generalises naturally to the infinite-dimensional setting. This means a diagonalisation of the operator with respect to a basis of eigenvectors. We then give a concise, constructive proof of the spectral decomposition theorem for linear, compact and symmetric operators on real Hilbert spaces, and apply the result to solve a class of elliptic boundary value problems.