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https://youtu.be/YrVF2_VsBUM

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.

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https://dibeos.com/wp-content/uploads/2025/12/the_most_powerful_theorem_in_operator_theory__free-2.pdf