Methods of Applied Mathematics I

This is the graduate-level toolkit course for doing math on infinite-dimensional spaces, you learn to treat functions as vectors, differential operators as linear maps, and ODE boundary-value problems as spectral problems in Hilbert space. Expect problem sets that move between abstract operator theory (Fredholm alternative, compact operators, distributions) and concrete machinery (Green's functions, Sturm-Liouville expansions, orthogonal polynomials), with Keener as the main reference. It's the foundation course before PDEs, asymptotic methods, and numerical analysis at the graduate level, and it's where the hand-wavy "expand in eigenfunctions" tricks from undergrad finally get rigorous justification.