Lecture, four hours; discussion, one hour; outside study, seven hours. Recommended preparation: multivariable calculus. Review of functional analysis concepts. Vector spaces, norms, convexity, convergence, continuity, Banach/Hilbert/Sobolev spaces. Linear functionals. Orthonormal sets, linear operators and their spectrum. Minimum distance problems, least squares. Lagrange multipliers, nonlinear duality, variational methods. Finite difference and finite element approximation of partial differential equations (PDEs). Letter grading.