1. Euclidean space: open and closed sets. Compactness and Heine-Borel's theorem. Sequences and limits. 2. Multivariate calculus a. Continuity: mean value theorem. b. Differentiability: differentiation rules, approximation by a linear functional, Taylor's theorem. c. Integration: integration rules, Fubini's theorem, change of variables, differentiation under the integral, exchange of integrals and limits. d. Some classical results: Stokes' theorem, Green's formula, inverse and implicit function theorem. 3. Introduction to Function Spaces a. Banach spaces and their duals: norms, completeness, (topological) dual. b. Lp (1 <= p<= infinity) spaces and their duals: classical inequalities, representation theorems. c. The space C0([0, 1]) and its dual. 4. Introduction to Function Spaces Continued a. Hilbert Spaces: inner products, orthogonality, Gram-Schmidt process, orthonormal basis. b. L2([0, 1]) and L2(R). c. Fourier series and the Fourier transform. 5. Introduction to Partial Differential Equations a. Laplace's and Poisson's equation: fundamental solution, variational methods.