The course is divided into two parts: A first part dedicated to foundational topics in real and functional analysis, and a second part focused on more advanced topics. Part 1: Introduction to Functional Analysis Normed spaces and Banach spaces. Examples. Finite-dimensional spaces. Lp spaces. Riesz–Fischer Theorem. Hahn–Banach Theorem and consequences. Reflexivity. Baire Category Theorem, Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem, with applications. Weak topologies and weak and weak-* convergence. Compactness theorems for sequences in weak topologies. Banach–Alaoglu Theorem. Hilbert spaces. Orthogonality and orthonormal bases. Riesz Representation Theorem and the dual of Hilbert spaces. Trigonometric polynomials and L2-theory of Fourier series: Bessel's inequality, Parseval’s identity, and Plancherel’s identity. Orthonormal bases in L2(−π,π). Convolution in RnRn, Minkowski’s integral inequality, and Young’s inequality. Regularizing kernels. Part 2: Depending on the composition of the class and the time available, topics will be selected from the following list: Geometric form of the Hahn–Banach Theorem. Strong and weak closures of convex sets. Mazur’s Lemma. Extreme points and the Krein–Milman Theorem. Reflexivity: Kakutani and Eberlein–Šmulian Theorems. Uniformly convex spaces. Milman–Pettis Theorem. Uniform convexity of Lp spaces. Topological vector spaces and seminormed spaces. Metrizability. Topology of spaces of differentiable functions and introduction to distributions. Linear operators and their adjoints. Projection, isometric, and unitary operators. Finite-rank and compact operators. Hilbert–Schmidt operators. Closed and closable operators. Symmetric, self-adjoint, and normal operators. Self-adjoint extensions of symmetric operators. Basic notions of spectral theory. Spectral theory of compact operators. Spectral theorem for self-adjoint and normal operators. Introduction to the theory of semigroups of operators. Semigroups and their generators. Hille–Yosida Theorem. Applications to differential equations. Further topics in measure theory: Construction of measures. Product measures. Tonelli–Fubini Theorem. Positive Borel measures. Riesz Representation Theorem. Continuity properties of measurable functions. Signed and complex measures: Total variation, Hahn and Lebesgue decomposition theorems. Radon–Nikodym Theorem. Characterization of the dual of Lp spaces. Hardy–Littlewood maximal function. Differentiation of measures. Lebesgue differentiation: Differentiation of monotone functions, functions of bounded variation, differentiation of an integral, and absolutely continuous functions. Characterization theorem for absolutely continuous functions. Introduction to interpolation: Riesz–Thorin and Marcinkiewicz Interpolation Theorems. Lp spaces: Density and separability properties. Compactness in Lp: Kolmogorov–Riesz Theorem. Additional topics on Fourier series: Dirichlet and Fejér kernels. Hausdorff–Young Theorem. Pointwise convergence, Dini’s Theorem, and absolute convergence, Bernstein’s Theorem. Fourier transform in Rn: L1 theory and inversion theorem; L2 theory: Plancherel’s identity and inversion theorem. Hausdorff–Young inequality.