Functional Analysis is a core course in any mathematics curriculum at the masters level. It has wide ranging applications in several areas of mathematics, especially in the modern approach to the study of partial differential equations. The proposed course will cover all the material usually dealt with in any basic course of Functional Analysis. Starting from normed linear spaces, we will study all the important theorems, with applications, in the theory of Banach and Hilbert spaces. One important feature of the proposed course is the detailed treatment of weak topologies. Prerequisites are familiarity with real analysis, topology and linear algebra. Knowledge of measure theory is desirable . INTENDED AUDIENCE :MSc (Mathematics) and above PREREQUISITES : BSc (Mathematics) Real Analysis, Topology, Linear Algebra, Measure TheoryINDUSTRIES SUPPORT :None
Week 1: Normed linear spaces, examples. Continuous linear transformations, examples.Week 2: Continuous linear transformations. Hahn-Banach theorem-extension form. Reflexivity.Week 3: Hahn-Banach theorem-geometric form. Vector valued integration.Week 4: Baire’s theorem,.Principle of uniform boundedness. Application to Fourier series. Open mapping and closed graph theorems.Week 5: Annihilators. Complemented subspaces. Unbounded operators, Adjoints.Week 6: Weak topology. Weak-* topology. Banach-Alaoglu theorem. Reflexive spaces.Week 7: Separable spaces, Uniformly convex spaces, applications to calculus of variations.Week 8: L^p spaces. Duality, Riesz representation theorem.Week 9: L^p spaces on Euclidean domains,.Convolutions. Riesz representation theorem.Week 10: Hilbert spaces. Duality, Riesz representation theorem. Application to the calculus of variations. Lax-Milgram lemma. Orthonormal sets.Week 11: Bessel’s inequality, orthonormal bases, Parseval identity, abstract Fourier series. Spectrum of an operator.Week 12: Compact operators, Riesz-Fredholm theory. Spectrum of a compact operator. Spectrum of a compact self-adjont operator.