Course Content
Sets and numbers. Limits: sequences, convergence, Cauchy sequences, limit points, limsup, liminf, subsequences. Convergence of inifnite series. Applications. Normed spaces, completeness, Banach space. Inner-product spaces, Hilbert space. Applications. Continuous functions: continuity, left/right continuity, uniform continuity, intermediate value theorem. Applications. Fundamentals of topological spaces, compactness, connectedness. Topological groups. Applications. Differentiation. Inverse function theorem. Applications. Integration: Riemann integration. Lebesgue measure, Lebesgue integration. Applications. Function spaces, convergence of functions. Applications.
Text / References
- 1 T. Tao, Analysis I and II, Second Edition, Hindustan Book Agency, 2009.K. Kuratowski, Introduction to Set Theory and Topology, Oxford, 1961.C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978.