Core course for M.Sc. Mathematics
Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 231/320 Real Analysis II (for undergraduates)
Overview: The aim of this course is to build a rigorous base for advanced topics such as Complex Analysis, Measure & Integration, Numerical Analysis, Functional Analysis, and Differential Equations. The course design also attempts to take into account the diverse backgrounds of our students.
Topics under each section are divided in two parts. Part (a) contains topics that will be covered only briefly whereas topics in part (b) will be done in detail.
- Real number system
- Archimedean property, density of rationals, extended real numbers, countable sets, uncountable sets.
- Cauchy completeness of reals, Axiom of Choice, Zorn’s Lemma, equivalence of AC & ZL.
- Metric spaces
- Definitions and examples, open sets, closed sets, limit points, closure,equivalent metrics, relative metric, product metric, convergence, continuity, connectedness, compactness.
- Uniform continuity, completion of a metric space, Cantor’s intersection property, finite intersection property, totally bounded spaces, characterization of compact metric spaces.
- Sequences and Series of Functions
- Pointwise and uniform convergence, uniform convergence and continuity, uniform convergence and integration, differentiation, Weierstrass M-test.
- Power series, exponential and logarithmic functions, Fourier series, equicontinuous family of functions, Stone-Weierstrass approximation theorem, Arzela-Ascoli theorem.
- Principles of Mathematical Analysis by Walter Rudin, Tata McGraw-Hill
- Mathematical Analysis by Tom M. Apostol, Narosa
- Topology of Metric Spaces by S. Kumaresan, Narosa
- Introduction to Topology & Modern Analysis by G. F. Simmons, Tata McGraw-Hill
- Real Analysis by N. L. Carothers, Cambridge University Press