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.

**Detailed Syllabus:**

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.

**References:**

*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