Real Analysis I | Department of Mathematics

# Real Analysis I

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Calculus I (MAT 101)

Overview: This course provides a rigorous base for the geometric facts and relations that we take for granted in one-variable Calculus. The main ingredients include sequences; series; continuous and differentiable functions on R; their various properties and some highly applicable theorems. This is the foundational course for further study of topics in pure or applied Analysis, such as Metric Spaces, Complex Analysis, Numerical Analysis, and Differential Equations.

Detailed Syllabus:

1. Fundamentals: Review of N, Z and Q, order, sup and inf, R as a complete ordered field, Archimedean property and consequences, intervals and decimals. Functions: Images and pre-images, Cartesian product, Cardinality.
2. Sequences: Convergence, bounded, monotone and Cauchy sequences, subsequences, lim sup and lim inf.
3. Series: Infinite Series: Cauchy convergence criterion, Infinite Series of non-negative terms, comparison and limit comparison, integral test, p-series, root and ratio test, power series, alternating series, absolute and conditional convergence,  rearrangement.
4. Continuity: Limits of functions, continuous functions, Extreme Value Theorem, Intermediate Value Theorem, monotonic functions, uniform continuity.
5. Differentiation: Differentiable functions on R, local maxima, local minima, Mean Value Theorems, L'Hopital's Rule, Taylor's Theorem.

References:

1. A Basic Course in Real Analysis by Ajit Kumar and S Kumaresan. CRC Press, 2014.
2. Introduction to Real Analysis by R G Bartle & D R Sherbert, John Wiley & Sons, Singapore, 2/e (or later editions), 1994.
3. Elementary Analysis: The Theory of Calculus by Kenneth A Ross. Springer India, 2004.
4. Analysis I by Terence Tao. Hindustan Book Agency. 2nd Edition, 2009.
5. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3rd Edition, 2006.
6. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002.
7. Calculus, Volume 1, by Tom Apostol, Wiley India. 2nd Edition, 2011.