Basic Real Analysis Course | IIT Bombay | Prof. I.K. Rana | 12-Week Guide
Course Details
| Exam Registration | 158 |
|---|---|
| Course Status | Ongoing |
| Course Type | Core |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Undergraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 19 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Unlocking the Rigor of Mathematics: Your Guide to Basic Real Analysis
For students aspiring to deepen their understanding of calculus and build a rigorous mathematical foundation, Real Analysis is the essential next step. It moves beyond computation to explore the why behind the concepts. This article provides a comprehensive overview of the acclaimed Basic Real Analysis course, meticulously designed and taught by the distinguished Prof. Inder K. Rana of IIT Bombay.
Meet the Instructor: A Legacy of Excellence
Learning from Prof. I. K. Rana is a privilege. An Emeritus Fellow at IIT Bombay's Department of Mathematics, he brings over 36 years of experience teaching undergraduate and postgraduate students. His expertise is recognized through multiple prestigious awards, including the C. L. Chandna Mathematics Award (2000), the IIT Bombay Excellence in Teaching Award (2004), and the Aryabhata Award (2012). Prof. Rana is also a prolific author, with seminal texts like "Introduction to Measure and Integration" and "From Numbers to Analysis" to his credit. His pedagogical clarity makes complex topics accessible.
Course Overview: Building the Bedrock of Analysis
This 12-week undergraduate course is designed as a first encounter with Real Analysis. It aims to solidify your grasp of the real number system and build towards advanced concepts in calculus.
Intended Audience: Students from any discipline with a solid foundation in basic calculus.
Prerequisites: Exposure to fundamental calculus concepts.
Level: Undergraduate
Duration: 12 Weeks
Industry Support: The logical thinking and problem-solving skills developed are valued across all technical and analytical industries.
Week-by-Week Journey Through Real Analysis
The course is structured to progressively build mathematical maturity. Here’s a detailed breakdown of the curriculum:
| Week | Core Topics Covered |
|---|---|
| Week 1 | Review of sequences and series of real numbers. |
| Week 2 | Tests for convergence of Series. Limit superior and limit inferior. |
| Week 3 | Cauchy sequences and completeness of R. |
| Week 4 | Basic notions of Metric Spaces (focus on Rⁿ). Connectedness, Compactness, Heine-Borel Theorem. |
| Week 5 | Continuity and Uniform continuity. |
| Week 6 | Monotone functions and functions of bounded variation. |
| Week 7 | Derivatives. Mean Value Theorem and its powerful applications. |
| Week 8 | Riemann-Stieltjes integral. Riemann's Criterion for integrability. Improper integrals and the Gamma function. |
| Week 9 | Sequences and series of functions. The crucial concept of Uniform convergence. |
| Week 10 | Functions of several variables: Directional, partial, and total derivatives. |
| Week 11 | Mean Value Theorem, Taylor's Theorem for multivariable functions, and applications to optimization (Maxima/Minima). |
| Week 12 | Double and triple integrals. Statement of Fubini's Theorem and the change of variable formula (with illustrations). |
Essential Reading: The Canon of Real Analysis
To complement the lectures, Prof. Rana recommends several classic textbooks that have guided generations of mathematicians:
- T. M. Apostol, Mathematical Analysis - A comprehensive and rigorous reference.
- R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis - Renowned for its clarity and excellent exercises.
- R. R. Goldberg, Methods of Real Analysis - A classic text with a problem-solving focus.
- K. A. Ross, Elementary Analysis: The Theory of Calculus - Perfect for a first course, bridging calculus and analysis seamlessly.
Why Study This Course?
This course is more than a syllabus; it's an initiation into rigorous mathematical thought. You will transition from using calculus as a tool to understanding its underlying theorems and proofs. Key takeaways include:
- A deep understanding of limits, continuity, and differentiability.
- The ability to work confidently with convergence of sequences and series.
- Mastery of integration theory beyond the basic Riemann integral.
- An introduction to the analysis of functions in multiple dimensions.
- Development of logical proof-writing and abstract reasoning skills critical for advanced STEM fields.
Whether you are a mathematics major, an engineering student, or a curious learner, this Basic Real Analysis course under the guidance of Prof. I. K. Rana offers an unparalleled opportunity to build a strong, rigorous, and intuitive foundation for all future advanced mathematics and its applications.
Enroll Now →