Lecture Format and Access to Resources. Lectures will be held on Microsoft Teams. All the resources related to the course (recordings of the lectures, notes, assignments, etc.) will be posted on the MS Team page titled MA 221 - August 2021. If you wish to attend this course, please add yourself to the MA 221 team using the team link/code available on the IISc intranet.
Instructor. Purvi Gupta (You may either email me at purvigupta(at)iisc(dot)ac(dot)in, or use the Teams chat to drop me a message.)

TAs. Agniva Chatterjee (agnivac(at)iisc(dot)ac(dot)in)
         Gouranga Mallik (gourangam(at)iisc(dot)ac(dot)in)

Lectures. Tu/Th 2:00 - 3:30 pm.
Office hours. TBA.

Course Description. The aim of this course is to introduce the language, rigour and techniques that form the foundation of mathematical analysis. It is a rite of passage for those looking to transition from merely using calculus to understanding why (and when) it works. Your ability to "do" a problem will be measured by the robustness of your arguments. Expect to encounter many inequalities, several notions of limits, and some long (but very important) constructions.

We will cover all the topics listed here; this essentially corresponds to Chapters 1-7, 9 in Rudin's book. In the calendar below, we will build a more precise list of topics over the course of the semester.

  • (Main) W. Rudin, Principles of Mathematical Analysis, 3rd edition.
  • (Additional) T. M. Apostol, Mathematical Analysis, 2nd edition.

Evaluation Scheme.
Course Calendar.

# Date (Day)               Topics               Assignments/Tests
Week 1 Assignment 1 (due on 15/08)
1. 03/08 (Tu) Introduction; basic set theory
2. 05/08 (Th) From $\mathbb N$ to $\mathbb Q$
Week 2
3. 10/08 (Tu) Ordered fields, the real line
4. 12/08 (Th) Construction of $\mathbb R$
Week 3 Assignment 2 (due on 30/08)
5. 17/08 (Tu) Properties of $\mathbb R$
6. 19/08 (Th) Cardinality
Week 4
7. 24/08 (Tu) Metric spaces; open and closed sets
8. 26/08 (Th) Compactness Test I (last 45 min. of class)
Week 5 Assignment 3 (due on 17/09)
9. 31/08 (Tu) Compactness (contd.), the Cantor set
10. 02/09 (Th) Connectedness
Week 6
11. 07/09 (Tu) Sequences: limits and subsequential limits
12. 09/09 (Th) Sequences: completeness, order and algebraic limit theorems
Week 7 Practice problems
13. 14/09 (Tu) Sequences: upper and lower limits, monotonicity; convergence of series
14. 16/09 (Th) Series: tests for positive series, p-series
Week 8
15. 21/09 (Tu) Series: power series, algebraic combinations, rearrangements
16. 23/09 (Th) Limits and continuity: basic properties and examples
Midterm Week (no classes): Test II on 28/09 at 2:00 pm
Week 9 Assignment 4 (due on 19/10)
17. 5/10 (Tu) Continuity and topology
18. 7/10 (Th) One-sided limits and discontinuities
Week 10
19. 12/10 (Tu) Differentiation: basic properties, mean value theorems
20. 14/10 (Th) Differentiation: mean value theorems, Taylor's theorem
Week 11 Assignment 5 (due on 06/11)
19/10 (Tu) No class due to holiday
21. 21/10 (Th) Riemann integration: some sufficient conditions for integrability
Week 12
22. 26/10 (Tu) The fundamental theorems of calculus, integration techniques
28/10 (Th) Test III
Week 13
23. 02/11 (Tu) Uniform convergence, the metric space of bounded continuous functions
04/11 (Th) No class due to holiday
Week 14 Assignment 6
24. 09/11 (Tu) Equicontinuity; the Arzela--Ascoli Theorem
25. 11/11 (Th) Uniform continuity, Integration and Differentiation
Week 15
26. 16/11 (Tu) The Weierstrass Approximation Theorem; Limits and continuity of multivariable functions
27. 18/11 (Th) Differentiability of functions of several variables
Week 16
28. 23/11 (Tu) Directional derivatives; Inverse Function Theorem (additional video will be posted)
25/11 (Th) Test IV