Course Description. The aim of this course is to introduce the language, rigour and techniques that form the foundation of mathematical analysis. MA 221 is a core course for first-year Int. Ph.D. students. However, you are welcome to register for this course, if you are looking to transition from merely using calculus to understanding why (and when) it works. In this course, your ability to "solve" a problem will be measured by the robustness of your arguments.

We will (roughly) cover Chapters 1-9 from Rudin's book (listed below). Lecture-wise topics will be collated in the calendar below.

References.
  • (Main) W. Rudin, Principles of Mathematical Analysis, 3rd edition.
  • (Additional) T. M. Apostol, Mathematical Analysis, 2nd edition.
Instructor. Purvi Gupta (You may either email me at purvigupta(at)iisc(dot)ac(dot)in, or us MS Teams to drop me a message.)

Lecture timings. MWF 2:00 - 3:00 pm
Venue. LH-4, Mathematics
Office hours. W 4:00 - 5:00 pm (in L-25).

TA. Sudip Dolai (sudipdolai(at)iisc(dot)ac(dot)in)
TA Office hours. M 5:30 - 6:30 pm
Location. X-20
Access to Resources. All the resources related to the course will be posted on the course MS Teams page. Please add yourself to the MA 221 team using the team link available on the IISc intranet.
Evaluation Scheme.
Homework submission. A new assignment will be posted on Teams (almost) every weekend. Some of the problems will be marked for submission. Please submit a hard copy of your solutions to those to Sudip Dolai by the deadline stated on the assignment. Your graded copies will be available for pick up from Sudip's office in a week or so. He will post an annoucement regarding their availability.
Course Calendar.
*Each week runs from Sunday to Sunday in this calendar.

# Day               Topics               Assignments/Tests
Week 1: July 28-Aug 03*
1. F Basic set theory
Week 2: Aug 04-Aug 10* Assignment 1 posted
2. M Gaps in Q; ordered fields
3. W A construction of R
4. F Some important properties of R
Week 3: Aug 11-Aug 17* Assignment 2 posted
5. M The extended real line, the complex field Assignment 1 is due
6. W Cardinality
F No class due to institute holiday
Week 4: Aug 18-Aug 24* Assignment 3 posted
7. M Metric spaces Assignment 2 is due
8. W Metric topology
9. F Compactness
Week 5: Aug 25-Aug 31* Assignment 4 posted
10. M Connectedness and relative topology
11. W Perfect sets Assignment 3 is due
12. F Convergent sequences
Week 6: Sept 01-Sept 07* Assignment 5 TBP
13. M Completeness
14. W Completeness (contd.) Assignment 4 is due
15. F Upper and lower limits
Week 7: Sept 08-Sept 14
16. M Series
17. W Series (contd.)
18. F Review Assignment 5 is due
Week 8: Sept 16-Sept 22
M Institute Holiday
W No class due to midterm week
F Mid-sem I from 2-4 pm
Course dropping (without mention): Sept. 21-23
Week 9: Sept 23-Sept 29
19. M Power series, products
20. W Rearrangements Assignment 6 TBP
21. F Functional limits and continuity
Week 10: Sept 30-Oct 06
22. M Continuity and topology
W Institute Holiday Assignment 7 posted
23. F Uniform continuity Assignment 6 is due (in class)
Week 11: Oct 07-Oct 13
M Class cancelled
24. W Linear transformations
25. F Differentiability of functions (of one and several variables)
Week 12: Oct 14-Oct 20 Assignment 8 posted
26. M Differentiability (contd.) Assignment 7 is due (in class)
27. W Some properties of differentiable functions
28. F Directional derivatives
Week 13: Oct 21-Oct 27
29. M Mean value theorems
30. W Partial derivatives, L'Hopital's rule Assignment 8 is due (in class)
F Mid-sem II from 4-6 pm
Week 14: Oct 28-Nov 03
31. M Inverse Function Theorem Assignment 9 is posted
32. W Riemann Integration
F Institute Holiday
Week 15: Nov 04-Nov 10
33. M Sufficient conditions for Riemann integrability
34. W Fundamental theorems of calculus Assignment 9 is due
35. F Sequences and series of functions
Week 16: Nov 11-Nov 17
36. M Uniform continuity
37. W Equicontinuity
F Insitute holiday
Week 17: Nov 18-Nov 22
38. M EXTRA CLASS: Wrap-up Assignment 10 is due
Week 18: Nov 25-Nov 29
Tu FINAL EXAMINATION from 9-noon on November 26