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.

# | 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 |