Gautam Bharali

               Department of Mathematics

                 Indian Institute of Science

                 Bangalore 560012


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  • Announcements on special measures related to COVID-19 concerning lectures/tutorials

    Lectures and tutorials will be conducted online and in real-time until further instructions from the Institute.

    The first lecture will be on January 5, 11:00 a.m.–noon.

    Online lectures will be conducted via MS Teams. To reiterate: the lectures will be in real-time, and interactive (to the extent that the technology permits). A recording of each online lecture will be available for students to re-watch or to download.

    Students, whether they are taking the course for credit or auditing it, must join the UM204 Team. A joining code will be posted on this page by 11:59 p.m. on Jan 3, 2022 (to access it, you must be logged into your IISc Microsoft Outlook account in a separate tab in the same browser). Students not from IISc should e-mail me to receive the joining code.

  • Meeting times

    Lectures: Mondays: 11:00–11:55 a.m., Wednesdays and Fridays, 11:00 a.m.–noon

    Tutorials: Mondays, 12:05–1:00 p.m.

  • All about this course: Offline-specific protocols/information coming soon!

  • Recommended books

    Walter Rudin, Principles of Mathematical Analysis, 3rd Ediition, McGraw-Hill International Editions, 1976.

    Terence Tao, Analysis–I, 3rd Ediition, TRIM Series, Hindustan Book Agency, 2014.

    T.M. Apostol, Mathematical Analysis, 2nd Ediition, Narosa, 1996.

  • Teaching Assistants (replace «...» by in the addresses below)

    (1) Kiran Kumar Behera: (kiranbehera@«...»)

    Tutorial room: G-21, Old Physics Building

    (2) Kartick Ghosh: (kartickghosh@«...»)

    Tutorial room: G-01, Old Physics Building

  • Documents

    Hints to solving the problems in the mid-semester exam

  • Syllabus ( the topics below comprise the syllabus of the final exam )

    Your lecture notes will cover all the material (except for those results assigned for self-study) in the syllabus. The occasional chapter references below are to a more extensive treatment of the topic in question and indicate the primary source of the material presented in the lectures.

    Aspects of the theory of sets, the axioms of specification and union, De Morgan's laws

    Two-fold cartesian products, relations and functions, equivalence relations

    The natural numbers, Peano's axioms, mathematical induction, Peano arithmetic

    The integers: the definition/construction of the set of integers and integer arithmetic

    Fields, the rational numbers: the definition/construction of the set of rationals and rational arithmetic

    Ordered sets, the "usual order" on the rationals, ordered fields

    (The treatment of the above topics follows, although selectively, that of Chapters 1–4 of Tao's Analysis 1)

    The least upper bound property, the definition/meaning of the system of real numbers

    Dedekind cuts, construction of the real line (Chapter 1: Appendix of Rudin's Principles)

    The Archimedean property of the real line

    Metric spaces, open and closed sets in metric spaces and associated concepts, the closure of a set

    Open and closed sets relative to a metric subspace

    Compact sets in a metric space (Chapter 2 of Rudin's Principles)

    The characterisation of compact subsets of Euclidean spaces

    Countable and uncountable sets, the Cantor set (and its uncountability from first principles)

    Sequences and convergence

    Subsequences, subsequential limits

    The extended real number system, limits at infinity, upper and lower limits (i.e., limsup and liminf)

    Topics listed up to this point comprise the syllabus of the mid-term examination. They will also be a part of the syllabus of the final examination.

    Extracting convergent subsequences and the role of compactness

    Cauchy sequences, the definition of completeness

    Sufficient conditions for completeness

    Infinite series and their convergence, the Cauchy criterion

    Convergence tests for non-negative series, criterion for summability of the series of pth powers

    Absolute convergence, the Ratio and Root Tests, convergence tests for non-negative series as tests for absolute convergence, conditional convergence

    Self-study: The limits of special sequences (the section Some Special Sequences in Chapter 3 of Rudin's Principles)

    The limit of a function: various equivalent definitions, the algebra of limits

    Continuity at a point, continuous functions

    Continuity and compactness, attainment of extreme values, uniform continuity

    Connectedness, the characterisation of all connected subsets of the real line

    Continuity and connectedness, the Intermediate Value Theorem

    Differentition in one variable

    Review (self-study): The algebra of derivatives, the chain rule for differentiation, the relation between critical points and points of local maximum/minimum, Rolle's theorem, Lagrange's mean value theorem and its applications (pages 104–108 of Rudin's Principles)

    Higher-order derivatives, Taylor's theorem

    Integration: motivation, the Riemann integral, and a characterisation of Riemann integrability

    Riemann integrability of continuous functions

    (The treatment of the above topics in the theory of the Riemann integral is that of Chapter 6 of Rudin's Principles, with the following major difference: the gauge α in Rudin's discussion of the Riemann–Stieltjes integral is just id[a, b] for this course.)

    The first Fundamental Theorem of Calculus

    Primitives/antiderivatives, the second Fundamental Theorems of Calculus

    Techinques of integration: integration by parts (the change-of-variables theorem, which you know from UM101, will be of no relevance in our present treatment of integration)

    Sequences of functions, motivations for uniform convergence and some examples

    Uniform convergence and Riemann integration

    The sup-metric, the relationship between uniform convergence and convergence relative to the sup-metric

    The space Cb(X;V) and its completeness (when V is a complete normed vector space)

    Equicontinuity, the Arzela–Ascoli Theorem (note: proof not included, see the Announcements section for more on this)

    The interpretation of the Arzela–Ascoli Theorem in terms of compact sets in Cb(X;Rk)

  • Announcements

    Apr. 13: Our treatment of the Arzela–Ascoli Theorem differs considerably from that given in Rudin's book, the principal differences being: (i) no single non-technical result summarising the results on pages 155–158 is presented in the book (perhaps because any version of the latter result has a lengthy proof); and (ii) no connection with compact sets in Cb(X;Rk) is given in the book. Therefore, to the students who were absent from the last lecture: please consult the notes of someone who attended class on April 13!

    Apr. 11: The end-semester examination is scheduled for April 19, 9:30 a.m. Venue: Room G-01 in the Old Physics Building, Duration: stay tuned!

    Mar. 24: The second of the two make-up lectures will be on March 26 during the time-slot agreed upon. Venue: Room G-01 in the Old Physics Building.

    Feb. 26: The mid-semester examination is scheduled for March 1, 10:00–11:30 a.m. Venue: Room G-01 in the Old Physics Building.

    Feb. 11: The mid-semester exams have been postponed to the period February 28–March 5. These will be in-person exams!

    Feb. 11: In-person lectures will begin from Wednesday, February 23. Location TBA Location: Room G-01 in the Old Physics Building. (In-person tutorials, therefore, will only commence after the mid-semester examination week.)

    Jan. 9: The time-slot for the two make-up lectures has been fixed at 3:00–4:00 p.m. on Saturday. The first make-up lecture will be on February 12.

    Jan. 9: The mid-semester exams are scheduled for the period February 21–26. Stay tuned for the date and time of the UM204 mid-semester exam.

  • Homework assignments

    Homework 13

    Homework 12

    Homework 11

    Homework 10

    Homework 9

    Homework 8

    Homework 7

    Homework 6

    Homework 5

    Homework 4

    Homework 3

    Homework 2

    Homework 1

  • Quiz solutions

    The solution to Quiz 8

    The solution to Quiz 7

    The solution to Quiz 6

    The solution to Quiz 5

    The solution to Quiz 4

    The solution to Quiz 3

    The solution to Quiz 2

    The solution to Quiz 1


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Page last updated on April 16, 2022