Gautam Bharali

               Department of Mathematics

                 Indian Institute of Science

                 Bangalore 560012


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  • Meeting times

    Lectures: Mondays, Wednesdays and Fridays 11:05 a.m.–12:05 p.m.

    Tutorials: Fridays 12:05 p.m.–1:00 p.m.

  • All about this course (PDF)

  • 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)

    G-02 GROUP

    Teaching assistant: Prateek Kumar Vishwakarma (prateekv@«...»), Location: Room G-02, Old Physics Building

    G-20 GROUP

    Teaching assistant: Ramesh Chandra Sau (rameshsau@«...»), Location: G-20, Old Physics Building

  • Documents

  • Syllabus (tentative: the list below will grow as the semester progresses )

    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.

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

    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 integers: the definition/construction of the set of integers and integer arithmetic, ordering the integers

    The rational numbers, fields, ordered fields and the "usual order" on the rationals

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

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

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

    The Archimedean property of the real line

    Countable and uncountable sets

    Metric spaces, open and closed sets in metric spaces and associated concepts

    Closure and interior of a set in a metric space, and their relative analogues

    Compact metric spaces (Chapter 2 of Rudin's Principles)

    The characterisation of compact subsets of Euclidean spaces

    Properties of compact subsets of a metric space, the Cantor set

    Sequences and convergence

    Cauchy sequences, the definition of completeness, completeness of the real line

    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.

    Sufficient conditions for completeness

    Subsequences, subsequential limits

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

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

    Infinite series and their convergence, criteria for convergence

    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

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

    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, and applications

    Differentition in one variable

    The chain rule for differentiation in one variable

    Geometric interpretations of the derivative, the relation between critical points and points of local maximum/minimum

    Rolle's theorem, Lagrange's mean value theorem and its applications

    Higher-order derivatives, Taylor's theorem

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

    Riemann integrability of continuous functions, functions with discontinuities

    (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 and second Fundamental Theorems of Calculus

    Techinques of integration: the change-of-variable formulas for the Riemann integral, integration by parts

    Definition and properties of the logarithm

    Sequences of functions, examples and motivations for uniform convergence

    Uniform convergence and Riemann integration

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

    The space Cb(X;Y) and its completeness (when Y is complete)

    Equicontinuity, compact sets in C(K;Rn): the Arzela–Ascoli Theorem (proof not included)

    The Weierstrass approximation theorem (Chapter 7 of Rudin's Principles. Note: we omit the Stone–Weierstrass theorems.)

  • Announcements

    Apr. 30: The graded answer-scripts of the end-semester exam can be viewed this afternoon from 4:30 to 5:30 p.m. Venue: Lecture Hall 4, Department of Mathematics.

    Apr. 21: The end-semester exam will be held on April 27 at 9:30 a.m. Venue: Room G-21 in the Old Physics Building.

    Apr. 4: There will be an extra classe on April 6 from 3:15–4:15 p.m. Venue: Room G-02 in the Old Physics Building.

    Mar. 8: The lecture on March 11 will begin at 11:10 a.m. (owing to one of the mandatory Safety Workshops being organised by the Institute).

    Feb. 20: The mid-semester examination will be held in Room G-21 in the Old Physics Building.

    Feb. 13: The mid-semester examination is scheduled for February 23 at 9:30 a.m. Venue: TBA

    Feb. 13: The undergraduate mid-semester examination week is February 15–23. There will be no lectures during this period. HOWEVER we shall meet for a pre-examination tutorial session at the usual tutorial time on February 15, the first 10 minutes of which would be devoted to winding-up the discussion started on February 13. Venue for the February 15 tutorials: Room T-01 (for the G-02 group) and Room T-02 (for the G-20 group) in the UG Administrative Block.

    Feb. 1: The venue of the make-up lecture on February 2 is Lecture Hall 4 at the Department of Mathematics.

    Jan. 25: There will be a make-up lecture on February 2. Time: 12:10 p.m. to 1:10 p.m., Venue: TBA.

    Jan. 23: Tutorials begin from February 1. Locations TBA.

    Jan. 18: The form for finding a time for the tutorial for the course must be filled by 11:59 p.m. of January 21.

    Jan. 11: In view of the request made by students taking DS289, we shall meet from 11:05 a.m. to 12:05 p.m. (at our usual venue) from January 16.

  • 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 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


  • UNDERGRADUATE ANALYSIS & LINEAR ALGEBRA (UM101)   [Autumn 2015, Autumn 2017]


  • ANALYSIS–II: MEASURE AND INTEGRATION (MA222)  [Spring 2012, Spring 2017 ]

  • COMPLEX ANALYSIS (MA224)  [ Spring 2016 ]

  • TOPICS IN COMPLEX ANALYSIS (MA324)  [Spring 2014]

  • INTRODUCTION TO SEVERAL COMPLEX VARIABLES (MA328-329)  [experimentally as a "topics course" (MA329) in Autumn 2014 ]


Page last updated on April 30, 2019