There will be 33 Lectures, 90-120 minutes each.

Syllabus (Page 155)

Evaluation Plan

• End-Semester Examination: 80 marks

• Assignments: 10 marks

• Class Assessment: 10 marks

Office hours: Monday to Friday, 4 to 5 PM.

     Topics covered in

• Lecture 01 : Well ordering principle, Finite and infinite sets, The set of all natural numbers is not finite.

• Lecture 02 : Countable and uncountable sets-definitions and examples.

• Lecture 03 : The set of all rational numbers is countable.

• Lecture 04 : Countable union of countable sets is a countable set, Cantor's theorem.

• Lecture 05 : Supremum, Infimum.

• Lecture 06 : Completeness property of ℝ.

• Lecture 07 : Completeness gives the Archimedean property which in turn gives : existence of nth square roots of positive real numbers, and the density of ℚ  in ℝ.

• Lecture 08 : Existence of nth square roots of positive real numbers.

• Lecture 09 : The density of ℚ  (and its complement) in ℝ.

• Lecture 10 : Metric spaces and exampes.

• Lecture 11 : Examples of metrics and visualization of unit spheres and unit balls with different metrics on ℝ^2.

• Lecture 12 : Diameter of a set, Bounded sets; given a metric space (X,d) another metric may be defined on X which makes X a bounded set.

• Lecture 13 : Distance of a point from a set; Open and closed balls; open and closed set; boundary of a set; closed sets.

• Lecture 14 : Limit point; interior and closure of a set.

• Lecture 15 : Dense subsets and nowhere dense sets; separble space; sequences and their convergence.

• Lecture 16 : Open set relative to a subspace; subsequences.

• Lecture 17 : Cauchy seuences and Complete metric space, (ℝ, usual metric) is complete.

• Lecture 18 : Series.

• Lecture 19 : Nested set theorem, Baire Category theorem and its applications.

• Lecture 20 : Limit.

• Lecture 21 : Continuity.

• Lecture 22 : Uniform continuity, Basic properties of compact sets.

• Lecture 23 : Finite intersection property, Continuous functions on compact sets.

• Lecture 24 : Total boundedness.

• Lecture 25 : Separated sets and connected subsets.

• Lecture 26 : Basic properties of connected sets.

• Lecture 27 : Continuous functions on connected sets; Path connected sets.

• Lecture 28 : Sequence of functions and their pointwise and uniform convergence.

• Lecture 29 : More on uniform convergence; Dini's theorem.

• Lecture 30 : Weierstrass approximation theorem; Series of functions.

• Lecture 31 : Power series; Continuous function which is nowhere differentiable; Introduction to Darboux integral.

• Lecture 32 : Riemann integral.

• Lecture 33 : Fundamental theorem of calculus

• Lecture 34* : For a compact metric space X there is a tool to find whether a subspace of C(X), with supremum metric, is compact: Arzela-Ascoli's theorem.

References

• Gerald B. Folland, Real analysis, second ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication.

• Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis.

• Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.