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