MAL402 (Real Analysis)

There will be 49 (42+7) Lectures, 50-55 minutes each.

Syllabus (Page 155)

Evaluation Plan

• End-Semester Examination: 80 marks

• Assignments: 10 marks

• Class assessment: 10 marks

     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; dense subsets and nowhere dense sets; separble space; open set in subspace.

• Lecture 15 : sequences.

• Lecture 16 : series

• Lecture 17 : limit and continuity

• Lecture 18 : Complete metric space

• Lecture 19 : Nested set theorem

• Lecture 20 : Baire Category theorem, Applications of Baire category theorem

• Lecture 21 : Basic properties of compact set

• Lecture 22 : Total boundedness

• Lecture 23 : Finite intersection property

• Lecture 24 : Continuous functions on compact sets

• Lecture 25 : Uniform continuity

• Lecture 26 : Basic properties of connected sets

• Lecture 27 : Continuous functions on connected sets

• Lecture 28 : Path connected sets

• Lecture 29 : Riemann integral

• Lecture 30 : 

• Lecture 31 :

• Lecture 32 :Fundamental theorem of calculus

• Lecture 33 : Pointwise convergence of functions

• Lecture 34 : Uniform convergence of functions

• Lecture 35 : Series of functions

• Lecture 36 : Power series

• Lecture 37 : Power series

• Lecture 38 : Dini's theorem

• Lecture 39 : 

• Lecture 40 : Arzela-Ascoli's theorem

• Lecture 41 : Continuous function which is nowhere differentiable

• Lecture 42 : Weierstrass approximation 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.