MAL402 (Real Analysis)

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

Syllabus (Page 155)

     Topics covered in

• Lecture 01 : Well ordering principle, Countable and uncountable sets.

• Lecture 02 : Countable union of countable sets is a countable set, Cantor's theorem.

• Lecture 03 : Supremum, Infimum, Completeness property of ℝ.

• Lecture 04 : 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 05 : Existence of nth square roots of positive real numbers.

• Lecture 06 : Metric spaces and exampes, Open sets.

• Lecture 07 : Closed sets, Sequences

• Lecture 08 : Series

• Lecture 09 : Limit and continuity

• Lecture 10 : Complete metric space 

• Lecture 11 : Nested set theorem 

• Lecture 12 : Baire Category theorem

• Lecture 13 : Applications of Baire category theorem

• Lecture 14 :

• Lecture 15 : Basic properties of compact set

• Lecture 16 : Total boundedness

• Lecture 17 : Finite intersection property

• Lecture 18 : Continuous functions on compact sets

• Lecture 19 : Uniform continuity

• Lecture 20 :

• Lecture 21 :

• Lecture 22 : Basic properties of connected sets

• Lecture 23 : Continuous functions on connected sets

• Lecture 24 : 

• Lecture 25 : Path connected sets

• Lecture 26 : Riemann integral

• Lecture 27 : Riemann integral

• Lecture 28 : Fundamental theorem of calculus

• Lecture 29 : Fundamental theorem of calculus

• Lecture 30 : Fundamental theorem of calculus

• Lecture 31 :

• Lecture 32 :

• Lecture 33 : Pointwise convergence of functions

• Lecture 34 : Pointwise convergence of functions

• Lecture 35 : Uniform convergence of functions

• Lecture 36 : Uniform convergence of functions

• Lecture 37 : Series of functions

• Lecture 38 : Series of functions

• Lecture 39 : Power series

• Lecture 40 : Power series

• Lecture 41 : Power series

• Lecture 42 : Dini's theorem

• Lecture 43 : Dini's theorem

• Lecture 44 : 

• Lecture 45 : Arzela-Ascoli's theorem

• Lecture 46 : Arzela-Ascoli's theorem

• Lecture 47 : Continuous function which is nowhere differentiable

• Lecture 48 : Weierstrass approximation theorem

• Lecture 49 : 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.