Unit 1

Metric Spaces – Definition and examples, Open and Closed Sets, Limit points, interior points, Open Balls and Open Sets, Convergent Sequences, Limit Points, Bounded sets, Dense Sets, Boundary of a set. (Sections 2.15 to 2.30)

Unit 2

Compact spaces: Definition and Examples, Compact subspaces of R and Heine Borel Theorem, Characterization of Compact Metric Spaces, Connected Sets. (Sections 2.31 to 2.47)

Unit 3

Convergent Sequences, Cauchy Sequence, Complete Metric Spaces, Continuous Functions on a Metric Space, Continuity and Compactness, Continuity and Connectedness. (Sections 3.1 to 3.12, 4.1 to 4.9, 4.13 to 4.19, 4.22, 4.23)

Unit 4

Riemann-Stieltjes Integral, Definition and Existence of the Integral, Properties of the Integral, Integration by parts, Sufficient and Necessary Conditions for Existence of Riemann Integrals, Fundamental Theorem of Calculus. (Sections 6.1 to 6.13, 6.20, 6.21)

Unit 5

Sequences and Series of Functions: Sequence of functions and its point-wise limit, Discussion of main problems, Uniform convergence, Uniform convergence and continuity, Uniform convergence and Integration, Uniform convergence and Differentiation, Equicontinuity and Stone-Weierstrass Theorem, Power Series. (Chapter 7, 8.1)

Course outcomes

CO1 To understand the basics of Real analysis and apply the acquired knowledge in signals and Systems, Digital Signal Processing. Etc

CO2 Knowledge and Understanding: Learn the theory of Riemann-Stieltjes integrals, to be acquainted with the ideas of the total variation and to be able to deal with functions of bounded variation.

CO3 Intellectual Skills: Develop a reasoned argument in handling problems about functions, especially those that are of bounded variation.

CO4 General and Transferable Skills: Develop the ability to reflect on problems that are quite significant in the field of real analysis.

CO5 Develop the ability to consider problems that could be solved by implementing concepts from different areas in mathematics and identify, formulate, and solve problems.

Textbooks

1. W. Rudin, Principles of Mathematical Analysis, Tata McGraw-Hill.

References

1. Apostol, Mathematical Analysis, Narosa.
2. S.R. Ghorpade and B.V. Limaye, A Course in Multivariable Calculus and Analysis, Springer.