COURSE SUMMARY
Course Title:
Real Analysis
Course Code:
18MAT301
Year Taught:
2018
2019
Semester:
5
Degree:
Integrated Degree
School:
School of Arts and Sciences
Campus:
Mysuru

'Real Analysis' is a course offered at the School of Arts and Sciences, Amrita Vishwa Vidyapeetham, Mysuru campus.

#### Scope and Objectives

To enable students to understand the basic properties of the field of real numbers and understand notion of continuous functions and their properties.

#### Syllabus

Unit I:

Review: Sets and Functions, Mathematical Induction, Finite and Infinite Sets.
The Real Numbers: The Algebraic and Order Properties of R, Absolute Value and the Real Line, The Completeness Property of R, Applications of the Supremum Property.
Chapter-2 (Sec.2.1-2.3)

Unit II:

Sequences and Series:
Sequences and Their Limits, Limit Theorems, Monotone Sequences, Subsequences and the Bolzano-Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences, Introduction to Infinite Series.
Chapter-3 (Sec.3.1-2.6)
Review of Limits: Limits of Functions, Limit Theorems. (Chapter-4, review only)

Unit III:

Continuous Functions: Continuous Functions, Combinations of Continuous Functions,Continuous Functions on Intervals, Uniform Continuity.
Chapter-5 (Sec.5.1-5.4)

Unit IV:

Differentiation:
The Derivative, The Mean Value Theorem, L'Hospital's Rules, Taylor's Theorem.
Chapter-6 (Sec.6.1-6.4)

Unit V:

The Riemann Integral: Riemann Integral, Riemann Integrable Functions, The Fundamental Theorem.
Chapter-7 (Sec.7.1-7.4)

#### Textbook

1. Introduction to Real Analysis, by Robert Gardner Bartle, Donald R. Sherbert, Fourth Edition, John Wiley and Sons, 2011.

#### References

1. Mathematical Analysis by Tom M. Apostol, Second Edition, Narosa publishing house, New Delhi,1989.
2. Principles of Mathematical Analysis by Rudin.W, Third Edition, McGraw-Hill International Editions, 1976.
3. Real Analysis by H.L. Royden and P.M.Fitzpatrick.Fourth Edition. Pearson Education Asia Limited, 2010.