MA5330 Real Analysis

Course Details

Unit I - Metric spaces
Review of the real number system, Basic concepts/ definitions and examples, continuous functions, completeness, Baire category theorem, contraction mapping theorem, connectedness, compactness, HeineBorel theorem. (14 lectures)
Unit II - The Riemann-Stieltjes integral
The Riemann-Stieltjes integral and its properties, integrals of continuous and monotone functions, fundamental theorems of calculus, integration by parts, change of variables formula. (13 lectures)
Unit III - Uniform convergence
Sequences and series of functions, Weierstrass M-test, uniform convergence and its relation to continuity, differentiation and integration, Weierstrass approximation theorem, equicontinuity, Arzela-Ascoli theorem. (13 lectures)

Course References:

Text Books:
1. W. Rudin - Principles of mathematical analysis, Mcgraw-Hill 1976.
2. Terence Tao - Analysis I and II, with Chapter 11 from volume I and Chapters 1,2 and 3 from Volume II, trim series 37 and 38, Hindustan Book Agency.

Reference Books:
1. C. C. Pugh, Real Mathematical Analysis, Springer, 2002.
2. T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.
3. G. F. Simmons, Topology and Modern Analysis, Kreiger, 2003.