Real Analysis II

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Real Analysis I (MAT 220)

Overview: Continuing the work done in MAT 220 of  understanding  the rigor behind one-variable Differential Calculus, this course dwells on various aspects of Integration as well as functions on higher dimensional spaces. We discuss sequences and series of functions; uniform convergence and consequences; some important approximation theorems for continuous functions; rigorous discussions of some special functions; and finally the world of functions of several variables.

Detailed Syllabus:

1. Integration: Upper and lower Riemann integrals, basic properties of Riemann integral, Riemann integrability of continuous and monotone functions, non-Riemann integrable functions, Fundamental Theorem of Calculus and consequences.
2. Sequences and Series of Functions: Pointwise and uniform convergence, uniform convergence and continuity, series of functions, Weierstrass M-test,  uniform convergence and integration, uniform convergence and differentiation,  equicontinuous families of functions, Stone-Weierstrass Theorem.
3. Some Special Functions: Power Series, the exponential, logarithmic and trigonometric functions.
4. Topology of Rⁿ: Open and closed sets, continuous functions, completeness, compactness, connectedness.
5. Functions of Several Variables: Derivatives, partial and directional derivatives, Chain Rule, Inverse Function Theorem. 

References:

1. Analysis II by Terence Tao. Hindustan Book Agency. 2^nd Edition, 2009.
2. Principles of Mathematical Analysis by Walter Rudin. McGraw-Hill. 3^rd Edition, 2006.
3. Real Mathematical Analysis by Charles C Pugh. Springer India. 2004.
4. Mathematical Analysis by Tom Apostol, Narosa, New Delhi, 2/e, 2002.
5. Calculus, Volume 2, by Tom Apostol, Wiley India. 2nd Edition, 2011.