Syllabus (MATHEMATICS)

Course Type: MAJ-7

Semester: 5

Course Code: BMTMMAJ07T

Course Title: Real Analysis-II

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: CO1: Upon successful completion, the students will be familiar with open cover and functions of bounded variation in R. CO2: Objective of this course is the introduction of some new

Syllabus:

Real Analysis-II [Credit: 5, 75L]

• Compactness in R: Concepts of open cover of a set. Compact set in IR, Heine-Borel theorem (Proof not required).
• Functions of bounded variation (BV): Definition and examples. Monotone function is of BV. If f is on BV on [a,b] then f is bounded on [a,b]. Examples of functions of BV which are not continuous and continuous functions not of BV. Bounded variation functions and their properties. Necessary and sufficient condition for a function f to be of BV on [a,b] is that f can be written as the difference of two monotonic increasing functions on[a,b].
• Riemann integration: Partition and refinement of partition of a closed and bounded interval. Upper and lower sums, Darbaux integration, Darbaux theorem, Riemann conditions of integrability, Riemann sum and definition of Riemann integral through Riemann sums, equivalence of two Definitions. Riemann integrability of monotone and continuous functions, Properties of the Riemann integral; definition and integrability of piecewise continuous and monotone functions. Intermediate Value theorem for Integrals. Antiderivative (primitive or indefinite integral). Fundamental theorem of Integral Calculus. First Mean Value theorem of integral calculus. Statement of second mean value theorems of integrals calculus (both Bonnet’s and Weierstrass’ form) and simple problems.
• Riemann-Stieltjes integral: Upper and lower Riemann-Stieltjes integral, Rectifiable Curves. Change of variable in a Riemann-Stieltjes integral, Reduction to Riemann integral, necessary as well as sufficient conditions for existence of Riemann-Stieltjes integrals.
• Improper integrals: Types of improper integrals with examples, convergence of improper integrals with examples. Convergence of Beta and Gamma functions.
• Sequence of Functions: Pointwise and uniform convergence of sequence of functions. Theorems on continuity, derivability and integrability of the limit function of a sequence of functions.
• Series of functions: Theorems on the continuity and derivability of the sum function of a series of functions; Cauchy criterion for uniform convergence and Weierstrass M-Test.
• Fourier series: Definition of Fourier coefficients and series, Reimann-Lebesgue lemma, Bessel's inequality, Parseval's identity, Dirichlet's condition. Examples of Fourier expansions and summation results for series.
• Power series: Radius of convergence. Differentiation and integration of power series; Abel’s Theorem; Weierstrass Approximation Theorem (statement only).

Reading References:

1. S.K. Mapa; Introduction to Real analysis; Levant.
2. G.B. Thomas and R.L. Finney; Calculus, 9th Ed.; Pearson Education, Delhi; 2005.
3. Tom M. Apostol; Mathematical Analysis; Narosa Publishing House.
4. Courant and John; Introduction to Calculus and Analysis, Vol II; Springer.
5. W. Rudin; Principles of Mathematical Analysis; Tata McGraw-Hill.
6. S.N. Mukhopadhyay, S. Mitra; Mathematical Analysis, Vol-II; U.N Dhur & Sons Pvt. Ltd; 2014.
7. S.C. Malik & Savita Arora; Mathematical Analysis; New Age International (P) Limited.
8. Steen, L., Seebach, J.; Counter Examples in Topology; Holt, Reinhart and Winston, New York.
9. Hocking, J., Young, G.; Topology; Addison-Wesley Reading; 1961.
10. Kelley, J.L.; General Topology; Van Nostrand Reinhold Co., New York; 1995.
11. Simmons, G.F.; Introduction to Topology and Modern Analysis; McGraw Hill, 1963.
12. Dugundji, J.; Topology; Allyn and Bacon, 1966.
13. Munkres, J.R.; Topology: A First Course; Prentice Hall of India Pvt. Ltd., New Delhi; 2000.