Real Analysis-I

Syllabus Details

• Unit I:(Foundations and Metric Spaces) Real number system and its structure, and preliminaries of metric spaces: Archimedean property; infimum and supremum; inequalities such as Bernoulli inequality and Cauchy–Schwarz inequality; countable and uncountable sets; metric spaces; open balls, bounded sets, and sequences in metric spaces; closure, interior, and boundary of a set; dense subset; distance between a point and a set; distance between two sets; nested interval theorem.
• Unit II:(Continuity and Compactness) More of elementary metric space theory relevant to the study of Euclidean spaces: continuous functions between metric spaces; uniform and Lipschitz continuity; totally bounded sets; compactness, Heine–Borel theorem; continuous functions and compactness; connectedness; intermediate value theorem.
• Unit III:(Sequences and Series of Numbers) Sequences and series of real numbers and their convergence: sequences and subsequences of real numbers; convergence and limit of some standard sequences; sequences related to exponential and log functions; monotone sequences; series of real numbers and their convergence; absolute and unconditional convergence; geometric series; tests for convergence such as comparison test, ratio test, and root test; power series and radius of convergence.
• Unit IV:(Differentiation) Differentiation of one-variable functions: properties of differentiation; product rule and chain rule; Darboux theorem; mean value theorems; test for local extrema; L’Hospital’s rule; Taylor’s theorem; convex functions and differentiation.
• Unit V:(Sequences and Series of Functions) Sequences of functions; pointwise and uniform convergence; uniform convergence on compact sets; pointwise boundedness and uniform boundedness; Dini’s theorem; equicontinuous family; Arzelà–Ascoli theorem; series of real functions and their convergence; Weierstrass M-test; continuity of power series; power series expansion for the exponential function.

Course Learning Objective

1. Explain elementary metric space theory relevant to the study of Euclidean spaces and solve elementary problems about metric spaces.
2. Explain the behavior of sequences and series of real numbers as well as functions in terms of their convergence, and solve elementary problems about their convergence.
3. Explain basic techniques in one-variable calculus and apply those to solve elementary problems in calculus.
4. Demonstrate the results in Real Analysis with a critical mind, and use those results and proof-techniques to analyze and decide — with suitable justification — whether a given statement in Real Analysis is true or false.
5. Explain the theory and theoretical structure of Real Analysis as a prerequisite for future study in subjects such as Topology, Complex Analysis, Measure Theory, and Functional Analysis.

Recommended Books

1. Goldberg, Richard R. Methods of Real Analysis. 2nd ed. John Wiley & Sons, 1976
2. Rudin, Walter. Principles of Mathematical Analysis. 3rd ed. International Series in Pure and Applied Mathematics, 1976
3. Bartle, Robert G. The Elements of Real Analysis. 2nd ed. John Wiley & Sons, 1976
4. Ross, Kenneth A. Elementary Analysis: The Theory of Calculus. 2nd ed., with Jorge M. Lopez. Undergraduate Texts in Mathematics, Springer, 2013
5. Apostol, Tom M. Mathematical Analysis. 2nd ed. Addison-Wesley, Reading,1974