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It is a first level course on Functional Analysis. The motto is to familiarize the students with basic concepts, principles and methods of Functional analysis and its applications.

 Module Learning Units Lectures Module I 1. Dedekind Theory of Irrational numbers:- Rational numbers, section of Rational numbers, Irrational numbers, real Numbers, Dedekind Theorem, The Continuum Exercise- Tutorial 2. Cantor’s Theory of Irrational numbers:- Cantor’s Theory, Convergent sequence of real numbers, Equivalence of the definition of Dedekind & Cantor 3. Sets of Points- The upper & lower bounds, l.u.b. & g.l.b. of sets, limiting point, Weierstrass Theorem, Derived sets, Countable & Non constable sets, Cardinal numbers, Open & Closed sets, Closure of a set, Perfect set, Heine-Borel Theorem 14 Module II 1. Limit of Sequences of Real Numbers:- Bounded sequences, Null sequences, Monotone sequences, Convergent sequences, Fundamental theorems on limit, limit sup, limit inf of sequences, Ratio Test & other Tests, Cauchy theorems, Cauchy Convergence Criteria Exercises- Tutorial 2. Infinite Series of Real numbers:- Introduction of infinite series, Tests for its convergence, Absolute convergence, Conditional convergence 3. Limit of functions Concepts of Limit of functions, Limit Theorems, Some extension of Limit Concepts, Exercises- Tutorials 13 Module III 1. Continuity of Functions:- Cauchy’s  and Heine’s definitions of continuity, Properties of Continuous functions, Uniform continuity, Absolute continuity, Discontinuous Functions, Types of Discontinuities 2. Differentiability:- Concept of Derivatives, Rolle’s theorem, Mean value theorem,  L’ Hospital Rule, Taylors Theorem Exercises- Tutorial 9 Module IV 1. Riemann Integration / Reimann- Stieltjes Intergral:- The Upper and lower R-integrals, Integrable ( R ) functions, Properties of definite and indefinite integral, Mean value theorems, Absolute convergence, convergence, Test for improper integrals. Definition & Existence of the Reimann- Stieltjes Integral & its properties Exercise, Tutorial 8
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1. W. Rudin - Principles of Mathematica Analysis - Mc. Graw Hill Int. Edition (3rd)

2. Robert G. Bartle and Donald R. Shebert - Introduction to Real Analysis - Wiley India, 3rd ed.

3. Sterling K. Berberian - A First course in Real Analysis - 1994, Springer Verlag, Ny. Inc.

4. N. Saran - Theory of Function of Real Variable

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