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

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

Nil.

W. Rudin - Principles of Mathematica Analysis - Mc. Graw Hill Int. Edition (3rd)

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

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

N. Saran - Theory of Function of Real Variable

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