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Course Co-ordinated by IIT Kharagpur
Prof. P.D. Srivastava
IIT Kharagpur


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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.



Learning Units


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


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


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


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


  • Nil.

  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

  • Nil.

  • Nil.

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