# Mathematics - A Basic Course in Real Analysis

Indian Institute of Technology Kharagpur

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.

Course topics:

• Dedekind Theory of Irrational numbers
• Cantorâ€™s Theory of Irrational numbers
• Sets of Points
• Limit of Sequences of Real Numbers
• Infinite Series of Real numbers
• Limit of functions
• Continuity of Functions
• Differentiability
• Riemann Integration / Reimann- Stieltjes Intergral
• ##### Mod-01 Lec-01 Rational Numbers and Rational Cuts
Prof. P.D. Srivastava
• ##### Mod-02 Lec-02 Irrational numbers, Dedekind&#39;s Theorem
Prof. P.D. Srivastava
• ##### Mod-03 Lec-03 Continuum and Exercises
Prof. P.D. Srivastava
• ##### Mod-03 Lec-04 Continuum and Exercises (Contd.)
Prof. P.D. Srivastava
• ##### Mod-04 Lec-05 Cantor&#39;s Theory of Irrational Numbers
Prof. P.D. Srivastava
• ##### Mod-04 Lec-06 Cantor&#39;s Theory of Irrational Numbers (Contd.)
Prof. P.D. Srivastava
• ##### Mod-05 Lec-07 Equivalence of Dedekind and Cantor&#39;s Theory
Prof. P.D. Srivastava
• ##### Mod-06 Lec-08 Finite, Infinite, Countable and Uncountable Sets of Real Numbers
Prof. P.D. Srivastava
• ##### Mod-07 Lec-09 Types of Sets with Examples, Metric Space
Prof. P.D. Srivastava
• ##### Mod-08 Lec-10 Various properties of open set, closure of a set
Prof. P.D. Srivastava
• ##### Mod-09 Lec-11 Ordered set, Least upper bound, greatest lower bound of a set
Prof. P.D. Srivastava
• ##### Mod-10 Lec-12 Compact Sets and its properties
Prof. P.D. Srivastava
• ##### Mod-11 Lec-13 Weiersstrass Theorem, Heine Borel Theorem, Connected set
Prof. P.D. Srivastava
• ##### Mod-12 Lec-14 Tutorial - II
Prof. P.D. Srivastava
• ##### Mod-13 Lec-15 Concept of limit of a sequence
Prof. P.D. Srivastava
• ##### Mod-14 Lec-16 Some Important limits, Ratio tests for sequences of Real Numbers
Prof. P.D. Srivastava
• ##### Mod-15 Lec-17 Cauchy theorems on limit of sequences with examples
Prof. P.D. Srivastava
• ##### Mod-16 Lec-18 Fundamental theorems on limits, Bolzano-Weiersstrass Theorem
Prof. P.D. Srivastava
• ##### Mod-17 Lec-19 Theorems on Convergent and divergent sequences
Prof. P.D. Srivastava
• ##### Mod-18 Lec-20 Cauchy sequence and its properties
Prof. P.D. Srivastava
• ##### Mod-19 Lec-21 Infinite series of real numbers
Prof. P.D. Srivastava
• ##### Mod-20 Lec-22 Comparison tests for series, Absolutely convergent and Conditional convergent series
Prof. P.D. Srivastava
• ##### Mod-21 Lec-23 Tests for absolutely convergent series
Prof. P.D. Srivastava
• ##### Mod-22 Lec-24 Raabe&#39;s test, limit of functions, Cluster point
Prof. P.D. Srivastava
• ##### Mod-23 Lec-25 Some results on limit of functions
Prof. P.D. Srivastava
• ##### Mod-24 Lec-26 Limit Theorems for functions
Prof. P.D. Srivastava
• ##### Mod-25 Lec-27 Extension of limit concept (one sided limits)
Prof. P.D. Srivastava
• ##### Mod-26 Lec-28 Continuity of Functions
Prof. P.D. Srivastava
• ##### Mod-27 Lec-29 Properties of Continuous Functions
Prof. P.D. Srivastava
• ##### Mod-28 Lec-30 Boundedness Theorem, Max-Min Theorem and Bolzano&#39;s theorem
Prof. P.D. Srivastava
• ##### Mod-29 Lec-31 Uniform Continuity and Absolute Continuity
Prof. P.D. Srivastava
• ##### Mod-30 Lec-32 Types of Discontinuities, Continuity and Compactness
Prof. P.D. Srivastava
• ##### Mod-31 Lec-33 Continuity and Compactness (Contd.), Connectedness
Prof. P.D. Srivastava
• ##### Mod-32 Lec-34 Differentiability of real valued function, Mean Value Theorem
Prof. P.D. Srivastava
• ##### Mod-33 Lec-35 Mean Value Theorem (Contd.)
Prof. P.D. Srivastava
• ##### Mod-34 Lec-36 Application of MVT , Darboux Theorem, L Hospital Rule
Prof. P.D. Srivastava
• ##### Mod-35 Lec-37 L&#39;Hospital Rule and Taylor&#39;s Theorem
Prof. P.D. Srivastava
• ##### Mod-36 Lec-38 Tutorial - III
Prof. P.D. Srivastava
• ##### Mod-37 Lec-39 Riemann/Riemann Stieltjes Integral
Prof. P.D. Srivastava
• ##### Mod-38 Lec-40 Existence of Reimann Stieltjes Integral
Prof. P.D. Srivastava
• ##### Mod-39 Lec-41 Properties of Reimann Stieltjes Integral
Prof. P.D. Srivastava
• ##### Mod-39 Lec-42 Properties of Reimann Stieltjes Integral (Contd.)
Prof. P.D. Srivastava
• ##### Mod-40 Lec-43 Definite and Indefinite Integral
Prof. P.D. Srivastava
• ##### Mod-41 Lec-44 Fundamental Theorems of Integral Calculus
Prof. P.D. Srivastava
• ##### Mod-42 Lec-45 Improper Integrals
Prof. P.D. Srivastava
• ##### Mod-43 Lec-46 Convergence Test for Improper Integrals
Prof. P.D. Srivastava