infocobuild

Introductory Course in Real Analysis

Introductory Course in Real Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This is a basic course in Real Analysis which is a back bone of any course on pure and applied Mathematics and Statistics. This is a very useful course for any branch of science and engineering. The present course has been designed to introduce the subject to undergraduate/postgraduate students in science and engineering. The course contains a good introduction to each topic and an advance treatment of theory at a fairly understandable level to the students at this stage. Each concept has been explained through examples and application oriented problems. (from nptel.ac.in)

Countable and Uncountable Sets


Lecture 01 - Countable and Uncountable Sets
Lecture 02 - Properties of Countable and Uncountable Sets
Lecture 03 - Examples of Countable and Uncountable Sets
Lecture 04 - Concepts of Metric Space
Lecture 05 - Open Ball, Closed Ball, Limit Point of a Set
Lecture 06 - Tutorial I
Lecture 07 - Some Theorems on Open and Closed Sets
Lecture 08 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set
Lecture 09 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set (cont.)
Lecture 10 - Compact Set
Lecture 11 - Properties of Compact Sets
Lecture 12 - Tutorial II
Lecture 13 - Heine-Borel Theorem
Lecture 14 - Weierstrass Theorem
Lecture 15 - Cantor Set and its Properties
Lecture 16 - Derived Set and Dense Set
Lecture 17 - Limit of a Sequence, Monotone Sequence
Lecture 18 - Tutorial III
Lecture 19 - Some Important Limits of Sequences
Lecture 20 - Ratio Test, Cauchy's Theorems on Limits of Sequences of Real Numbers
Lecture 21 - Fundamental Theorems on Limits
Lecture 22 - Some Results on Limits and Bolzano-Weierstrass Theorem
Lecture 23 - Criteria for Convergent Sequences
Lecture 24 - Tutorial IV
Lecture 25 - Criteria for Divergent Sequence
Lecture 26 - Cauchy Sequence
Lecture 27 - Cauchy Convergence Criteria for Sequences
Lecture 28 - Infinite Series of Real Numbers
Lecture 29 - Convergence Criteria for Series of Positive Real Numbers
Lecture 30 - Tutorial V
Lecture 31 - Comparison Test for Series
Lecture 32 - Absolutely and Conditionally Convergent Series
Lecture 33 - Rearrangement Theorem and Test for Convergence of Series
Lecture 34 - Ratio and Integral Test for Convergence of Series
Lecture 35 - Raabe's Test for Convergence of Series
Lecture 36 - Tutorial VI
Lecture 37 - Limit of Functions and Cluster Point
Lecture 38 - Limit of Functions (cont.)
Lecture 39 - Divergence Criteria for Limit
Lecture 40 - Various Properties of Limit of Functions
Lecture 41 - Left and Right Hand Limits for Functions
Lecture 42 - Tutorial VII
Lecture 43 - Limit of Functions at Infinity
Lecture 44 - Continuous Functions (Cauchy's Definition)
Lecture 45 - Continuous Functions (Heine's Definition)
Lecture 46 - Properties of Continuous Functions
Lecture 47 - Properties of Continuous Functions (cont.)
Lecture 48 - Tutorial VIII
Lecture 49 - Boundedness Theorem and Max-Min Theorem
Lecture 50 - Location of Root and Bolzano's Theorem
Lecture 51 - Uniform Continuity and Related Theorems
Lecture 52 - Absolute Continuity and Related Theorems
Lecture 53 - Types of Discontinuities
Lecture 54 - Tutorial IX
Lecture 55 - Types of Discontinuities (cont.)
Lecture 56 - Relation between Continuity and Compact Sets
Lecture 57 - Differentiability of Real Valued Functions
Lecture 58 - Local Max-Min Cauchy's and Lagrange's Mean Value Theorem
Lecture 59 - Rolle's Mean Value Theorems and its Applications
Lecture 60 - Tutorial X
Lecture 61 - Applications of Derivatives
Lecture 62 - Applications of Mean Value Theorem and Darboux's Theorem
Lecture 63 - L'Hospital's Rule
Lecture 64 - Taylor's Theorem
Lecture 65 - Riemann/Riemann-Stieltjes Integral
Lecture 66 - Tutorial XI
Lecture 67 - Riemann/Riemann-Stieltjes Integral (cont.)
Lecture 68 - Existence of Riemann-Stieltjes Integral
Lecture 69 - Riemann-Stieltjes Integrable Functions
Lecture 70 - Properties of Riemann-Stieltjes Integral
Lecture 71 - Various Results of Riemann-Stieltjes Integral using Step Function
Lecture 72 - Some More Results on Riemann-Stieltjes Integral
Lecture 73 - Tutorial XII

References
Introductory Course in Real Analysis
Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This is a basic course in Real Analysis which is a back bone of any course on pure and applied Mathematics and Statistics.