Real Analysis with Prof. S.H. Kulkarni

Course Description

Real number system and its order completeness, sequences and series of real numbers. Metric spaces: Basic concepts, continuous functions, completeness, contraction mapping theorem, connectedness, Intermediate Value Theorem, Compactness, Heine-Borel Theorem. Differentiation, Taylor's theorem, Riemann Integral, Improper integrals Sequences and series of functions, Uniform convergence, power series, Weierstrass approximation theorem, equicontinuity, Arzela-Ascoli theorem.

Screenshot from Lecture 37: Differentiation of Vector Valued Functions
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Video Lectures & Study Materials

Visit the official course website for more study materials: http://nptel.ac.in/syllabus/111106053/

# Lecture Play Lecture
I. Review of Set Theory
1 Introduction to Real Analysis Play Video
2 Functions and Relations Play Video
3 Finite and Infinite Sets Play Video
4 Countable Sets Play Video
5 Uncountable Sets, Cardinal Numbers Play Video
II. Sequences and Series of Real Numbers
6 Real Number System Play Video
7 LUB Axiom Play Video
8 Sequences of Real Numbers Play Video
9 Sequences of Real Numbers - continued Play Video
10 Sequences of Real Numbers - continued... Play Video
11 Infinite Series of Real Numbers Play Video
12 Series of nonnegative Real Numbers Play Video
13 Conditional Convergence Play Video
III. Metric Spaces -- Basic Concepts
14 Metric Spaces: Definition and Examples Play Video
15 Metric Spaces: Examples and Elementary Concepts Play Video
16 Balls and Spheres Play Video
17 Open Sets Play Video
18 Closure Points, Limit Points and isolated Points Play Video
19 Closed sets Play Video
IV. Completeness
20 Sequences in Metric Spaces Play Video
21 Completeness Play Video
22 Baire Category Theorem Play Video
V. Limits and Continuity
23 Limit and Continuity of a Function defined on a Metric space Play Video
24 Continuous Functions on a Metric Space Play Video
25 Uniform Continuity Play Video
VI. Connectedness & Compactness
26 Connectedness Play Video
27 Connected Sets Play Video
28 Compactness Play Video
29 Compactness - Continued Play Video
30 Characterizations of Compact Sets Play Video
31 Continuous Functions on Compact Sets Play Video
32 Types of Discontinuity Play Video
VII. Differentiation
33 Differentiation Play Video
34 Mean Value Theorems Play Video
35 Mean Value Theorems - Continued Play Video
36 Taylor's Theorem Play Video
37 Differentiation of Vector Valued Functions Play Video
VIII. Integration
38 Integration Play Video
39 Integrability Play Video
40 Integrable Functions Play Video
41 Integrable Functions - Continued Play Video
42 Integration as a Limit of Sum Play Video
43 Integration and Differentiation Play Video
44 Integration of Vector Valued Functions Play Video
45 More Theorems on Integrals Play Video
IX. Sequences and Series of Functions
46 Sequences and Series of Functions Play Video
47 Uniform Convergence Play Video
48 Uniform Convergence and Integration Play Video
49 Uniform Convergence and Differentiation Play Video
50 Construction of Everywhere Continuous Nowhere Differentiable Function Play Video
51 Approximation of a Continuous Function by Polynomials: Weierstrass Theorem Play Video
52 Equicontinuous family of Functions: Arzela - Ascoli Theorem Play Video