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Real Analysis

Real Analysis. Instructor: Prof. S.H. Kulkarni, Department of Mathematics, IIT Madras. This course discusses the fundamental concepts in real analysis. 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. (from nptel.ac.in)

Introduction


Review of Set Theory
Lecture 01 - Introduction
Lecture 02 - Functions and Relations
Lecture 03 - Finite and Infinite Sets
Lecture 04 - Countable Sets
Lecture 05 - Uncountable Sets, Cardinal Numbers
Sequences and Series of Real Numbers
Lecture 06 - Real Number System
Lecture 07 - Least Upper Bound (LUB) Axiom
Lecture 08 - Sequences of Real Numbers
Lecture 09 - Sequences of Real Numbers (cont.)
Lecture 10 - Sequences of Real Numbers (cont.)
Lecture 11 - Infinite Series of Real Numbers
Lecture 12 - Series of Nonnegative Real Numbers
Lecture 13 - Conditional Convergence
Metric Spaces: Basic Concepts
Lecture 14 - Metric Spaces: Definition and Examples
Lecture 15 - Metric Spaces: Examples and Elementary Concepts
Lecture 16 - Balls and Spheres
Lecture 17 - Open Sets
Lecture 18 - Closure Points, Limit Points and Isolated Points
Lecture 19 - Closed Sets
Completeness
Lecture 20 - Sequences in Metric Spaces
Lecture 21 - Completeness
Lecture 22 - Baire Category Theorem
Limits and Continuity
Lecture 23 - Limit and Continuity of a Function Defined on a Metric Space
Lecture 24 - Continuous Functions on a Metric Space
Lecture 25 - Uniform Continuity
Connectedness and Compactness
Lecture 26 - Connectedness
Lecture 27 - Connected Sets
Lecture 28 - Compactness
Lecture 29 - Compactness (cont.)
Lecture 30 - Characterizations of Compact Sets
Lecture 31 - Continuous Functions on Compact Sets
Lecture 32 - Types of Discontinuity
Differentiation
Lecture 33 - Differentiation
Lecture 34 - Mean Value Theorems
Lecture 35 - Mean Value Theorems (cont.)
Lecture 36 - Taylor's Theorem
Lecture 37 - Differentiation of Vector Valued Functions
Integration
Lecture 38 - Integration
Lecture 39 - Integrability
Lecture 40 - Integrable Functions
Lecture 41 - Integrable Functions (cont.)
Lecture 42 - Integration as a Limit of Sum
Lecture 43 - Integration and Differentiation
Lecture 44 - Integration of Vector Valued Functions
Lecture 45 - More Theorems on Integrals
Sequences and Series of Functions
Lecture 46 - Sequences and Series of Functions
Lecture 47 - Uniform Convergence
Lecture 48 - Uniform Convergence and Integration
Lecture 49 - Uniform Convergence and Differentiation
Lecture 50 - Construction of Everywhere Continuous, Nowhere Differentiable Function
Lecture 51 - Approximation of a Continuous Function by Polynomials: Weierstrass Theorem
Lecture 52 - Equicontinuous Family of Functions: Arzela-Ascoli Theorem

References
Real Analysis
Instructor: Prof. S.H. Kulkarni, Department of Mathematics, IIT Madras. This course discusses the fundamental concepts in real analysis.