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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)

Lecture 42 - Integration as a Limit of Sum

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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