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Last updated:
September 23, 2023
Duration:
Unlimited Duration
FREE
This course includes:
Unlimited Duration
Badge on Completion
Certificate of completion
Unlimited Duration
Description
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, HeineBorel theorem. Differentiation, Taylor's theorem, Riemann integral, improper integrals, sequences and series of functions, uniform convergence, power series, Weierstrass approximation theorem, equicontinuity, ArzelaAscoli theorem. (from nptel.ac.in)
Course Curriculum

 Lecture 01 – Introduction Unlimited
 Lecture 02 – Functions and Relations Unlimited
 Lecture 03 – Finite and Infinite Sets Unlimited
 Lecture 04 – Countable Sets Unlimited
 Lecture 05 – Uncountable Sets, Cardinal Numbers Unlimited

 Lecture 06 – Real Number System Unlimited
 Lecture 07 – Least Upper Bound (LUB) Axiom Unlimited
 Lecture 08 – Sequences of Real Numbers Unlimited
 Lecture 09 – Sequences of Real Numbers (cont.) Unlimited
 Lecture 10 – Sequences of Real Numbers (cont.) Unlimited
 Lecture 11 – Infinite Series of Real Numbers Unlimited
 Lecture 12 – Series of Nonnegative Real Numbers Unlimited
 Lecture 13 – Conditional Convergence Unlimited

 Lecture 14 – Metric Spaces: Definition and Examples Unlimited
 Lecture 15 – Metric Spaces: Examples and Elementary Concepts Unlimited
 Lecture 16 – Balls and Spheres Unlimited
 Lecture 17 – Open Sets Unlimited
 Lecture 18 – Closure Points, Limit Points and Isolated Points Unlimited
 Lecture 19 – Closed Sets Unlimited

 Lecture 20 – Sequences in Metric Spaces Unlimited
 Lecture 21 – Completeness Unlimited
 Lecture 22 – Baire Category Theorem Unlimited

 Lecture 23 – Limit and Continuity of a Function Defined on a Metric Space Unlimited
 Lecture 24 – Continuous Functions on a Metric Space Unlimited
 Lecture 25 – Uniform Continuity Unlimited

 Lecture 26 – Connectedness Unlimited
 Lecture 27 – Connected Sets Unlimited
 Lecture 28 – Compactness Unlimited
 Lecture 29 – Compactness (cont.) Unlimited
 Lecture 30 – Characterizations of Compact Sets Unlimited
 Lecture 31 – Continuous Functions on Compact Sets Unlimited
 Lecture 32 – Types of Discontinuity Unlimited

 Lecture 33 – Differentiation Unlimited
 Lecture 34 – Mean Value Theorems Unlimited
 Lecture 35 – Mean Value Theorems (cont.) Unlimited
 Lecture 36 – Taylor’s Theorem Unlimited
 Lecture 37 – Differentiation of Vector Valued Functions Unlimited

 Lecture 38 – Integration Unlimited
 Lecture 39 – Integrability Unlimited
 Lecture 40 – Integrable Functions Unlimited
 Lecture 41 – Integrable Functions (cont.) Unlimited
 Lecture 42 – Integration as a Limit of Sum Unlimited
 Lecture 43 – Integration and Differentiation Unlimited
 Lecture 44 – Integration of Vector Valued Functions Unlimited
 Lecture 45 – More Theorems on Integrals Unlimited

 Lecture 46 – Sequences and Series of Functions Unlimited
 Lecture 47 – Uniform Convergence Unlimited
 Lecture 48 – Uniform Convergence and Integration Unlimited
 Lecture 49 – Uniform Convergence and Differentiation Unlimited
 Lecture 50 – Construction of Everywhere Continuous, Nowhere Differentiable Function Unlimited
 Lecture 51 – Approximation of a Continuous Function by Polynomials: Weierstrass Theorem Unlimited
 Lecture 52 – Equicontinuous Family of Functions: ArzelaAscoli Theorem Unlimited
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