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Created by:
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, 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)
Course Curriculum
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- 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
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- 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 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 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 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: Arzela-Ascoli Theorem Unlimited
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