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September 23, 2023

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FREE

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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

    • 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: Arzela-Ascoli Theorem Unlimited

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