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Course11110605352 lectures

Real Analysis

Prof. S.H. Kulkarni · IIT Madras

Start with lecture 1 At NPTEL

Lectures

MODULE 1: REVIEW OF SET THEORY

  1. Introduction Video
  2. Functions and Relations Video
  3. Finite and Infinite Sets Video
  4. Countable Sets Video
  5. Uncountable Sets, Cardinal Numbers Video

MODULE 2: SEQUENCES AND SERIES OF REAL NUMBERS

  1. Real Number System Video
  2. LUB Axiom Video
  3. Sequences of Real Numbers Video
  4. Sequences of Real Numbers - continued Video
  5. Sequences of Real Numbers - continued... Video
  6. Infinite Series of Real Numbers Video
  7. Series of nonnegative Real Numbers Video
  8. Conditional Convergence Video

MODULE 3: METRIC SPACES - BASIC CONCEPTS

  1. Metric Spaces: Definition and Examples Video
  2. Metric Spaces: Examples and Elementary Concepts Video
  3. Balls and Spheres Video
  4. Open Sets Video
  5. Closure Points, Limit Points and isolated Points Video
  6. Closed sets Video

MODULE 4: COMPLETENESS

  1. Sequences in Metric Spaces Video
  2. Completeness Video
  3. Baire Category Theorem Video

MODULE 5: LIMITS AND CONTINUITY

  1. Limit and Continuity of a Function defined on a Metric space Video
  2. Continuous Functions on a Metric Space Video
  3. Uniform Continuity Video

MODULE 6: CONNECTEDNESS AND COMPACTNESS

  1. Connectedness Video
  2. Connected Sets Video
  3. Compactness Video
  4. Compactness - Continued Video
  5. Characterizations of Compact Sets Video
  6. Continuous Functions on Compact Sets Video
  7. Types of Discontinuity Video

MODULE 7: DIFFERENTIATION

  1. Differentiation Video
  2. Mean Value Theorems Video
  3. Mean Value Theorems - Continued Video
  4. Taylor's Theorem Video
  5. Differentiation of Vector Valued Functions Video

MODULE 8: INTEGRATION

  1. Integration Video
  2. Integrability Video
  3. Integrable Functions Video
  4. Integrable Functions - Continued Video
  5. Integration as a Limit of Sum Video
  6. Integration and Differentiation Video
  7. Integration of Vector Valued Functions Video
  8. More Theorems on Integrals Video

MODULE 9: SEQUENCES AND SERIES OF FUNCTIONS

  1. Sequences and Series of Functions Video
  2. Uniform Convergence Video
  3. Uniform Convergence and Integration Video
  4. Uniform Convergence and Differentiation Video
  5. Construction of Everywhere Continuous Nowhere Differentiable Function Video
  6. Approximation of a Continuous Function by Polynomials: Weierstrass Theorem Video
  7. Equicontinuous family of Functions: Arzela - Ascoli Theorem Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Madras
Department
Mathematics
Teacher
Prof. S.H. Kulkarni
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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