Introduction to Real Analysis 
Introduction to Real Analysis
by IIT Madras
Video Lecture 1 of 52
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Date Added: April 7, 2016

Lecture Description

This video lecture, part of the series Real Analysis with Prof. S.H. Kulkarni by Prof. , does not currently have a detailed description and video lecture title. If you have watched this lecture and know what it is about, particularly what Mathematics topics are discussed, please help us by commenting on this video with your suggested description and title. Many thanks from,

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Course Index

  1. Introduction to Real Analysis
  2. Functions and Relations
  3. Finite and Infinite Sets
  4. Countable Sets
  5. Uncountable Sets, Cardinal Numbers
  6. Real Number System
  7. LUB Axiom
  8. Sequences of Real Numbers
  9. Sequences of Real Numbers - continued
  10. Sequences of Real Numbers - continued...
  11. Infinite Series of Real Numbers
  12. Series of nonnegative Real Numbers
  13. Conditional Convergence
  14. Metric Spaces: Definition and Examples
  15. Metric Spaces: Examples and Elementary Concepts
  16. Balls and Spheres
  17. Open Sets
  18. Closure Points, Limit Points and isolated Points
  19. Closed sets
  20. Sequences in Metric Spaces
  21. Completeness
  22. Baire Category Theorem
  23. Limit and Continuity of a Function defined on a Metric space
  24. Continuous Functions on a Metric Space
  25. Uniform Continuity
  26. Connectedness
  27. Connected Sets
  28. Compactness
  29. Compactness - Continued
  30. Characterizations of Compact Sets
  31. Continuous Functions on Compact Sets
  32. Types of Discontinuity
  33. Differentiation
  34. Mean Value Theorems
  35. Mean Value Theorems - Continued
  36. Taylor's Theorem
  37. Differentiation of Vector Valued Functions
  38. Integration
  39. Integrability
  40. Integrable Functions
  41. Integrable Functions - Continued
  42. Integration as a Limit of Sum
  43. Integration and Differentiation
  44. Integration of Vector Valued Functions
  45. More Theorems on Integrals
  46. Sequences and Series of Functions
  47. Uniform Convergence
  48. Uniform Convergence and Integration
  49. Uniform Convergence and Differentiation
  50. Construction of Everywhere Continuous Nowhere Differentiable Function
  51. Approximation of a Continuous Function by Polynomials: Weierstrass Theorem
  52. Equicontinuous family of Functions: Arzela - Ascoli Theorem

Course Description

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.

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