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Introductory Course in Real Analysis

Prof. P D Srivastava · IIT Kharagpur

Start with lecture 1 At NPTEL

About

This is a basic course in Real Analysis which is a back bone of any course on pure & applied Mathematics and Statistics. This is a very useful course for any branch of science and engineering. The present course has been designed to introduce the subject to undergraduate/postgraduate students in science and engineering. The course contains a good introduction to each topic and an advance treatment of theory at a fairly understandable level to the students at this stage. Each concept has been explained through examples and application oriented problems.

Lectures

Week 1

  1. Lecture 1: Countable and Uncountable sets YouTube
  2. Lecture 2: Properties of Countable and Uncountable sets YouTube
  3. Lecture 3: Examples of Countable and Uncountable sets YouTube
  4. Lecture 4: Concepts of Metric Space YouTube
  5. Lecture 5: Open ball, Closed ball, Limit point of a set YouTube
  6. Lecture 6: Tutorial - 1 YouTube

Week 2

  1. Lecture 7: Some theorems on Open and Closed sets YouTube
  2. Lecture 8: Ordered set, Least upper bound, Greatest lower bound of a set YouTube
  3. Lecture 9: Ordered set, Least upper bound, Greatest lower bound of a set (contd.) YouTube
  4. Lecture 10: Compact Set YouTube
  5. Lecture 11: Properties of Compact sets YouTube
  6. Lecture 12: Tutorial - 2 YouTube

Week 3

  1. Lecture 13: Heine Borel Theorem YouTube
  2. Lecture 14: Weierstrass Theorem YouTube
  3. Lecture 15: Cantor set & its properties YouTube
  4. Lecture 16: Derived set & Dense set YouTube
  5. Lecture 17: Limit of a sequence & monotone sequence YouTube
  6. Lecture 18: Tutorial III YouTube

Week 4

  1. Lecture 19 : Some Important Limits of Sequences YouTube
  2. Lecture 20 : Ratio Test Cauchy's Theorems on Limits of Sequences of Real Numbers YouTube
  3. Lecture 21: Fundamental Theorems on Limits YouTube
  4. Lecture 22: Some Results on Limits and Bolzano - Weierstrass Theorem YouTube
  5. Lecture 23: Criteria for Convergent Sequence YouTube
  6. Lecture 24: Tutorial - IV YouTube

Week 5

  1. Lecture 25 : Criteria for Divergent Sequence YouTube
  2. Lecture 26 : Cauchy Sequence YouTube
  3. Lecture 27 : Cauchy Convergence Criteria for Sequences YouTube
  4. Lecture 28 : Infinite Series of Real Numbers YouTube
  5. Lecture 29 : Convergence Criteria for Series of Positive Real Numbers YouTube
  6. Lecture 30 : Tutorial V YouTube

Week 6

  1. Lecture 31: Comparison Test for Series YouTube
  2. Lecture 32: Absolutely and Conditionally Convergent Series YouTube
  3. Lecture 33: Rearrangement Theorem and Test for Convergence of Series YouTube
  4. Lecture 34: Ratio and Integral Test for Convergence of Series YouTube
  5. Lecture 35: Raabe's Test for Convergence of Series YouTube
  6. Lecture 36 : Tutorial VI YouTube

Week 7

  1. Lecture 37 : Limit of functions and cluster point YouTube
  2. Lecture 38 : Limit of functions(contd.) YouTube
  3. Lecture 39 : Divergence criteria for limit YouTube
  4. Lecture 40 : Various properties of limit of functions YouTube
  5. Lecture 41 : Left and Right Hand Limits for functions YouTube
  6. Lecture 42 : Tutorial VII YouTube

Week 8

  1. Lecture 43 : Limit of Functions at Infinity YouTube
  2. Lecture 44 : Continuous Functions (Cauchy's Definition) YouTube
  3. Lecture 45: Continuous Functions (Heine's Definition) YouTube
  4. Lecture 46: Properties of Continuous Functions YouTube
  5. Lecture 47: Properties of Continuous Functions (Contd.) YouTube
  6. Lecture 48 : Tutorial VIII YouTube

Week 9

  1. Lecture 49: Boundness Theorem and Max-Min Theorem YouTube
  2. Lecture 50: Location of Root and Bolzano's Theorem YouTube
  3. Lecture 51: Uniform Continuity and Related Theorems YouTube
  4. Lecture 52: Absolute Continuity and Related Theorems YouTube
  5. Lecture 53 : Types of Discontinuities YouTube
  6. Lecture 54 : Tutorial IX YouTube

Week 10

  1. Lecture 55 : Types of Discontinuities (Contd.) YouTube
  2. Lecture 56 : Relation between Continuity and Compact Sets YouTube
  3. Lecture 57 : Differentiability of Real Valued Functions YouTube
  4. Lecture 58 : Local Max. - Min. Cauchy's and Lagrange's Mean Value Theorem YouTube
  5. Lecture 59 : Rolle's Mean Value Theorems and Its Applications YouTube
  6. Lecture 60 : Tutorial X YouTube

Week 11

  1. Lecture 61 : Applications of Derivatives YouTube
  2. Lecture 62 : Application of MVT & Darboux's theorem YouTube
  3. Lecture 63 : L' Hospital Rule YouTube
  4. Lecture 64 : Taylor's theorem YouTube
  5. Lecture 65 : Riemann/Riemann Steiltjes Integral YouTube
  6. Lecture 66 : Tutorial XI YouTube

Week 12

  1. Lecture 67 : Riemann/Riemann Steiltjes Integral (contd.) YouTube
  2. Lecture 68 : Existence of Riemann/Riemann Steiltjes Integral YouTube
  3. Lecture 69 : Riemann/Riemann Steiltjes Integrable functions YouTube
  4. Lecture 70 : Properties of Riemann Steiltjes Intergral YouTube
  5. Lecture 71 : Various Results of Riemann Stieltjes Integral using Step Function YouTube
  6. Lecture 72 : Some more results on Riemann Stieltjes Integral YouTube
  7. Lecture 73 : Tutorial XII YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Kharagpur
Department
Mathematics
Teacher
Prof. P D Srivastava
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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