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September 23, 2023

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Description

Introductory Course in Real Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur.

This is a basic course in Real Analysis which is a back bone of any course on pure and 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. (from nptel.ac.in)

Course Curriculum

  • Lecture 01 – Countable and Uncountable Sets Unlimited
  • Lecture 02 – Properties of Countable and Uncountable Sets Unlimited
  • Lecture 03 – Examples of Countable and Uncountable Sets Unlimited
  • Lecture 04 – Concepts of Metric Space Unlimited
  • Lecture 05 – Open Ball, Closed Ball, Limit Point of a Set Unlimited
  • Lecture 06 – Tutorial I Unlimited
  • Lecture 07 – Some Theorems on Open and Closed Sets Unlimited
  • Lecture 08 – Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set Unlimited
  • Lecture 09 – Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set (cont.) Unlimited
  • Lecture 10 – Compact Set Unlimited
  • Lecture 11 – Properties of Compact Sets Unlimited
  • Lecture 12 – Tutorial II Unlimited
  • Lecture 13 – Heine-Borel Theorem Unlimited
  • Lecture 14 – Weierstrass Theorem Unlimited
  • Lecture 15 – Cantor Set and its Properties Unlimited
  • Lecture 16 – Derived Set and Dense Set Unlimited
  • Lecture 17 – Limit of a Sequence, Monotone Sequence Unlimited
  • Lecture 18 – Tutorial III Unlimited
  • Lecture 19 – Some Important Limits of Sequences Unlimited
  • Lecture 20 – Ratio Test, Cauchy’s Theorems on Limits of Sequences of Real Numbers Unlimited
  • Lecture 21 – Fundamental Theorems on Limits Unlimited
  • Lecture 22 – Some Results on Limits and Bolzano-Weierstrass Theorem Unlimited
  • Lecture 23 – Criteria for Convergent Sequences Unlimited
  • Lecture 24 – Tutorial IV Unlimited
  • Lecture 25 – Criteria for Divergent Sequence Unlimited
  • Lecture 26 – Cauchy Sequence Unlimited
  • Lecture 27 – Cauchy Convergence Criteria for Sequences Unlimited
  • Lecture 28 – Infinite Series of Real Numbers Unlimited
  • Lecture 29 – Convergence Criteria for Series of Positive Real Numbers Unlimited
  • Lecture 30 – Tutorial V Unlimited
  • Lecture 31 – Comparison Test for Series Unlimited
  • Lecture 32 – Absolutely and Conditionally Convergent Series Unlimited
  • Lecture 33 – Rearrangement Theorem and Test for Convergence of Series Unlimited
  • Lecture 34 – Ratio and Integral Test for Convergence of Series Unlimited
  • Lecture 35 – Raabe’s Test for Convergence of Series Unlimited
  • Lecture 36 – Tutorial VI Unlimited
  • Lecture 37 – Limit of Functions and Cluster Point Unlimited
  • Lecture 38 – Limit of Functions (cont.) Unlimited
  • Lecture 39 – Divergence Criteria for Limit Unlimited
  • Lecture 40 – Various Properties of Limit of Functions Unlimited
  • Lecture 41 – Left and Right Hand Limits for Functions Unlimited
  • Lecture 42 – Tutorial VII Unlimited
  • Lecture 43 – Limit of Functions at Infinity Unlimited
  • Lecture 44 – Continuous Functions (Cauchy’s Definition) Unlimited
  • Lecture 45 – Continuous Functions (Heine’s Definition) Unlimited
  • Lecture 46 – Properties of Continuous Functions Unlimited
  • Lecture 47 – Properties of Continuous Functions (cont.) Unlimited
  • Lecture 48 – Tutorial VIII Unlimited
  • Lecture 49 – Boundedness Theorem and Max-Min Theorem Unlimited
  • Lecture 50 – Location of Root and Bolzano’s Theorem Unlimited
  • Lecture 51 – Uniform Continuity and Related Theorems Unlimited
  • Lecture 52 – Absolute Continuity and Related Theorems Unlimited
  • Lecture 53 – Types of Discontinuities Unlimited
  • Lecture 54 – Tutorial IX Unlimited
  • Lecture 55 – Types of Discontinuities (cont.) Unlimited
  • Lecture 56 – Relation between Continuity and Compact Sets Unlimited
  • Lecture 57 – Differentiability of Real Valued Functions Unlimited
  • Lecture 58 – Local Max-Min Cauchy’s and Lagrange’s Mean Value Theorem Unlimited
  • Lecture 59 – Rolle’s Mean Value Theorems and its Applications Unlimited
  • Lecture 60 – Tutorial X Unlimited
  • Lecture 61 – Applications of Derivatives Unlimited
  • Lecture 62 – Applications of Mean Value Theorem and Darboux’s Theorem Unlimited
  • Lecture 63 – L’Hospital’s Rule Unlimited
  • Lecture 64 – Taylor’s Theorem Unlimited
  • Lecture 65 – Riemann/Riemann-Stieltjes Integral Unlimited
  • Lecture 66 – Tutorial XI Unlimited
  • Lecture 67 – Riemann/Riemann-Stieltjes Integral (cont.) Unlimited
  • Lecture 68 – Existence of Riemann-Stieltjes Integral Unlimited
  • Lecture 69 – Riemann-Stieltjes Integrable Functions Unlimited
  • Lecture 70 – Properties of Riemann-Stieltjes Integral Unlimited
  • Lecture 71 – Various Results of Riemann-Stieltjes Integral using Step Function Unlimited
  • Lecture 72 – Some More Results on Riemann-Stieltjes Integral Unlimited
  • Lecture 73 – Tutorial XII Unlimited

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