Mathematics - A Basic Course in Real Analysis (English)

   Thursday, April 03, 2025   

Starting a new Lecture Notes Series on Mathematics - A Basic Course in Real Analysis

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Lecture 1: Mod-01 Lec-01 Rational Numbers and Rational Cuts

Lecture 2: Mod-02 Lec-02 Irrational numbers, Dedekind's Theorem

Lecture 3: Mod-03 Lec-03 Continuum and Exercises

Lecture 4: Mod-03 Lec-04 Continuum and Exercises (Contd.)

Lecture 5: Mod-04 Lec-05 Cantor's Theory of Irrational Numbers

Lecture 6: Mod-04 Lec-06 Cantor's Theory of Irrational Numbers (Contd.)

Lecture 7: Mod-05 Lec-07 Equivalence of Dedekind and Cantor's Theory

Lecture 8: Mod-06 Lec-08 Finite, Infinite, Countable and Uncountable Sets of Real Numbers

Lecture 9: Mod-07 Lec-09 Types of Sets with Examples, Metric Space

Lecture 10: Mod-08 Lec-10 Various properties of open set, closure of a set

Lecture 11: Mod-09 Lec-11 Ordered set, Least upper bound, greatest lower bound of a set

Lecture 12: Mod-10 Lec-12 Compact Sets and its properties

Lecture 13: Mod-11 Lec-13 Weiersstrass Theorem, Heine Borel Theorem, Connected set

Lecture 14: Mod-12 Lec-14 Tutorial - II

Lecture 15: Mod-13 Lec-15 Concept of limit of a sequence

Lecture 16: Mod-14 Lec-16 Some Important limits, Ratio tests for sequences of Real Numbers

Lecture 17: Mod-15 Lec-17 Cauchy theorems on limit of sequences with examples

Lecture 18: Mod-16 Lec-18 Fundamental theorems on limits, Bolzano-Weiersstrass Theorem

Lecture 19: Mod-17 Lec-19 Theorems on Convergent and divergent sequences

Lecture 20: Mod-18 Lec-20 Cauchy sequence and its properties

Lecture 21: Mod-19 Lec-21 Infinite series of real numbers

Lecture 22: Mod-20 Lec-22 Comparison tests for series, Absolutely convergent and Conditional convergent series

Lecture 23: Mod-21 Lec-23 Tests for absolutely convergent series

Lecture 24: Mod-22 Lec-24 Raabe's test, limit of functions, Cluster point

Lecture 25: Mod-23 Lec-25 Some results on limit of functions

Lecture 26: Mod-24 Lec-26 Limit Theorems for functions

Lecture 27: Mod-25 Lec-27 Extension of limit concept (one sided limits)

Lecture 28: Mod-26 Lec-28 Continuity of Functions

Lecture 29: Mod-27 Lec-29 Properties of Continuous Functions

Lecture 30: Mod-28 Lec-30 Boundedness Theorem, Max-Min Theorem and Bolzano's theorem

Lecture 31: Mod-29 Lec-31 Uniform Continuity and Absolute Continuity

Lecture 32: Mod-30 Lec-32 Types of Discontinuities, Continuity and Compactness

Lecture 33: Mod-31 Lec-33 Continuity and Compactness (Contd.), Connectedness

Lecture 34: Mod-32 Lec-34 Differentiability of real valued function, Mean Value Theorem

Lecture 35: Mod-33 Lec-35 Mean Value Theorem (Contd.)

Lecture 36: Mod-34 Lec-36 Application of MVT , Darboux Theorem, L Hospital Rule

Lecture 37: Mod-35 Lec-37 L'Hospital Rule and Taylor's Theorem

Lecture 38: Mod-36 Lec-38 Tutorial - III

Lecture 39: Mod-37 Lec-39 Riemann/Riemann Stieltjes Integral

Lecture 40: Mod-38 Lec-40 Existence of Reimann Stieltjes Integral

Lecture 41: Mod-39 Lec-41 Properties of Reimann Stieltjes Integral

Lecture 42: Mod-39 Lec-42 Properties of Reimann Stieltjes Integral (Contd.)

Lecture 43: Mod-40 Lec-43 Definite and Indefinite Integral

Lecture 44: Mod-41 Lec-44 Fundamental Theorems of Integral Calculus

Lecture 45: Mod-42 Lec-45 Improper Integrals

Lecture 46: Mod-43 Lec-46 Convergence Test for Improper Integrals