A Basic Course in Real Analysis

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Contents:
Dedekind Theory of Irrational numbers : Rational numbers, section of Rational numbers, Irrational numbers, real Numbers, Dedekind Theorem, The Continuum Exercise- Tutorial
Cantor’s Theory of Irrational numbers : Cantor’s Theory, Convergent sequence of real numbers, Equivalence of the definition of Dedekind & Cantor
Sets of Points : The upper & lower bounds, l.u.b. & g.l.b. of sets, limiting point, Weierstrass Theorem, Derived sets, Countable & Non constable sets, Cardinal numbers, Open & Closed sets, Closure of a set, Perfect set, Heine-Borel Theorem

Limit of Sequences of Real Numbers : Bounded sequences, Null sequences, Monotone sequences, Convergent sequences, Fundamental theorems on limit, limit sup, limit inf of sequences, Ratio Test & other Tests, Cauchy theorems, Cauchy Convergence Criteria
Exercises- Tutorial
Infinite Series of Real numbers : Introduction of infinite series, Tests for its convergence, Absolute convergence, Conditional convergence
Limit of functions : Concepts of Limit of functions, Limit Theorems, Some extension of Limit Concepts, Exercises- Tutorials

Continuity of Functions : Cauchy’s and Heine’s definitions of continuity, Properties of Continuous functions, Uniform continuity, Absolute continuity, Discontinuous Functions, Types of Discontinuities
Differentiability : Concept of Derivatives, Rolle’s theorem, Mean value theorem, L’ Hospital Rule, Taylors Theorem Exercises- Tutorial
Riemann Integration / Reimann- Stieltjes Intergral : The Upper and lower R-integrals, Integrable ( R ) functions, Properties of definite and indefinite integral, Mean value theorems, Absolute convergence, convergence, Test for improper integrals. Definition & Existence of the Reimann- Stieltjes Integral & its properties Exercise, Tutorial

Course Curriculum

Mod-01 Lec-01 Rational Numbers and Rational Cuts Details 52:37
Mod-02 Lec-02 Irrational numbers, Dedekind’s Theorem Details 54:42
Mod-03 Lec-03 Continuum and Exercises Details 56:11
Mod-03 Lec-04 Continuum and Exercises (Contd.) Details 0:55
Mod-04 Lec-05 Cantor’s Theory of Irrational Numbers Details 53:8
Mod-04 Lec-06 Cantor’s Theory of Irrational Numbers (Contd.) Details 55:6
Mod-05 Lec-07 Equivalence of Dedekind and Cantor’s Theory Details 54:37
Mod-06 Lec-08 Finite, Infinite, Countable and Uncountable Sets of Real Numbers Details 55:18
Mod-07 Lec-09 Types of Sets with Examples, Metric Space Details 55:2
Mod-08 Lec-10 Various properties of open set, closure of a set Details 55:20
Mod-09 Lec-11 Ordered set, Least upper bound, greatest lower bound of a set Details 56:22
Mod-10 Lec-12 Compact Sets and its properties Details 55:44
Mod-11 Lec-13 Weiersstrass Theorem, Heine Borel Theorem, Connected set Details 56:8
Mod-12 Lec-14 Tutorial – II Details 56:13
Mod-13 Lec-15 Concept of limit of a sequence Details 54:51
Mod-14 Lec-16 Some Important limits, Ratio tests for sequences of Real Numbers Details 51:48
Mod-15 Lec-17 Cauchy theorems on limit of sequences with examples Details 54:15
Mod-16 Lec-18 Fundamental theorems on limits, Bolzano-Weiersstrass Theorem Details 54:36
Mod-17 Lec-19 Theorems on Convergent and divergent sequences Details 52:42
Mod-18 Lec-20 Cauchy sequence and its properties Details 53:53
Mod-19 Lec-21 Infinite series of real numbers Details 53:16
Mod-20 Lec-22 Comparison tests for series, Absolutely convergent and Conditional convergent series Details 54:53
Mod-21 Lec-23 Tests for absolutely convergent series Details 53:1
Mod-22 Lec-24 Raabe’s test, limit of functions, Cluster point Details 57:20
Mod-23 Lec-25 Some results on limit of functions Details 53:36
Mod-24 Lec-26 Limit Theorems for functions Details 54:9
Mod-25 Lec-27 Extension of limit concept (one sided limits) Details 52:26
Mod-26 Lec-28 Continuity of Functions Details 54:22
Mod-27 Lec-29 Properties of Continuous Functions Details 54:7
Mod-28 Lec-30 Boundedness Theorem, Max-Min Theorem and Bolzano’s theorem Details 56:25
Mod-29 Lec-31 Uniform Continuity and Absolute Continuity Details 53:41
Mod-30 Lec-32 Types of Discontinuities, Continuity and Compactness Details 55:55
Mod-31 Lec-33 Continuity and Compactness (Contd.), Connectedness Details 55:59
Mod-32 Lec-34 Differentiability of real valued function, Mean Value Theorem Details 53:52
Mod-33 Lec-35 Mean Value Theorem (Contd.) Details 56:46
Mod-34 Lec-36 Application of MVT , Darboux Theorem, L Hospital Rule Details 52:54
Mod-35 Lec-37 L’Hospital Rule and Taylor’s Theorem Details 54:6
Mod-36 Lec-38 Tutorial – III Details 52:42
Mod-37 Lec-39 Riemann/Riemann Stieltjes Integral Details 53:3
Mod-38 Lec-40 Existence of Reimann Stieltjes Integral Details 55:39
Mod-39 Lec-41 Properties of Reimann Stieltjes Integral Details 54:35
Mod-39 Lec-42 Properties of Reimann Stieltjes Integral (Contd.) Details 56:45
Mod-40 Lec-43 Definite and Indefinite Integral Details 55:39
Mod-41 Lec-44 Fundamental Theorems of Integral Calculus Details 52:12
Mod-42 Lec-45 Improper Integrals Details 55:53
Mod-43 Lec-46 Convergence Test for Improper Integrals Details 53:47

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