Real Analysis I

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REVIEW OF SET THEORY : Operations on sets, family of sets, indexing set, functions, axiom of choice, relations, equivalence relation, partial order, total order, maximal element, Zorn’s lemma, finite set, countable set, uncountable set, Cantor’s theorem, cardinal numbers, Continuum Hypothesis – SEQUENCES AND SERIES OF REAL NUMBERS : Real Number System, algebraic properties, order properties, absolute value function, LUB axiom, Archemedian property – Sequences of real numbers, convergent sequence, subsequence, Sandwich theorem, monotonic sequence, limsup, liminf, Bolzano-Weierstrass Theorem, Cauchy sequence, infinite series, convergent series, absolute convergence, series of nonnegative real numbers, comparison test, root test, ratio test, power series, extended real line, conditional convergence, rearrangement.

METRIC SPACES – BASIC CONCEPTS : Metric, metric space, metric induced by norm, open ball, closed ball, sphere, interval, interior, exterior, boundary, open set, topology, closure point, limit point, isolated point, closed set, Cantor set – COMPLETENESS : Sequences in metric spaces, complete metric space, Cantor’s Intersection Theorem, Baire Category Theorem – LIMITS AND CONTINUITY : Limit and continuity of a function defined on a metric space, uniform continuity, homeomorphism, Lipschitz continuous function, contraction, isometry, Banach’s contraction mapping principle – CONNECTEDNESS AND COMPACTNESS : Connectedness: Connected set, interval, Intermediate Value Theorem, connected component, totally disconnected set – Compactness: Compact set, finite intersection property, totally bounded set, Bolzano – Weierstrass theorem, sequential compactness, Heine – Borel theorem, continuous functions on compact sets, types of discontinuity.

DIFFERENTIATION : Derivative, differentiable function, chain rule, derivative of a composite function, local minimum, local maximum, Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, indeterminate forms, L’ Hospital’s rule, intermediate value property, higher order derivatives, Taylor’s theorem, Taylor series, infinitely differentiable function, Maclaurin series, differentiation of vector valued functions – MODULE 8: INTEGRATION : Riemann integral, Riemann -Stieltjes integral, Riemann -Stieltjes integrable function, First Mean Value Theorem of Integral Calculus, Integration as a Limit of Sum, Fundamental Theorem of Integral Calculus, integration of vector valued function, function of bounded variation, total variation, Integration by part, second Mean Value Theorem of Integral Calculus, change of variable formula, improper integral – SEQUENCES AND SERIES OF FUNCTIONS : Sequence of functions, pointwise convergence, series of functions, uniform convergence, uniformly bounded sequence, Cauchy’s criterion for uniform convergence, uniformly Cauchy sequence, Weierstrass’ M test, Dini’s theorem, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, Weierstrass theorem, equicontinuous family of functions, Arzela – Ascoli Theorem.

Course Curriculum

Introduction Details
Functions and Relations Details 51:36
Finite and Infinite Sets Details
Countable Sets Details
Uncountable Sets, Cardinal Numbers Details
Real Number System Details 52:16
LUB Axiom Details
Sequences of Real Numbers Details 52:36
Sequences of Real Numbers – continued Details 52:23
Sequences of Real Numbers – continued… Details 50:59
Infinite Series of Real Numbers Details
Series of nonnegative Real Numbers Details
Conditional Convergence Details
Metric Spaces: Definition and Examples Details 52:56
Metric Spaces: Examples and Elementary Concepts Details 52:9
Balls and Spheres Details 52:3
Open Sets Details
Closure Points, Limit Points and isolated Points Details 52:20
Closed sets Details
Sequences in Metric Spaces Details 51:44
Completeness Details 49:20
Baire Category Theorem Details 53:38
Limit and Continuity of a Function defined on a Metric space Details
Continuous Functions on a Metric Space Details
Uniform Continuity Details 51:1
Connectedness Details
Connected Sets Details 54:53
Compactness Details 51:22
Compactness – Continued Details 51:59
Characterizations of Compact Sets Details 56:29
Continuous Functions on Compact Sets Details 53:20
Types of Discontinuity Details
Differentiation Details
Mean Value Theorems Details 50:19
Mean Value Theorems – Continued Details 51:35
Taylor’s Theorem Details 50:13
Differentiation of Vector Valued Functions Details 50:59
Integration Details 51:2
Integrability Details 50:43
Integrable Functions Details
Integrable Functions – Continued Details 52:33
Integration as a Limit of Sum Details 52:25
Integration and Differentiation Details 54:25
Integration of Vector Valued Functions Details 51:51
More Theorems on Integrals Details 52:35
Sequences and Series of Functions Details 51:34
Uniform Convergence Details
Uniform Convergence and Integration Details 52:50
Uniform Convergence and Differentiation Details 52:6
Construction of Everywhere Continuous Nowhere Differentiable Function Details 53:42

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