Real Analysis I

Overview

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, Zorns lemma, finite set, countable set, uncountable set, Cantors 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, Cantors 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, Banachs 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, Rolles theorem, Lagranges mean value theorem, Cauchys mean value theorem, indeterminate forms, L Hospitals rule, intermediate value property, higher order derivatives, Taylors 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, Cauchys criterion for uniform convergence, uniformly Cauchy sequence, Weierstrass M test, Dinis theorem, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, Weierstrass theorem, equicontinuous family of functions, Arzela - Ascoli Theorem.