This note is an activity-oriented
companion to the study of real analysis. It is intended as a pedagogical
companion for the beginner, an introduction to some of the main ideas in real
analysis, a compendium of problems, are useful in learning the subject, and an
annotated reading or reference list. Topics covered includes: Sets, Functions,
Cardinality, Groups, Vector Spaces, And Algebras, Partially Ordered Sets, The
Real Numbers, Sequences And Indexed Families, Categories, Ordered Vector Spaces,
Topological Spaces, Continuity And Weak Topologies, Normed Linear Spaces,
Differentiation, Complete Metric Spaces, Algebras And Lattices Of Continuous
Functions.
This
note covers the following topics: Construction of the Real Line, Uniqueness of R
and Basic General Topology, Completeness and Sequential Compactness, Convergence
of Sums, Path-Connectedness, Lipschitz Functions and Contractions, and Fixed
Point Theorems, Uniformity, Normed Spaces and Sequences of Functions,
Arzela-Ascoli, Differentiation and Associated Rules, Applications of
Differentiation, The Riemann Integral, Limits of Integrals, Mean Value Theorem
for Integrals, and Integral Inequalities, Inverse Function Theorem, Implicit
Function Theorem and Lagrange Multipliers, Multivariable Integration and Vector
Calculus
This note covers the following topics: Numbers, Real (R) and
Rational (Q), Calculus in the 17th and 18th Centuries, Power Series, Convergence
of Sequences and Series, The Taylor Series, Continuity, Intermediate and Extreme
Values, From Fourier Series back to the Real Numbers.
This note explains
the following topics: Preliminaries: Proofs, Sets, and Functions, The Foundation
of Calculus, Metric Spaces, Spaces of Continuous Functions, Modes of continuity,
Applications to differential equations, Applications to power series.
This note explains the following topics: Logic and Methods of
Proof, Sets and Functions , Real Numbers and their Properties, Limits and
Continuity, Riemann Integration, Introduction to Metric Spaces.
This book is a one
semester course in basic analysis.It should be possible to use the book for both
a basic course for students who do not necessarily wish to go to graduate school
but also as a more advanced one-semester course that also covers topics such as
metric spaces. Topics covered includes: Real Numbers, Sequences and Series,
Continuous Functions, The Derivative, The Riemann Integral, Sequences of
Functions and Metric Spaces.
The
subject of real analysis is concerned with studying the behavior and properties
of functions, sequences, and sets on the real number line, which we denote as
the mathematically familiar R. This note explains the following topics:
Continuous Functions on Intervals, Bolzano’s Intermediate Value Theorem, Uniform
Continuity, The Riemann Integrals, Fundamental Theorems Of Calculus, Pointwise
and Uniform Convergence, Uniform Convergence and Continuity, Series Of
Functions, Improper Integrals of First Kind, Beta and Gamma Functions.
This note covers the following topics: Sequences
and Series of Functions, Uniform Convergence, Power series, Linear
transformations, Functions of several variables, Jacobians and extreme value
problems, The Riemann-Stieltjes integrals, Measure Theory.
Author(s): Guru Jambheshwar University of
Science and Technology, Hisar
This note explains the following topics:
Set Theory and the Real Numbers, Lebesgue Measurable Sets, Measurable Functions,
Integration, Differentiation and Integration, The Classical Banach Spaces, Baire
Category, General Topology, Banach Spaces, Fourier Series, Harmonic Analysis on
R and S and General Measure Theory.
This is a text in elementary real analysis. Topics covered includes:
Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions,
Differentiation, Riemann-Stieltjes Integration, Unifom Convergence and
Applications, Topological Results and Epilogue.