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Basic Real Analysis by Anthony W. Knapp
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About this book :-
Basic Real Analysis written by
Anthony W. Knapp
This book and its companion volume, Advanced Real Analysis, systematically develop concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. The two books together contain what the young mathematician needs to know about real analysis in order to communicate well with colleagues in all branches of mathematics.
The books are written as textbooks, and their primary audience isstudents who are learning the material for the first time and who are planning a career in which they will use advanced mathematics professionally. Much of the material in the books corresponds to normal course work. Nevertheless, it is often the case that core mathematics curricula, time-limited as they are, do not include all the topics that one might like. Thus the book includes important topics that may be skipped in required courses but that the professional mathematician will ultimately want to learn by self-study.
Book Detail :-
Title: Basic Real Analysis
Author(s): Anthony Knapp
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About Author :-
Author Anthony W. Knapp (born 1941, New Jersey) is an American mathematician at the State University of New York, Stony Brook working on representation theory, who classified the tempered representations of a semisimple Lie group. He won the Leroy P. Steele Prize for Mathematical Exposition in 1997. He became a fellow of the American Mathematical Society in 2012.
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• Download PDF Basic Real Analysis by Anthony Knapp
• Download PDF Advance Real Analysis (Digital Second Edition) by Anthony Knapp
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Book Contents :-
Basic Real Analysis written by
Anthony W. Knapp
cover the following topics.
THEORY OF CALCULUS IN ONE REAL VARIABLE
Review of Real Numbers, Sequences, Continuity, Interchange of Limits, Uniform Convergence, Riemann Integral, Complex-Valued Functions, Taylor’s Theorem with Integral Remainder, Power Series and Special Functions, Summability, Weierstrass Approximation Theorem, Fourier Series, Problems
Definition and Examples, Open Sets and Closed Sets, Continuous Functions, Sequences and Convergence, Subspaces and Products, Properties of Metric Spaces, Compactness and Completeness, Connectedness, Baire Category Theorem, Properties of C(S) for Compact Metric S, Completion, Problems
THEORY OF CALCULUS IN SEVERAL REAL VARIABLES
Operator Norm, Nonlinear Functions and Differentiation, Vector-Valued Partial Derivatives and Riemann Integrals, Exponential of a Matrix, Partitions of Unity, Inverse and Implicit Function Theorems, Definition and Properties of Riemann Integral, Riemann Integrable Functions, Fubini’s Theorem for the Riemann Integral, Change of Variables for the Riemann Integral, Arc Length and Integrals with Respect to Arc Length, Line Integrals and Conservative Vector Fields, Green’s Theorem in the Plane, Problems
THEORY OF ORDINARY DIFFERENTIAL EQUATIONS AND SYSTEMS
Qualitative Features and Examples, Existence and Uniqueness, Dependence on Initial Conditions and Parameters, Integral Curves, Linear Equations and Systems, Wronskian, Homogeneous Equations with Constant Coefficients, Homogeneous Systems with Constant Coefficients, Series Solutions in the Second-Order Linear Case, Problems
LEBESGUE MEASURE AND ABSTRACT MEASURE THEORY
Measures and Examples, Measurable Functions, Lebesgue Integral, Properties of the Integral, Proof of the Extension Theorem, Completion of a Measure Space, Fubini’s Theorem for the Lebesgue Integral, Integration of Complex-Valued and Vector-Valued Functions, L1, L2, L∞, and Normed Linear Spaces, Arc Length and Lebesgue Integration, Problems
MEASURE THEORY FOR EUCLIDEAN SPACE
Lebesgue Measure and Other Borel Measures, Convolution, Borel Measures on Open Sets, Comparison of Riemann and Lebesgue Integrals, Change of Variables for the Lebesgue Integral, Hardy–Littlewood Maximal Theorem, Fourier Series and the Riesz–Fischer Theorem, Stieltjes Measures on the Line, Fourier Series and the Dirichlet–Jordan Theorem, Distribution Functions, Problems
DIFFERENTIATION OF LEBESGUE INTEGRALS ON THE LINE
Differentiation of Monotone Functions, Absolute Continuity, Singular Measures, and Lebesgue Decomposition, Problems
FOURIER TRANSFORM IN EUCLIDEAN SPACE
Elementary Properties, Fourier Transform on L1, Inversion Formula, Fourier Transform on L2, Plancherel Formula, Schwartz Space, Poisson Summation Formula, Poisson Integral Formula, Hilbert Transform, Problems
L p SPACES
Inequalities and Completeness, Convolution Involving L p, Jordan and Hahn Decompositions, Radon–Nikodym Theorem, Continuous Linear Functionals on L p, Riesz–Thorin Convexity Theorem, Marcinkiewicz Interpolation Theorem, Problems
Open Sets and Constructions of Topologies, Properties of Topological Spaces, Compactness and Local Compactness, Product Spaces and the Tychonoff Product Theorem, Sequences and Nets, Quotient Spaces, Urysohn’s Lemma, Metrization in the Separable Case, Ascoli–Arzela` and Stone–Weierstrass Theorems, Problems
INTEGRATION ON LOCALLY COMPACT SPACES
Setting, Riesz Representation Theorem, Regular Borel Measures, Dual to Space of Finite Signed Measures, Problems
HILBERT AND BANACH SPACES
Definitions and Examples, Geometry of Hilbert Space, Bounded Linear Operators on Hilbert Spaces, Hahn–Banach Theorem, Uniform Boundedness Theorem, Interior Mapping Principle, Problems
APPENDIX A. BACKGROUND TOPICS
Sets and Functions, Mean Value Theorem and Some Consequences, Inverse Function Theorem in One Variable, Complex Numbers, Classical Schwarz Inequality, Equivalence Relations, Linear Transformations, Matrices, and Determinants, Factorization and Roots of Polynomials, Partial Orderings and Zorn’s Lemma, Cardinality
APPENDIX B. ELEMENTARY COMPLEX ANALYSIS
Complex Derivative and Analytic Functions, Complex Line Integrals, Goursat’s Lemma and the Cauchy Integral Theorem, Cauchy Integral Formula, Taylor’s Theorem, Local Properties of Analytic Functions, Logarithms and Winding Numbers, Operations on Taylor Series, Argument Principle, Residue Theorem, Evaluation of Definite Integrals, Global Theorems in Simply Connected Regions, Global Theorems in General Regions, Laurent Series, Holomorphic Functions of Several Variables, Problems
CONTENTS OF ADVANCED REAL ANALYSIS
Introduction to Boundary-Value Problems, Compact Self-Adjoint Operators, Topics in Euclidean Fourier Analysis, Topics in Functional Analysis, Distributions, Compact and Locally Compact Groups, Aspects of Partial Differential Equations, Analysis on Manifolds, Foundations of Probability, Introduction to Wavelets,
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