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Real Analysis with Real Applications

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ISBN-10: 0130416479

ISBN-13: 9780130416476

Edition: 2002

Authors: Kenneth R. Davidson, Allan P. Donsig

List price: $120.00
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Description:

Using a progressive but flexible format, this book contains a series of independent chapters that show how the principles and theory of real analysis can be applied in a variety of settingsin subjects ranging from Fourier series and polynomial approximation to discrete dynamical systems and nonlinear optimization. Users will be prepared for more intensive work in each topic through these applications and their accompanying exercises. Chapter topics under the abstract analysis heading include: the real numbers, series, the topology of R^n, functions, normed vector spaces, differentiation and integration, and limits of functions. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. For math enthusiasts with a prior knowledge of both calculus and linear algebra.
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Book details

List price: $120.00
Copyright year: 2002
Publisher: Prentice Hall PTR
Publication date: 12/20/2001
Binding: Hardcover
Pages: 624
Size: 7.25" wide x 9.25" long x 1.25" tall
Weight: 2.398
Language: English

Background
The Language of Mathematics
Sets and Functions
Calculus
Linear Algebra
The Role of Proofs
Appendix: Equivalence Relations
Abstract Analysis
The Real Numbers
An Overview of the Real Numbers
Infinite Decimals
Limits
Basic Properties of Limits
Upper and Lower Bounds
Subsequences
Cauchy Sequences
Appendix: Cardinality
Series
Convergent Series
Convergence Tests for Series
The Numbere
Absolute and Conditional Convergence
The Topology of R^n
n-dimensional Space
Convergence and Completeness inR n
Closed and Open Subsets ofR n
Compact Sets and the Heine-Borel Theorem
Functions
Limits and Continuity
Discontinuous Functions
Properties of Continuous Functions
Compactness and Extreme Values
Uniform Continuity
The Intermediate Value Theorem
Monotone Functions
Normed Vector Spaces
Differentiable Functions
The Mean Value Theorem
Riemann Integration
The Fundamental Theorem of Calculus
Wallis's Product and Stirling's Formula
Measure Zero and Lebesgue's Theorem
Differentiation and Integration
Definition and Examples
Topology in Normed Spaces
Inner Product Spaces
Orthonormal Sets
Orthogonal Expansions in Inner Product Spaces
Finite-Dimensional Normed Spaces
TheL P Norms
Limits of Functions
Limits of Functions
Uniform Convergence and Continuity
Uniform Convergence and Integration
Series of Functions
Power Series
Compactness and Subsets ofC(K)
Metric Spaces
Definitions and Examples
Compact Metric Spaces
Complete Metric Spaces
Connectedness
Metric Completion
TheL P Spaces and Abstract Integration
Applications
Approximation by Polynomials
Taylor Series
How Not to Approximate a Function
Bernstein's Proof of the Weierstrass Theorem
Accuracy of Approximation
Existence of Best Approximations
Characterizing Best Approximations
Expansions Using Chebychev Polynomials
Splines
Uniform Approximation by Splines
Appendix: The Stone-Weierstrass Theorem
Discrete Dynamical Systems
Fixed Points and the Contraction Principle
Newton's Method
Orbits of a Dynamical System
Periodic Points
Chaotic Systems
Topological Conjugacy
Iterated Function Systems and Fractals
Differential Equations
Integral Equations and Contractions
Calculus of Vector-Valued Functions
Differential Equations and Fixed Points
Solutions of Differential Equations
Local Solutions
Linear Differential Equations
Perturbation and Stability of Des
Existence without Uniqueness
Fourier Series and Physics
The Steady-State Heat Equation
Formal Solution
Orthogonality Relations
Convergence in the Open Disk
The Poisson Formula
Poisson's Theorem
The Maximum Principle
The Vibrating String (Formal Solution)
The Vibrating String (Rigourous Solution)
Appendix: The Complex Exponential
Fourier Series and Approximation
Least Squares Approximations
The Isoperimetric Problem
The Riemann-Lebesgue Lemma
Pointwise Convergence of Fourier Series
Gibbs's Phenomenon
Cesaro Summation of Fourier Series