Real Analysis and Applications: Theory in Practice: Undergraduate Texts in Mathematics

Analysis.- Review.- The Real Numbers.- Series.- Topology of.- Functions.- Differentiation and Integration.- Norms and Inner Products.- Limits of Functions.- Metric Spaces.- Applications.- Approximation by Polynomials.- Discrete Dynamical Systems.- Differential Equations.- Fourier Series and Physics.- Fourier Series and Approximation.- Wavelets.- Convexity and Optimization.

From the reviews:
Real Analysis with Real Applications:
"A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.;-- and yet, the new books that hit the market don't always hit the mark: The balance between theory and applications, --between technical proofs and intuitive ideas,--between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark. The writing is both systematic and engaged.- Refreshing! Novel: includes wavelets, approximation theory, discrete dynamics, differential equations, Fourier analysis, and wave mechanics." (Palle E. T. Jorgenson, Review from Amazon.com)
“In this exceptionally rich work, Davidson (Univ. of Waterloo, Canada) and Donsig (Univ. of Nebraska, Lincoln) meld material that is found in standard introductory real analysis works … and applications from both the mathematical and ‘real’ worlds … . The volume contains a substantial collection of exercises. Summing Up: Recommended. Upper-division undergraduates and graduate students.” (D. Robbins, Choice, Vol. 47 (10), June, 2010)
“This book is intended to provide an introduction both to real analysis and to a range of important applications in various fields. … In summary, this book is well conceived, well executed, and richer than many recent volumes in the same field. Teachers wanting a solid and interesting treatment that goes right to the point and does not bore good students with verbose explanations will find this book much to their liking. The volume also contains an adequate supply of exercises.” (Teodora-Liliana Rădulescu, Zentralblatt MATH, Vol. 1179, 2010)

This new approach to real analysis stresses the use of the subject in applications, showing how the principles and theory of real analysis can be applied in various settings. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. Each chapter has many useful exercises.
 
The treatment of the basic theory covers the real numbers, functions, and calculus, while emphasizing the role of normed vector spaces, and particularly of Rⁿ. The applied chapters are mostly independent, giving the reader a choice of topics. This book is appropriate for students with a prior knowledge of both calculus and linear algebra who want a careful development of both analysis and its use in applications.
 
Review of the previous version of this book, Real Analysis with Real Applications:
 
"A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.; ---and yet, the new books that hit the market don't always hit the mark: the balance between theory and applications, ---between technical proofs and intuitive ideas, ---between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark."
 
Palle E. T. Jorgenson, Review from Amazon.com
 
Kenneth R. Davidson is University Professor of Mathematics at the University of Waterloo. Allan P. Donsig is Associate Professor of Mathematics at the University of Nebraska-Lincoln.

Includes applications that cover: Approximation by polynomials Discrete dynamical systems Differential equations Fourier series and physics Fourier series and approximation Convexity and optimization Appropriate for math enthusiasts with a prior knowledge of both calculus and linear algebra Includes supplementary material: sn.pub/extras