What do you think?
Rate this book
Real Analysis With Real Applications
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 settings—in 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.
624 pages, Hardcover
First published December 30, 2001
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 of 1 review
November 13, 2020
I had the honour to be in Professor Davidson's Real Analysis class at University of Waterloo from September to December 2019. If what he told me is true, this term would be his second-last teaching term before his retirement.
From one of the most formidable analysis professors at UW, the course was a blast. Many interesting ideas were introduced at a rapid pace. Basic concepts such as completeness and compactness were accompanied by fairly involved examples, with considerable time devoted to the field of p-adic numbers and properties of the general Cantor set.
This book was used as a textbook along with a few others throughout the term. Although Prof. Davidson was the author of this book, the contents of it was not faithfully followed until after we had introduced metric completion and the Baire Category Theorem. From that point on, the lectures on the Weierstrass Approximation theorem, the Stone-Weierstrass Theorem, the Banach Contraction Principle, and the theory of differential equations, were nearly identical to the respective chapters in this book.
Two things from this book marks it apart from the other textbooks that we used. One is the proof of the Weierstrass approximation theorem, which was from Bernstein. Of course, other, more efficient proofs were introduced later after the chapters on Fourier series. Another unique aspect of this book is its chapters on the theory of differential equations. Little application is found in the pages, with theories concerning the existence of solutions to differential solutions dominating. Proofs of Picard's theorems were given in full both in the book and in lectures, along with proofs for the Continuation Theorem, Peano's Theorem, and theorems on Perturbations of DE solutions. No techniques of solving differential equations were presented.
The chapter on differential equations, for me, was the highlight of the course as well as the book. To the best of my limited knowledge, there do not exist many books out there devoted to the theory of differential equations.
From one of the most formidable analysis professors at UW, the course was a blast. Many interesting ideas were introduced at a rapid pace. Basic concepts such as completeness and compactness were accompanied by fairly involved examples, with considerable time devoted to the field of p-adic numbers and properties of the general Cantor set.
This book was used as a textbook along with a few others throughout the term. Although Prof. Davidson was the author of this book, the contents of it was not faithfully followed until after we had introduced metric completion and the Baire Category Theorem. From that point on, the lectures on the Weierstrass Approximation theorem, the Stone-Weierstrass Theorem, the Banach Contraction Principle, and the theory of differential equations, were nearly identical to the respective chapters in this book.
Two things from this book marks it apart from the other textbooks that we used. One is the proof of the Weierstrass approximation theorem, which was from Bernstein. Of course, other, more efficient proofs were introduced later after the chapters on Fourier series. Another unique aspect of this book is its chapters on the theory of differential equations. Little application is found in the pages, with theories concerning the existence of solutions to differential solutions dominating. Proofs of Picard's theorems were given in full both in the book and in lectures, along with proofs for the Continuation Theorem, Peano's Theorem, and theorems on Perturbations of DE solutions. No techniques of solving differential equations were presented.
The chapter on differential equations, for me, was the highlight of the course as well as the book. To the best of my limited knowledge, there do not exist many books out there devoted to the theory of differential equations.
Displaying 1 of 1 review