Advanced Calculus: An Introduction to Linear Analysis

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Wiley-Interscience, Apr 14, 2008 - Mathematics - 387 pages
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Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra

"Advanced Calculus" reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis.

Following an introduction dedicated to writing proofs, the book is divided into three parts:

Part One explores foundational one-variable calculus topics from the viewpoint of linear spaces, norms, completeness, and linear functionals.

Part Two covers Fourier series and Stieltjes integration, which are advanced one-variable topics.

Part Three is dedicated to multivariable advanced calculus, including inverse and implicit function theorems and Jacobian theorems for multiple integrals.

Numerous exercises guide readers through the creation of their own proofs, and they also put newly learned methods into practice. In addition, a "Test Yourself" section at the end of each chapter consists of short questions that reinforce the understanding of basic concepts and theorems. The answers to these questions and other selected exercises can be found at the end of the book along with an appendix that outlines key terms and symbols from set theory.

Guiding readers from the study of the topology of the real line to the beginning theorems and concepts of graduate analysis, "Advanced Calculus" is an ideal text for courses in advanced calculus and introductory analysis at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for engineers, scientists, and mathematicians.

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This is another example of someone who either refuses to or simply cannot write clearly.
Just to focus on 1.7, the Heine-Borel Covering Theorem which is representative. His opening is to say the
least baffling to someone without at least some introduction to topology and this is made worse by the deliberate absence of meaningful examples of open and closed sets. His exposition on Theorem 1.7.1, that the union of open sets is an open set is just as baffling, as he refers the reader to a later exercise and then starts the proof in medias res. This is simply bad teaching, bad presentation. In Definition 1.7.2, he defines an open cover in as incomprehensibly a way as possible and then makes matters worse by providing no meaningful example. When he finally gets to the-Borel Covering Theorem, as he does not clearly explain what it stands for, his proof comes out as a pile of gibberish. There is simply no excuse for this.
Many of the problems do not clearly follow from the exposition and there are far too many of the "give an example" type. Richardson is the professor; he should be providing the examples; he should be facilitating comprehension, not deliberately obviating it.
Why Richardson and others take the byzantine approach to composing mathematics texts, as opposed to the down-to-earth approach, is beyond me. Are they fearful that too many people will grasp the subject too quickly? Are they afraid that if they write in an immediately comprehensible manner that students will simply read their books, not attend class and put them out of a job?
I have no idea why Richardson even wrote this book (except for the obvious and vulgar reason)for it is no better--and in some ways, it is considerably worse--than the texts out there. This book is beyond disappointing and I resent having paid nearly $100.00 for it and gotten next to nothing out of it. Richardson should be ashamed for even writing it.
 

Contents

Real Numbers and Limits of Sequences
3
Continuous Functions
39
Riemann Integral
69
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