From the reviews:
BULLETIN OF MATHEMATICS BOOKS
"This text is written with the usual elegance and orderly style that we have come to expect from this author. Topics are presented in a clear and logical manner, and the text is rich with important and interesting content…This text is well worth considering in any program in applied mathematics, and also by anyone seeking a valuable reference in analysis and its applications."
"The book succeeds well in providing a sound basis in analysis beyond that which most students will have received as undergraduates…Overall, this text serves as a wonderful introduction to ‘useful’ analysis for new applied mathematicians."
"The author describes this marvelous book as designed for beginning graduate students in mathematics – in particular for those who intend to specialize in applied mathematics, and for graduate students in other disciplines such as engineering, physics and computer science…Those who are familiar with the author’s earlier books will not be surprised by its excellence. It is business-like and will be found to be demanding, but it is user-friendly. It is the reviewer’s opinion that it will be extremely useful and popular as a text; institutions that do not already require their students to take such a course no longer have an excuse, and should immediately organize one based on this book…The author gives a rigorous discussion of a very large number of topics. Careful definitions and clearly stated theorems are given…The writing in clear and readable throughout, with a pleasant and friendly style…it is the reviewer’s view that the author has presented a rich harvest of useful and interesting analysis in a splendid manner."
"This book evolved from a course for graduate students of mathematics ... . In the chapters, after introducing the topic in the necessarily rigorous way ... an excellent survey of more recent developments follows. ... In my opinion the book is splendidly written, the material is not too dense, the remarks are very informative and also the layout is nice. It was a pleasure to read the book and I think that it can be highly recommended to both students or lecturers in mathematics ... ." (H. Winkler, Nieuw Archief voor Wiskunde, Vol. 5 (2), 2004)
"This is a top-down book. Effort is made not just to be correct but to elucidate. Attention is given to motivation and to the relations between topics. Overview is not lost despite the presence of considerable detail. ... The book enjoys a clear visual layout and has excellent references and index. There is a rich supply of exercises. The style of presentation is pleasant." (Charles Pearce, The Australian Mathematical Society Gazette, Vol. 31 (1), 2004)
"The author gives a rigorous discussion of a very large number of topics. Careful definitions and clearly stated theorems are given. Most results are proved in detail ... . Almost every subsection contains exercises, some contain a large number. The writing is clear and readable throughout, with a pleasant and friendly style. ... it is the reviewer’s view that the author has presented a rich harvest of useful and interesting analysis in a splendid manner." (R. G. Bartle, Mathematical Reviews, Issue 2002 e)
"The book succeeds well in providing a sound basis in analysis beyond that which most students will have received as undergraduates. ... throughout the text, Cheney maintains a nice balance between mathematical theory, a discussion motivating applications of the theory, and concrete examples of the analysis in action. ... Particularly nice is a section on ‘analytic pitfalls’ ... a warning to all of us to beware of the details. Overall, this text serves as a wonderful introduction to ‘useful’ analysis for new applied mathematicians." (Karl R. Knaub, SIAM Review, Vol. 44 (3), 2002)
"This text is written with the usual elegance and orderly style that we have to come to expect from this author. Topics are presented in a clear and logical manner, and the text is rich with important and interesting content. ... This text is well worth considering in any program in applied mathematics, and also by anyone seeking a valuable reference in analysis and its applications." (William Gearhart, The Bulletin of Mathematics Books, Issue 41, 2002)
"Analysis for applied mathematicians grew out of a beginning graduate course for intending applied mathematicians at the University of Texas. ... There are extensive cross-references to standard sources listed in the 8-page bibliography and there is an excellent array of accessible exercises provided at the end of each section. ... This is a very readable, well-organised text: the applied context gives a unifying narrative thrust with a pragmatic emphasis on what the key ideas are and what they may be used for." (Nick Lord, The Mathematical Gazette, Vol. 87 (509), 2003)
"The book is primarily written for preliminary graduate students of applied mathematics and is suitable also for courses offered to the students of Engineering, Physics and Computer Science. ... The book is adequately well-written and arranged with proper care to suit junior graduate students of applied mathematics. ... The bibliography is exhaustive and is suitable for those who desire to pursue their further studies in this branch of applied mathematics." (B. K. Lahiri, Journal of the Indian Academy of Mathematics, Vol. 24 (1), 2002)
"The book certainly will bridge a gap between the basic education of an applied mathematician and the needs of modern applied mathematics. The text contains a plenty of exercises and examples. The great experience of the author can be felt everywhere. Thus the book is of great value both for students and lecturers." (R. Schumann, Internationale Mathematische Nachrichten, Issue 190, 2002)
"The book is very well written, and the presentation is clear and interesting. It is certainly a valuable complementation to the vast literature ... and may be recommended as additional literature to graduate and PhD students in mathematics." (Jürgen Appell, Zentralblatt MATH, Vol. 984, 2002)