Beginning Functional Analysis (Hardcover)

~ (Author) "The idea of an inner product space is to describe an abstract structure with the desirable properties of Euclidean space: a distance-measuring device and a..." (more)
Key Phrases: invariant subspace problem, complete orthonormal sequence, parallelogram equality, Per Enflo, Frigyes Riesz, John von Neumann (more...)

Editorial Reviews

Review

From the reviews:

"As an introductory text, Beginning functional analysis delivers what it says: a fast moving introduction into this world of abstract analysis. … There is a great deal to recommend this book: it is clearly and concisely written, and works to overcome some of the dryness from which many similar books suffer. … In reviewing it, I found it an enjoyable read, which, more than once, found its way to my bedside table." (James Gazet, The Mathematical Gazette, March, 2005)

"According to the author, her primary goal was to write a text for (advanced undergraduates or beginning graduate) students that would introduce them to ‘the beautiful field of functional analysis’. This objective is certainly achieved with this attractive monograph. … In conclusion: a good companion for students interested in functional analysis." (Lucas Quarta, Belgian Mathematical Society-Simon Stevin Bulletin, Vol. 10 (2), 2003)

"There are really two levels to this book. First, it is an introductory text. Second, it provides a glimpse into the history of functional analysis and the personalities who developed the subject. Saxe has succeeded in presenting both of these subjects. … This is a good book for students to learn functional analysis. It is also one that students will enjoy using and out of which they will get more than just the nuts and bolts of the topics." (Robert W. Vallin, MAA Online, September, 2002)

"This is an excellent introductory text in functional analysis; it is more than just another book on the subject because of the author’s pleasant and vivid manner of writing and her original pedagogical point of view. The material has been adequately selected, each chapter is supplied with problems of different grades of difficulty and the bibliography contains the best books in the area. … Containing ‘what everyone should know about functional analysis’, this book has all the qualities to ensure a pedagogical success.” (Ioana Cioranescu, Mathematical Reviews, Issue 2002 m)

"The author offers a course of functional analysis for beginners in less than 200 pages: the prerequisites are a basic knowledge of linear algebra and real analysis. … The author’s style is stimulating, and she includes numerous comments on the development of the discipline." (European Mathematical Society Newsletter, September, 2002)

"The text is carefully written throughout. Many exercises varying from straightforward to challenging, are useful for self-study or independent study of the book. … The text contains many interesting remarks on the historical origins of mathematical concepts and theorems, and biographical information on mathematicians who contributed to the theories presented (e.g. M. Fréchet, F. Riesz, P. Enflo, M. Stone). The book is highly recommendable for a first course in functional analysis." (Joachim Naumann, Zentralblatt MATH, Vol. 1002 (2), 2003)

"The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis." (L’Enseignement Mathematique, Vol. 48 (1-2), 2002)

"The book is accessible to students who have completed first courses in linear algebra and real analysis. … Beginning Functional Analysis is designed as a text for a first course on functional analysis for advanced undergraduates or for beginning graduate students. … It can also be used for self-study or independent study. … It contains information about the historical development of the material and biographical information of key developers of the theories." (Karen Saxe, Wordtrade, 2009)

Product Description

The unifying approach of functional analysis is to view functions as points in some abstract vector space and the differential and integral operators relating these points as linear transformations on these spaces. The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis. Topics are developed in their historical context, with accounts of the past - including biographies - appearing throughout the text. The book offers suggestions and references for further study, and many exercises.

Karen Saxe is Associate Professor of Mathematics at Macalester College in St. Paul, Minnesota. She received her Ph.D. from the University of Oregon. Before joining the faculty at Macalester, she held a two-year FIPSE post-doctoral position at St. Olaf College in Northfield, Minnesota. She currently serves on the editorial board of the MAA's College Mathematics Journal. This is her first book.

Product Details

• Hardcover: 216 pages
• Publisher: Springer; 1 edition (December 7, 2001)
• Language: English
• ISBN-10: 0387952241
• ISBN-13: 978-0387952246
• Product Dimensions: 9.5 x 6.1 x 0.8 inches
• Shipping Weight: 1 pounds (View shipping rates and policies)
• Average Customer Review: 3.3 out of 5 stars  See all reviews (7 customer reviews)
• Amazon.com Sales Rank: #596,564 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

7 of 7 people found the following review helpful:

5.0 out of 5 stars Lucid, March 25, 2004

By	Anthony Varilly (Cambridge, England) - See all my reviews

This book is fantastic! It is an extremely readable account of the basics of the subject. I thought the Chapter on Measure Theory and Lebesgue integration were particularly well organized. Every definition was well motivated and the theorems were arranged in a very natural progression.

One thing I especially enjoyed about this book is that most of the proofs are done only for special cases of theorems, without loss of generality. For example, the Arzela-Ascoli theorem is proved for the function space C([a,b],R) (R = real numbers), but then Saxe points out what makes the proof 'tick' so that the reader may easily modify it to a more general setting (she always states the more precise versions of such theorems as well). This is great because it helps one's intuition without getting short-changed.

Finally, the book has a great wealth of historical notes and biographies which are rich in mathematical content (e.g., Saxe explains that Frechet was the first person to define a metric space even though he called it 'une class E'; Hausdorff gave it its modern name in 1914). The reader can in this way appreciate how the subject slowly developed into its present form.

This book is a jewel! I myself am not the biggest fan of functional analysis, but this book made me really appreciate the subject.

4 of 4 people found the following review helpful:

3.0 out of 5 stars Makes the material unnecessarily dry and boring, August 9, 2006

By	Alexander C. Zorach "Alex Zorach" (Lancaster, PA) - See all my reviews (REAL NAME)

I do not agree with the other readers that say this book moves too quickly or is too difficult--your first couple encounters with the UTM series often always seem this way. However, I don't think this is the best introductory book on the subject material for a number of reasons. The other reviewers make clear that less experienced students do not find this book easy--I am pointing out that a more experienced student finds this book boring.

I feel that functional analysis, like linear alegbra, is one subject in mathematics that is often not sufficiently motivated, and as a result comes across as both difficult and boring. While I think this book takes some of the difficulty out of things through its straightforward and clear explanations of various concepts, and through its relatively easy exercises which reinforce the material, I find this book to be exceptionally boring, especially in the initial few chapters. The book is too concise for a UTM text; it's hard for students to understand why to get through the earlier material unless you give more interesting examples and talk a little bit more about where you're going and why anyone would want to learn this stuff.

I would recommend the book by Kreyszig as a supplement to anyone stuck reading this one; it doesn't cover the same material (it tends to be a little less advanced, omitting discussion of measures), and it's a very different sort of book, but especially in the earlier few chapters it will provide a much needed energy, more interesting examples.

11 of 15 people found the following review helpful:

5.0 out of 5 stars The Lebesgue integral and more, May 1, 2003

By	Raymond Jensen - See all my reviews (REAL NAME)

These past two semesters I've been enrolled in a graduate-level analysis course. The book we used, by Folland (see my review) was not a good book, in my opinion. After reading the first several chapters of Folland, I just did not understand what the point was to Lebesgue integration, or why we had to develop all this machinery that goes along with it.

Sometime during the semester, I got hold of this book, by Saxe, and started reading the chapters on Lebesgue integration. After doing that, I began to develop an understanding of what it was, how it was used, and why it was necessary to cover all these theorems. The book gave me perspective on the subject; (and hence motivation) something which Folland did not do.

Saxe's book isn't without it's faults; I had some issues with her proof of the Baire Category theorem (in this case, I actually found Follands proof much more believable) and she got the year of Hermann Minkowski's death wrong. Other than that, I could not find any problems with the book.

In summary, this book fills a much needed void in the literature: a readable book which introduces the student to functional analysis beyond the undergraduate "advanced calculus level." If you are in a graduate-level real analysis course and haven't a clue what a sigma algebra is or why you should care (but would like to), then buy this book.