This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other.
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44 of 46 people found the following review helpful
Format:Hardcover
In the author's words in the preface, the dominant theme of this book is continuity and linearity, and its goal is to illuminate the meanings of these words and their relations to each other. The book, he says, belongs to the type of pure mathematics that is concerned with form and structure, and such a body of mathematics must be judged by its high aesthetic quality, and should exalt the mind of the reader.
The author's attitude can only be characterized as magnificent, and, if one is to judge his utterances in the preface by what is found after it, one will indeed find perfect evidence of his delight in mathematics and his high competence in elucidating very abstract concepts in topology and real analysis. Indeed, this has to be the best book ever written for mathematics at this level. It is a book that should be read by everyone that desires deep insights into modern real and functional analysis.
After a brief and informal overview of set theory, the author moves on to the theory of metric spaces in chapter 2. His emphasis is on the idea that metric spaces are easy to find, since every non-empty set has the discrete metric, and that metric spaces are good motivation for the more general idea of a topological space. The Cantor set, ubiquitous in measure theory, dynamical systems, and fractal geometry, is constructed as the most general closed set on the real line, i.e. one obtained by removing from the real line a countable disjoint class of open intervals. Continuity of mappings between metric spaces is defined, and also the concept of uniform continuity, the latter of which is motivated very nicely by the author. Then, the author takes the reader to a higher level of abstraction, wherein he asks the reader to consider all of the continuous functions on a metric space, and turn this collection into a metric space of a special type called a normed linear space, and, more specifically, a Banach space. Thus the author introduces the reader to the field of functional analysis.
A lengthy introduction to topological spaces follows in chapter 3. The author motivates well the idea of an open set, and shows that one could just as easily use closed sets as the fundamental concept in topology. And, most important for functional analysis, he introduces the weak topology, and shows how to obtain the weakest topology for a collection of mappings from a topological space to a collection of other topological spaces. The reader can see clearly that the weaker the topology on a space the harder it is for mappings to be continuous on the space.
Compactness, so essential in all areas of mathematics that make use of topology, is discussed in chapter 4. It is motivated by an abstraction of the Heine-Borel theorem from elementary real analysis, and the author shows how well-behaved things are on compact topological spaces. Some important theorems are proved in this chapter, namely Tychonoff's theorem, the Lebesgue covering lemma, and Ascoli's theorem.
Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces. Chapter 5 is an in-depth discussion of separation, and the reader again confronts function spaces, and their ability (or non-ability) to separate the points of a topological space. Spaces that allow such separation to occur are called completely regular, and this property has far-reaching consequences in analysis and other areas of mathematics. The Stone-Cech compactification is discussed as an imbedding theorem for completely regular spaces, analogous to one for normal spaces.
The intuitive idea of a space being connected is given rigorous treatment in chapter 6. Certain pathologies can of course arise when discussing connectedness, and the author shows this by discussing totally disconnected spaces, remarking that such spaces are very important in dimension theory and representation theory. Indeed, computational and fractal geometry is much harder to study because of the existence of these spaces.
Chapter 7 is important to all working in numerical analysis, wherein the author discusses approximation theory. The Weierstrass approximation and the Stone-Weierstrass theorems are discussed in detail.
A slight detour through algebra is given in chapter 8. Groups, rings, and fields are given a minimal treatment by the author, discussing only the basic rudiments that are needed to get through the rest of the book.
Banach spaces make their appearance in chapter 9, with the three pillars of the theory proven: the Hahn-Banach, the open mapping, and the uniform boundedness theorems. These theorems guarantee that the study of Banach spaces is worth doing, and that there are analogs of the finite dimensional theory in the (infinite)-dimensional context of Banach spaces. The theory of Banach spaces is very extensive, but this chapter gives a peek at this very interesting area of mathematics.
Banach spaces with an inner product are considered in chapter 10. These of course are the familiar Hilbert spaces, so important in physics and the subject of a huge amount of research in mathematics. The presence of the inner product allows constructions familiar from ordinary finite-dimensional vector spaces to carry over to the inifinite-dimensional setting, one example being the transpose of a matrix, which is replaced in the Hilbert space setting by a self-adjoint operator.
As a warm-up to the infinite-dimensional theory, finite-dimensional spectral theory is considered in chapter 11. The famous spectral theorem is proven. Then in chapter 12, the reader enters the world of "soft" analysis, wherein topological and algebraic constructions are used to study linear operators on spaces of infinite dimensions. Putting an algebraic structure on a Banach space gives a Banach algebra, and then the trick is deal with the spectrum of an element of this algebra. The reader can see the interplay between algebra, topology, and analysis in this chapter and the next one on commutative Banach algebras. Indeed, the Gelfand-Naimark theorem, that essentially states that elements of a commutative Banach *-algebra act like the functions on its maximal ideal space, has to rank as one of the most interesting results in the book, and indeed in all of mathematics.
23 of 23 people found the following review helpful
Format:Hardcover
I became aquainted with this book many years ago and I still read it ... and send students off to read it. The book is written by an incredible expositor who was and still may be at Colorado College in Colorado. It is always the book that first comes to mind when someone asks for a reference on any of the subjects it covers. These include point set topology, analysis (Not including integration or measure theory), and operator theory. It is introductory. This merely makes you wish the author would have written several advanced sequels to this amazing book.
This book has my highest recommendation. Every mathematics student should own a copy ...
12 of 12 people found the following review helpful
Format:Textbook Binding
The first part of this book that deals with topology is a pedagogical masterpiece. After motivating the key concepts of compactness and continuity in the relatively concrete setting of metric spaces, the book goes on to abstract topological spaces, a beautiful section on compactness including the tychonoff theorem, and an extremely lucid development of the separation axioms and the proof of the urysohn imbedding theorem and the stone-cech compactification. I personally find the chapter on connectedness to be the weak link in this part of the book. Wherever possible, Simmons provides an exhaustive list of examples (especially when introducing the various types of spaces) that aids comprehension. Moreover, some of the central concepts (product topology) and deeper results such as the Stone-Cech compactification are easier to appreciate because the author has a section on topological properties of the relevant function spaces couple of chapters ahead and several exercises along the way. All in all, a highly recommended intro to the subject.
Most Recent Customer Reviews
A must-have
For both undergrads and grad students, Introduction to Topology and Modern Analysis is a must-have. It's very readable and very reader-friendly and gives a great introduction to... Read more
Published 12 months ago by Matt Stevenson
Fantastically clear
I used this book recently for self-study. I found it to be very good despite the fact that it's relatively dated. Read more
Published on January 18, 2009 by Heraclitus Junior
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Published on August 8, 2006 by Athena
Good Classical Introduction to Banach Algebras
This is a fine book, but not quite in the 5-star league. Let me elaborate. The book is divided into three parts: general topology, the theory of Banach and Hilbert spaces, and... Read more
Published on February 19, 2002 by A. Ali
Topology Classic
This book was recommended for our analysis course (final year at Adelaide University). It helped me pass the course but more importantly, gave me an interest in metric spaces and... Read more
Published on July 5, 2001
Superb style!
This book was on a recommended list for a grad course in Analysis (we followed Rudin's book) that I took several years ago. Read more
Published on April 3, 2000 by Kiran Garimella
The best
This is the best book on analysis and point-set topology I know of. It is a model of mathematical exposition: completely rigorous but always letting the reader in on the thought... Read more
Published on March 26, 2000
Wonderful writing on wonderful topics.
I used this "text book" in a course in Topology at Swarthmore College in 1964. The writing is so good that I still enjoy picking it up - surprisingly often - and reading... Read more
Published on December 7, 1998 by peter_heisen@mercer.com
damn good!
The best book at this level (junior/senior). Topics have been chosen with taste, and treated with clarity. Read more
Published on March 15, 1998
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