Introduction to Topology and Modern Analysis by George F. Simmons (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Look, this is a preliminary review. I enjoy “pure mathematics” because I have strong mathematical intuition–what is sometimes called “sophistication”–but am far from strong at the technicalities of proof. So I need to search for just the right item, and this is one of the few I’ve found that I can–so far–recommend without reservation. There’s zip about applications here, it’s “all math all the way” but throughout there’s more than the clunky proof-driven boredom that passes for rigor. Everything is so clear. You get a real sense for why a topological space is more general than a metric space. And when Simmons talks about groups, you begin to see why they’re valuable and interesting, not just part of a system. Also it’s what we in the world of high literature call “good wrote”–in other words, Simmons can write a sentence. This is so important in math writing, not for the obvious reason, but because any deviation from clear and exact expression is simply going to mislead the reader! Is it meaningless to remark that the book is well-made? It’s Smyth-sewn in signatures, old-school, so it should prove durable, as durable as the pure mathematics contained within. A pleasure. For me this is beach reading, because it’s so very far away from it all . . .

⭐Complete to introduce into Modern Analysis

⭐A real great intro to topology and analysis!!!

⭐The service overall was very good:i) The item was as described, andii) It was shipped quickly

⭐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.

⭐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 Banach algebras. The first two parts lead, by way of synthesis, to the last part, where some interesting but elementary results are proved about Banach algebras in general and C*-algebras in particular. I might mention, for example, the Spectral theorem for compact self-adjoint operators, the Stone representation theorem, and the Gelfand-Naimark theorem.I can attest from personal experience that the book is well-written; indeed I worked through it chapter by chapter. But today there do exist a plethora of other treatments that can at least rival this text in lucidity, organisation and coverage. For example, for general topology, there is an excellent text by Willard titled ‘General Topology’,as well as Hocking and Young’s old ‘Topology’. Both of these go much further in the realm of point-set topology than Simmons. Similarly there are any number of well-written texts on functional analysis that cover the subject of Banach spaces, Hilbert spaces and self-adjoint operators very clearly. Indeed in some respects I feel the Simmons book was inadequate by itself and needed to be supplemented by a text on linear algebra; self-adjoint operators — and by implication, the Spectral theorem — need to be seen and manipulated in the finite-dimensional version before one examines their infinite-dimensional generalisation. The Simmons book is a bit weak here; one needs to be playing with matrices.These are, however, minor quibbles. The book can be recommended to a junior- or senior-level undergraduate.

⭐This was the low quality Indian version not to be sold outside India. For almost 60£ I expect a copy with high quality pages that I cannot see through. Will not order from Denver Bookz again.

⭐Good

⭐The book is good but not for the beginners. I referred “Topology without tears” along with it. I did basics and examples from “Topology without tears” then started this book. I think having many examples makes learning topology easier, which is where this book becomes a somewhat difficult read.All in all a good book but difficult and not beginner friendly.

⭐This is the most lucid and comprehensive book on the subject I came across. It complements well with Munkres as two books have alternate definitions which help in understanding the subject. I strongly recommend this one.

⭐very good book to all pg level students!it is very useful to cover csir exam in two topics one is functional analysis and the another one is topology!thanks to amazon such the very product with good packaging quality and the time of delivery

Keywords

Free Download Introduction to Topology and Modern Analysis in PDF format
Introduction to Topology and Modern Analysis PDF Free Download
Download Introduction to Topology and Modern Analysis 2003 PDF Free
Introduction to Topology and Modern Analysis 2003 PDF Free Download
Download Introduction to Topology and Modern Analysis PDF
Free Download Ebook Introduction to Topology and Modern Analysis