Name: Introduction to topology and modern analysis
ISBN: 9780070856950

What Hacker News says

euler1729Nov 2022

While I admire the intention of this book's author to make topology accessible earlier in someone's mathematical education, I believe that if you are at the intended level of this book, then perhaps topology is not something you should study right now. Studying abstract topological spaces without studying metric spaces or even analysis over Euclidean spaces first is doing injustice to your intuition and provides very less motivation to appreciate the need for topological abstraction. The book "Introduction to topology and modern analysis" by Simmons is perhaps a better introduction with this viewpoint in mind. Usually subjects like combinatorics and algebra also provide the necessary mathematical maturity to handle more abstract subjects like topology. And once someone is mature enough Topology by Munkres is the best introduction.

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thraxilJun 2021

Does math count as "technical?"

I was originally a Physics major and lately I've been on a kick of filling in the mathematics that was used in my Physics classes but that I feel like was never really gone into in much depth.

My current reading list is:

- "Analysis I" (and II) by Terence Tao (I finished the first volume and am now on the second, but I consider them really one book)

- "Understanding Analysis" by Stephen Abbott

- "Topology Through Inquiry" by Starbird and Su

- "Introduction To Topology And Modern Analysis" by George F. Simmons

The Terence Tao books are amazing so far. Extremely readable introduction to Real Analysis. Abbott also came highly recommended and from reading the first couple chapters I can see why as it also seems to be very readable. I don't know if it would be a better introduction than Tao, but it covers mostly the same material and I think having two different perspectives will really help solidify things for me.

Once I finish those, I'll see whether I want to go deeper into Topology or move to Complex Analysis or Differential Geometry.

I also have a copy of Emily Riehl's Category Theory in Context. I've read some Category Theory before and have a basic grasp, but after reading a few pages of her book, I put it aside until I feel like I'm much more well versed in Topology (the content looks amazing and I really want to read it, but it relies on Algegraic Topology more heavily than other Category Theory material I've seen). So I'll see where I'm at after getting through those Topology books.

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graycatFeb 2020

Simmons? WOW! He's a serious mathematician, e.g., as in his

George F. Simmons, Introduction to Topology and Modern Analysis.

I studied it one summer in an NSF program at Vanderbilt. Then I concluded that he is one heck of a good math writer.

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yomritoyjApr 2018

Always within arm's reach at both home and work:

Rudin, Real and Complex Analysis

Halmos, Finite Dimensional Vector Spaces

Billingsley, Probability and Measure

Kelley, General Topology

Lang, Undergraduate Algebra

Simmons, Introduction to Topology and Modern Analysis

Parthasarathy, Probability Measures on Metric Spaces

Whinston, Green and Mas-Colell, Microeconomic Theory

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graycatJul 2016

Herstein is a bit obscure for a first course. Once I was pushed into a course from Herstein. At first it was in group theory below what I'd already done. I asked the prof if he was going to get to group representation theory. He said that that topic was "deep". I told him it was my ugrad honors paper. The course was beneath me, and I saved time and didn't go. For the final exam, he gave me an oral. He asked me something about Galois theory, which is in Herstein and which I learned from there for his exam. His other question was about duality. He wanted me to show that the dual space of the dual space is isomorphic to the original space. That's true and easy to show, mostly just pushing symbols around, and the notation can get a little confusing, but is not true in some cases.

Some point set topology courses consider duals of duals, and when there is equality say that the original space is reflexive. Then they want to define the weak topology and the weak* topology, etc. Can see some of this in, say, G. Simmons, Introduction to Topology and Modern Analysis, super nicely written book although I have never seen any applications for it.

Pushing around duality in linear algebra can get to adjoint transformations. To me it turns out to be a lot of careful notation for something that turns out to be nearly trivial.

Yes, on problems in Herstein, there was one on rings. It took me a while but I got it. I showed it to the prof, and he sent me to another prof who thought that solving that problem was good. He gave me a more difficult related problem, and I made a mess out of it. The problem needed transfinite induction, and I was supposed to understand that but didn't understand it very well. So, I got pissed off: I should have been studying stuff I didn't know, e.g., transfinite induction, instead of wasting time in the Herstein course that had, say, only Galois theory I didn't already know.

At that school same thing happened in a topology course: The book was Kelley, famous. Heck as an ugrad, I'd taken that book and given lectures once a week to a prof and did a stack of the exercises. I did all but the chapter on compactness at which time I dropped everything to finish my honors paper. For me to sit in that topology course was stupid.

I went to the department chair and tried to get into some material I didn't know but wanted to, especially differential geometry for relativity. He told me no.

So, I did my teaching, started violin, met my wife, and left for a good job!

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graycatDec 2013

Two of the biggest 'branches' of mathematics are analysis and algebra. Essentially the first course in 'analysis' is calculus.

As in Simmons (in this list), the two most important pillars of analysis are continuity and linearity.

The first place to learn linearity well is 'linear algebra' -- or, much the same thing, matrix theory or finite dimensional vector spaces.

Then the algebraic aspects can be emphasized and can lead to applications in, say, the theory of error correcting codes.

For the analysis aspects, can continue with Hilbert and Banach spaces and more and also various applications.

Much of the mathematics in applied mathematics, science, engineering, and technology is linear algebra.

Here is a list of books relevant to linear algebra and/or to the course I took from Roger Horn. This list also contains maybe 60% of what I'd worked from before Horn's course that let me do well in that course.

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Linear Algebra

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Horn's course was close to

Roger A. Horn, Charles R. Johnson, 'Matrix Analysis', 0-521-38632-2, Cambridge University Press, 1990.

with also a few topics from

Roger A. Horn, Charles R. Johnson, 'Topics in Matrix Analysis', 0-521-46713-6, Cambridge University Press, 1994.

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The course also used for reference and some topics and exercises

Richard Bellman, 'Introduction to Matrix Analysis: Second Edition', McGraw-Hill, New York, 1970.

This book is just packed with little results; at some point can get the impression that the author went on and on ... writing. Bellman was a very bright guy in mathematics, engineering, and medicine.

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Relatively easy to read and relatively close to applications, and another book Horn's course used as a reference, is

Ben Noble, 'Applied Linear Algebra', Prentice-Hall, Englewood Cliffs, NJ, 1969.

Some edition of this book may be a good place to start for a student interested in applications now.

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About the easiest reading in this list, and my first text on linear algebra, was

D. C. Murdoch, 'Linear Algebra for Undergraduates', John Wiley and Sons, New York, 1957.

This book is awfully old, but what it has are still the basics.

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For my undergraduate honors paper I made some use of, and later read carefully nearly all of,

Evar D. Nering, 'Linear Algebra and Matrix Theory', John Wiley and Sons, New York, 1964.

The main part of this book is a relatively solid start, maybe a bit terse and advanced for a first text. The book also has in the back a collection of advanced topics, some of which might be quite good to know at some point and difficult to get elsewhere.

One of the topics in the back is linear programming, and for that I'd recommend something else, e.g., Chv'atal and/or Bazaraa and Jarvis in this list.

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Likely the crown jewel of books on linear algebra is

Paul R. Halmos, 'Finite-Dimensional Vector Spaces, Second Edition', D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958.

Halmos wrote this in about 1942 when he was an assistant to von Neumann at the Institute for Advanced Study. The book is intended to be a finite dimensional introduction to Hilbert space theory, or how to do linear algebra using mostly only what also works in Hilbert space. It's likely fair to credit von Neumann with Hilbert space.

The book is elegant.

Apparently at one time Harvard's course Math 55, with a colorful description at,

http://www.american.com/archive/2008/march-april-magazine-co...

used this text by Halmos and also, as also in this list, Rudin's 'Principles' and Spivak.

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Long highly regarded as a linear algebra text is

Hoffman and Kunze, 'Linear Algebra, Second Edition', Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

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Numerical Methods

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If want to take numerical computations in linear algebra seriously, then consider the next book or something better if can find it

George E. Forsythe and Cleve B. Moler, 'Computer Solution of Linear Algebraic Systems', Prentice-Hall, Englewood Cliffs, 1967.

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Multivariate Statistics

My main start with multivariate statistics was

N. R. Draper and H. Smith, 'Applied Regression Analysis', John Wiley and Sons, New York, 1968.

Apparently later editions of this book remain of interest.

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A relatively serious book on 'regression analysis' is

C. Radhakrishna Rao, 'Linear Statistical Inference and Its Applications: Second Edition', ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967.

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Three famous, general books on multivariate statistics are

Maurice M. Tatsuoka, 'Multivariate Analysis: Techniques for Educational and Psychological Research', John Wiley and Sons, 1971.

William W. Cooley and Paul R. Lohnes, 'Multivariate Data Analysis', John Wiley and Sons, New York, 1971.

Donald F. Morrison, 'Multivariate Statistical Methods: Second Edition', ISBN 0-07-043186-8, McGraw-Hill, New York, 1976.

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Analysis of Variance

A highly regarded first book on analsis of variance and experimental design is

George W. Snedecor and William G. Cochran, 'Statistical Methods, Sixth Edition', ISBN 0-8138-1560-6, The Iowa State University Press, Ames, Iowa, 1971.

and a famous, more mathematical, book is

Henry Scheff'e, 'Analysis of Variance', John Wiley and Sons, New York, 1967.

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Linear Optimization

A highly polished book on linear programming is

Vav sek Chv'atal, 'Linear Programming', ISBN 0-7167-1587-2, W. H. Freeman, New York, 1983.

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Nicely written and with more emphasis on the important special case of network flows is

Mokhtar S. Bazaraa and John J. Jarvis, 'Linear Programming and Network Flows', ISBN 0-471-06015-1, John Wiley and Sons, New York, 1977.

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A grand applied mathematics dessert buffet, based on Banach space and the Hahn-Banach theorem is

David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969.

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Mathematical Analysis Relevant to Understanding Linearity

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Long the first place a math student gets a fully serious encounter with calculus and closely related topics has been

Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York, 1964.

The first chapters of this book do well as an introduction to metric spaces, and that work applies fully to vector spaces.

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A nice place to get comfortable doing mathematics in several dimensions is

Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts, 1965.

Some of the material here is also good for optimization.

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Another place to get comfortable doing mathematics in several dimensions is

Michael Spivak, {\it Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus,\/} W.\ A.\ Benjamin, New York, 1965.\ \

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The first half, the 'real' half of the next book has polished introductions to Hilbert and Banach spaces which are some of the most important vector spaces

Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York, 1966.

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An elegant introduction to how to get comfortable in metric space is

George F. Simmons, 'Introduction to Topology and Modern Analysis', McGraw Hill, New York, 1963.

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Ordinary Differential Equations

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Linear algebra is important, as some points crucial, for ordinary differential equations, a polished introduction from a world expert is

Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961.

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