Introduction to topology and modern analysis
by George F. Simmons · first published 1963
100% recommend
Why HN recommends it
Gentle, well-motivated introduction to metric spaces and topology that builds intuition from analysis before abstraction; a standard reference for undergraduates and graduate students preparing for analysis-heavy work.
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…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…
on “Ask HN: Technical Summer Reading List” ↗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.
on “Mathematics for the Adventurous Self-Learner” ↗…and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York, 1966. ---------------------------------------- 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. ======================================== Ordinary Differential Equations ---------------------------------------- Linear algebra is important, as some points crucial, for…
on “Linear algebra tutorial in four pages” ↗…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…
on “The best linear algebra books” ↗…pure math major. If somewhere take a course in abstract algebra, e.g., with a little group theory, then that might help! Graduate School. Might learn a little more about topology, say, from Simmons, 'Introduction to Topology and Modern Analysis'. So, get good with metric spaces and get started on duality. The next big step is a course in measure theory and functional analysis. The Simmons work will help. Baby Rudin will be crucial; Halmos…
on “Ask HN: I want to learn statistics and data mining” ↗…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
on “Ask HN: What books do you keep on your desk?” ↗…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…
on “Topology Without Tears” ↗…proofs. So, I'd pass it up and return later when know enough real analysis to guess or understand most of the proofs easily. Note: In (the relatively elegant and easy to read) George F.\ Simmons, {\it Introduction to Topology and Modern Analysis,\/} McGraw Hill, New York, 1963.\ \ the flat statement IIRC "The two pillars of analysis are linearity and continuity." Well, for the linearity, really need a good background in linear algebra. For this,…
on “Ask HN: Resources to learn real analysis?” ↗"Sometimes"? Yup! "Tedious"? Yup! Can use Dedekind cuts or maybe something called the normal completion, and especially the first is darned tedious! There is a good math writer G. F. Simmons, of Introduction to Topology and Modern Analysis, who stated that the two pillars of analysis were linearity and continuity -- nice remark. He also stated that really to understand, have to chew on all the arguments, etc. or some such. Then I decided…
on “Why Mathematics is Boring (2007) [pdf]” ↗