Schaum Real Analysis Pdf

[Yi Pressimone]

Fortopology, I hear that Munkres is standard. However, if you are concerned about prices, Willard's General Topology is a (mostly) excellent Dover book. And for Algebraic, Alan Hatcher has an Algebraic Topology book that is freely available on his website. You could give your students select chapters from these and save them the exorbitant price tag on Munkres.

Not sure if it fits into a course, but for learning to do proofs, Velleman's text on proofs is excellent for those who "math for econ" but little more formal math. I worked through it summer before grad school, and it really helped me do very well at comps.

I'd like some feedback on what math books to buy. I teach our graduate intro to mathematical tools for econ class and while I have taken plenty of math as an undergraduate (Eurobro here, so we are talking semi-serious math), when I took those classes we used mostly esoteric non-English-language sources, often very old lecture notes.

Now I'd like to buy few books to have standard references for myself/students who need help. What would be the books you'd recommend for:

1) Advanced Calculus: i.e. the undergraduate class before real analysis. Some of our weaker grad students need help with this stuff (yes, MRM/LRM faculty).

2) Real Analysis: both intro and more advanced. Cue in the jokes. Is Ruding/Baby-Rudin the way to go?

3) Topology

Any feedback/smart-ass comments will be appreciated.

Calculus

1) Spivak Calculus. Gentle but not uncompromising starting point for someone who hasn't done any abstract math or encountered a rigorous German-style definition-theorem-proof text. Definitely the best bridge to analysis.

2) Spivak Calculus on Manifolds or Munkres Analysis on Manifolds. Spivak is a little terse and less accessible, but either is a better introduction to multivariate or vector calculus than what's contained in the usual engineering-oriented books.

Real Analysis

1) Pugh Real Mathematical Analysis - More talkative than Baby Rudin, but that's a good thing for a reference book for students to consult for a theorem or the intuition behind an idea rather than a book for careful self-study. Even as a book for self-study, this book managers to provide some motivation for students -- mainly with historical background and interesting pictures -- without watering down the material like Abbott or Bartles and Sherbert. It doesn't hurt either that the proofs are fairly accessible unlike Rudin's.

2) Ok Real Analysis with Economic Applications - Although Papa Rudin, Folland, or Royden would be natural choices as the text for a second year analysis course for mathematicians, Ok's book is plainly better-written. Although it lacks measure theory, I don't think that the presentation of measure theory in any of these three books is ideal for econ students.

3) Dudley Real Analysis and Probability - Essentially two books. First half contains measure theory at the level of Royden, but without the frustrating presentation. Second half is the ideal measure-theoretic probability reference for your students.

4) Aliprantis Border Infinite Dimensional Analysis: A Hitchhiker's Guide - Probably the definitive (functional) analysis reference for grad students. You didn't quite ask for this, but it's invaluable.

Note: Someone else suggested Willard's Topology book as a good cheap Dover alternative to Munkres. You may also want to consider Rosenlicht's Analysis text for the same reason (although I think Pugh's book is not much more expensive and is a better reference for students).

In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm fascinated with this subject. I think William Dunham's book 'Journey through Genius ' has launched this endless fascination.

I can't wait another whole year, however, following the regular school-curriculum and not learning anything like the things Dunham describes in his book. Our mathematics-book at school is a very 'calculus-orientated' one, I think. I don't think it's 'boring', but it's not a lot of fun either, compared to the evalutation of $\zeta(2)$, for example. Which is why I took up a 'job' as as a tutor for younger children to help them pass examinations. I wanted to make money (I've gathered about 300 euros so far) to buy some new math-books. I have already decided to buy the book ' Introductory Mathematics: Algebra and Analysis' which should provide me with some knowledge on the basics of Linear Algebra, Algebra, Set Theory and Sequences and Series. But what should I read next? What books should I buy with this amount of money in order to acquire a firm mathematical basis? And in what order? (The money isn't that much of a problem, though, I think my father will provide me with some extra money if I can convince him it's a really good book). Should I buy separate books on Linear Algebra, Algebra and a calculus book, like most university web-pages suggest their future students to buy?

Notice that it's important for me that the books are self-contained, i.e. they should be good self-study books. I don't mind problems in the books, either, as long as the books contain (at least a reasonable portion) of the answers (or a website where I can look some answers up).

I'm not asking for the quickest way to be able to acquire mathematical knowledge at (graduate)-university level, but the best way, as Terence Tao once commented (on his blog): "Mathematics is not a sprint, but a marathon".

Last but not least I'd like to add that I'm especially interested in infinite series. A lot of people have recommended me Hardy's book 'Divergent Series' (because of the questions I ask) but I don't think I posess the necessary prerequisite knowledge to be able to understand its content. I'd like to understand it, however!

I'm not a big fan of full roadmaps and reading lists. Exploring mathematics is something that can be totally different depending on where and who you are. Any serious roadmap needs to be flexible and take account of the course: reading maths is a skill (one you seem well on your way to learning btw! but still...), initially you may find actual teaching easier to grasp- and your reading should work along side that. So here's my attempt at a flexible roadmap:

1) Buy some very carefully chosen books and read them cover to cover: There's a lot of baffling books out there- even some that look really UG friendly can have you weeping by page 5 in your first year- and you only want 4-5 to start with (any more will just be too expensive and you won't get round to reading all of them- top up via the library). My recommendations are: 'Naive set theory- Halmos', 'Finite dimensional vector spaces- Halmos', 'Principles of mathematical analysis- Rudin' and 'Proofs from the book- Aigner and Ziegler'.

[These, ostensibly, cover the exact same material as the book you have decided to buy- but to develop quickly I urge you to buy more mathematical texts like these: Halmos' and Rudin's writing styles are very clear but technical (in a way that the book you are interested in will not be), and will make you a better mathematician faster than any book that tries to 'bridge the gap' ever would. I also seconded Owen's call for proofs from the book: it is simultaneously inspiring and useful as a way of seeing 'advanced' topics in action- it's something you'll keep coming back to, right up to your third year!]

2) Do all of the excercises: Or as much as you can bear to- even if it looks like it's beneath you (if you're half decent- a lot of first year will!) you will be surprised as to how much it helps with your mathematical development (and the crucial high mark you'll need for a good PhD placement). This applies to classes and your 4-5 text books.

3) Ask your tutor about doing some modules from the year above: If you've read all of those textbooks and done all of the excercises, you will be ready. Get some advice from your tutor about what would be best and roll with it (most unis won't make you take the exam, so if you don't feel comfortable you're fine). Taking something like metric spaces or group theory in your first year will put you top of the pile.

4) Keep doing all of these things: Immerse yourself in maths- keep on MO, meet likeminded people and no matter how slow the course seems to be moving, no matter the allure of apathy: keep at it. Advice for later books would be pointless now, but there will be people who can give it to you there and then (use maths forums if you want). Oh, and never rule out an area- you never know where intrigue will come from...

There are lots of good answers here, so I'm not going to add any additional book recommendations. I just want to warn you of one misconception I had when I was in your position. It is best illustrated by example but don't worry if you don't understand all the terms. That's part of the point.

A vector space is mathematical structure defined in terms of another called a field: for example the real numbers are a field and the plane is a vector space. Now, a field is a special case of another structure called a commutative ring. A field is just a commutative ring in which you can do division; the integers are an example of a commutative ring. Now, commutative rings are built out of abelian groups, which are themselves a certain kind of group.

My reaction to seeing such definitions was to assume that the best way to learn about the ones at the top of the hierarchy (e.g. vector spaces) was to develop a solid understanding of those at the bottom (e.g. groups). This seems natural because math is supposed to be a very methodical thing and logically if B is defined in terms of A you might expect you'd want to understand A first.

It turns out that this is for the most part wrong. The reason is that there are all sorts of crazy groups out there, but the abelian ones are some of the simplest and easiest to understand. This type of reasoning can be applied at each level, and when you get all the way up to vector spaces, you get a family of objects which behave very nicely, having eliminated some complicated behavior at each stage.

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