Textbooks similar to Harvard Math 55

[nextstep]

Hello,

I would like to strengthen my math knowledge by going through a series of basic pure math topics. Sort of what Math 55 does at Harvard, which covers linear algebra (Axler) and analysis (Rudin). I have a MSc in Computer Science, so I will not be going through these topics for the first time.

I read many book recommendation threads here and elsewhere, but I still have some doubts concerning how to put all pieces together:

- At this level, shall I treat topology as a separate subject? If so, what book would you recommend?
- Would you rather use Halmos instead of Axler for linear algebra?
- Would you consider something else than Rudin? I'm quite fond of Hubbard, but it covers a different range of topics.

Thanks

 

[verty]

I'll try to answer your questions based on what I have observed about those subjects.


1) Topology as a separate subject, I wouldn't. To me, topology seems to be a spin-off of the geometrical investigations of the 1800's. There are alternative geometries, hyperbolic and all that but also inversive. Inversive geometry treats lines and circles in the same way because an inversive transformation transforms lines into circles (and vice versa). So there is a kind of forgetting that lines and circles are distinct; if we pretend they are the same, what insights can be found or what savings can be made?

And topology seems to be a similar thing, we forget that a cup is not like a donut, they both have a hole and one continuous body, let's pretend they are the same. Now what insights can be had, etc. So essentially one is taking about really esoteric things that are far removed from any immediate purpose.

And for example, what worries me is reading that knots are related to topology. Knots were obviously used throughout history by sailors, and renowned sailors like William Parry, probably the most respected English sailor in history, set out on incredibly dangerous journeys for years at a time depending on their knowledge and use of knots. And obviously that was long before there were any topologists. So what could topology possibly add of value to that subject?

Of course I realize that one never knows when something in pure math will become useful. Cryptography is the most well-known example, that number theory has become very useful indeed, even critically important for banking and all that. And clearly there has been a use of manifolds for general relativity and cosmology. But I'm of the opinion that the pure mathematicians should deal with the pure math while the rest of us, until that math becomes applicable, should leave it up to them.

For these reasons, I wouldn't learn topology as a separate subject. There is also the reason that I often see given on this site, that topology is complete, half a century old and stagnant. So surely it is better learn it as you need it in other subjects.


2) This https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20 book looks interesting to me. I always like advanced books that are clear, they usually work well for a survey because they will say things like "We take for granted that ... (*), now ..." where the starred content forms a very nice survey.

But I think that is too advanced because you have chosen two undergrad books. Axler I know, it is very dry indeed. If you try to prove all the theorems and do whatever research you need to achieve that, you will get a lot out of it. And many people do like it. But it's black text on a white page unrelentingly, with little variation.

The Halmos book ("Finite-Dimensional Vector Spaces") I don't know but it looks well written, and I know that Halmos will be very clear with his definitions, so probably I would choose that one without much hesitation, it will be good and has many glowing recommendations.


3) Would I choose something other than Rudin? Yes, absolutely and without reservation. Rudin is a wasted exercise. I think he was trying to write a Bourbaki-style hyperelegant book, something for really bright students, but I think he failed in his aim. What he ended up creating was math written on the surface of Swiss cheese, where the holes obscure the meaning.

The limited experience I had looking at his Principles book was the following: looking at a proof, deciding to study it in detail, realizing that there were definitions unknown, looking back to find the definitions, seeing that they relied on other definitions, looking back to find those other definitions and finding that they were taken for granted. It's just too patchy to be anywhere close to usable for self-study.

What book though, this one looks nice to me: https://www.amazon.com/dp/0387974377/?tag=pfamazon01-20. It includes metric spaces but not immediately, as well as some n-variable content and Fourier series. If that is too slow for you, I don't know what to suggest. If I think of something, I may reply at some later point.

 

[jbunniii]

[edit] Sorry, I didn't notice this sentence in your post:

I have a MSc in Computer Science, so I will not be going through these topics for the first time.

In that case, you are probably well prepared to read Axler and Rudin, and whether these are the best choice depends on your taste.

I like Axler quite a bit, but he omits a fair number of important topics. A good alternative would be Lang's Linear Algebra.

For analysis, maybe consider one of these books instead of Rudin:

* Elementary Real Analysis, Bruckner/Thomson
* Elements of Real Analysis, Bartle
* Real Analysis, Carothers

I think all three of the above are much better motivated and just as rigorous as Rudin.

You might also consider Pugh's Real Mathematical Analysis. I don't like that book's exposition as much as the others (a bit too sloppy for my taste), but it has an amazing collection of fairly challenging exercises.

 

[nextstep]

Thanks for the extremely elaborate replies. I'd need some time to digest all the information.

I went through 1/3 of Axler last Summer without much trouble. I think it's a great book. I like what I've seen in Halmos too, so that front is well covered. A minor complaint about Halmos is the horrible 1940s typesetting it has compared to how neat Axler looks thanks to TeX. Someone should edit it, it's a shame.

With regards to analysis, I've heard good things about 2 books not mentioned so far, apart from some you've mentioned: Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach; and Analysis by Zorich. The former is covering an incredible amount of material, but perhaps orthogonal to using Axler or Halmos. The latter was recommended by V. Arnold to replace Rudin, which he dismissed as Bourbakian propaganda.

 

[verty]

Zorich looks decent but it is two books instead of one. Whereas for example, the book I recommended has most of that content in one book (nothing on forms though). But I suppose, if one was going to choose a second book on differential forms, using Zorich's two books would have the same effect.

Something to consider is that there are some very cheap books on differential forms, for example:

https://www.amazon.com/dp/0817637079/?tag=pfamazon01-20
https://www.amazon.com/dp/0521468000/?tag=pfamazon01-20

So how you get it is up to you.

-- but the Zorich books look very good, I should have made that clear.

 

[mathwonk]

The original text for math 55 harvard is available here free:

http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf

 

[nextstep]

Thanks mathwonk. Do you have any other book suggestions? I'm asking cause I've seen many wise book recommendations by you in the forum.

 

[kostas230]

I would suggest Apostol's Mathematical Analysis book for analysis or Kolmogorov's Introductory Real Analysis. Since the latter is really cheap, I would suggest getting both of them.