@Srivatsan: I agree, nicely formulated precise and to the point. I've never heard of the Kumon method, though, and I'm a bit suspicious if I would send my 5-year old to such a program.
and yes :)
Still not that old that I can't do that and I don't have many obligations today...
I know you didn't mean it literally when you said about 5-year old, but it still makes me curious. (Hopefully it's not inappropriate.) Do you have a kid? :)
I thought about offering one, but then decided against it. I haven't seen many book recommendation threads and suggestions that I found convincing and helpful, but maybe this would be a different case?
In particular, jokes about the axiom of choice, forcing, set theory, and never ever ever algebraic geometry (unless it is a general chat about mathematics in which I claim that I do not like alg. geo.)
Let $X$ be complete separable metric space and $A\subset X$ is open. Does it mean that there is a compact subset of $A$? My solution is the following: since $A$ is open there is $B(x,r)\subset A$, then $\overline{B(x,r/2)}\subset A$ where the closed ball is compact.
Well, yeah. The days are fine. My nose is a bit runny, and my throat is slightly sore. However the question is how am I going to feel later this evening.
Well, one for getting back to the original construction, two for showing a whole other thing cannot happen, three for making it even shorter, and two more because it means I can correct the mistake in the proof I wrote as well.
@JM Just wanted to say thanks again for your help on that problem. I don't know if you saw my edit, but it seems that Apostol probably meant the standard form only all along.
@pro: I did see, I take the time to read backlogs (as long as we're still not on SO-level activity). i have to wonder why Apostol wasn't more explicit there...
Only one course was a literal translation, that was measure theory. He just read Folland's book and took the exercises (including midterm and finals) from the book.
The only course I have heard given by my department in which such book is used is some low level course in linear algebra. They use a free .pdf book written by one of the professors, and even then I think that the exercises are given separately.
although now that I've been self-studying I've found a lot of value in learning directly from the book, whereas for most courses I never had a need to read the book
I took Fourier analysis with professor A, and the course was dead easy. One year before (and one year after) it was given by professor I which made it extremely hard.
@JM I used to take notes in class, but just never felt like I needed to read the book. I did do the exercises, but that was about it. I did fairly well with this approach.
Of course now that I'm really reading Apostol, it's clear that none of the courses covered the full depth of the material
@AsafKaragila That's a common situation here as well, even with the books.
Yes, just reading the contents it's clear that it's basically all the math I ever learned in college - they could just have all Math students buy and study from this one book
My plan is to finish Apostol's Vol I and Vol II, and then I'm not sure what I would like to specialize in. What would you consider your area of specialty?
Actually, I'd recommend seeing a nice math survey like Gowers's book, so you can pick what tickles your fancy. You don't necessarily have to take the same route as I did, as I did bump on dead ends for some time.
I'm pretty interested in Calculus of Variations (and I guess as a generalization, Morse Theory) but it seems that not many professors at universities near me specialize in that.
Okay, I was lying a bit there. You could have fun with minimal surfaces with just soap. :) But the differential geometry should help you understand what's going on.
yes, I think that's probably the next area I'll work on after I get through Apostol's books. I have do Carmo's book, and the library near me has Spivak's 5 volume treatise. Hopefully Apostol will bring me up to the level that I need for Spivak.
Let $X$ be complete separable metric space and $A\subset X$ is open. Does it mean that there is a compact subset of $A$? My solution is the following: since $A$ is open there is $B(x,r)\subset A$, then $\overline{B(x,r/2)}\subset A$ where the closed ball is compact.
