It's like saying "dogs and cats are the most violent animals among all - and I know so because I studied both" whilst not having studied crocodiles and other animals.
@Huy: In response to your question, you need to make them readable and usable by people other than yourself in classes. I've had to be sure to add more examples (which I might myself do in class separate from the text/notes) and clear explanations. And, to me, the most important part of any math text is the exercises. That's what I take the greatest pride in.
@DanielFischer: I can't find anything useful about the more general version of Riesz' representation theorem. In a proof, it is being used for $L^p$ spaces where $p \neq 2$ but it is stated that it does not work for $p = \infty$. Why is that? What conditions are there for the more general theorem?
LOL ... Both classes put together, about a dozen. A number of the business majors bailed out of my probability. I think they were given bad advice by the actuarial guy: Because the math stat course had been full for months, they decided that people wanting the actuarial certificate (indeed, preparing for the first exam) could take probability instead. I think that's a poor decision, as the statistics is crucial ... and the probability class in the math department is much more mature/demanding.
@Huy Which do you mean? The $(L^p)^\ast \cong L^q$ for conjugate exponents $p,q$? That works for $p < \infty$, though it might be that for $p = 1$ you need a $\sigma$-finite measure space, I don't remember exactly.
But you're a math guy, @Huy. These are business majors wanting to be actuaries ... for that statistics is essential, along with the rudiments of probability. But, yes, we teach much more challenging classes in the math department than the stat folks do.
You don't need an inner product to have a reflexive Banach space, @Huy :D
Definitely Griffiths and Harris Algebraic Geometry, the collected works of Chern and Griffiths (but not of Cartan or Weil ... much as I've loved having them on my shelves) ... a few advanced diff geo texts. I'll certainly keep Artin and Dummit and Foote.
@skull: It's first come first serve. @Mike wants dibs on some ... I don't know if he expects me to schlep them to shipping.
@TedShifrin: Do you have any advice on how to think about reflexive spaces? The definition is so abstract to me and I can't really imagine anything if I'm given a reflexive space.
I must admit that I've never thought about how to think of reflexive spaces. I always just considered either the definition, or the useful fact that all bounded subsets are relatively weakly compact in reflexive spaces.
“Professor Eddington, you must be one of the three persons in the world who understand general relativity.” To which Eddington, unruffled, replies, “On the contrary, I am trying to think who the third person is!”
I've got a quick question on something I heard a while back. A friend told me that he heard of finding the $\pi$th derivative of a function. How is that even possible?
First of all, it's only defined for $x>0$, and, no, it's not smooth at $0$, so if you try to glue to get a globally defined function, you're going to have bad news at $0$.
P.S. @hippa: $\sqrt x$ is not differentiable, let alone smooth, at $0$.
@TedShifrin Oh, I asked in chat whether it is important to study infinite-dimensional manifolds in for example Lang's Fundamentals of Differential Geometry where he defines Banach and Hilbert manifolds.
But the proof of the inverse function theorem that I prefer (and have in my text) works fine in Banach spaces. One of many reasons I prefer it to the proofs in Spivak's Calculus on Manifolds and Munkres's Analysis on Manifolds.
One of my friends/former students almost killed his car driving it without the fan working and it just died, overheated. Luckily, he didn't blow the head gasket. But close. Similar issues with computers, I fear ...
@TedShifrin On a related note, is it important to study integrals of Banach-valued functions instead of just real or complex valued ones? Lang uses this approach in his real analysis text.
It's no big deal, @Jasper, once you understand what an integral is. You can add vectors in a vector space just as you can add real numbers. You just don't have inequalities. But, again, I think that's not worth worrying about at the outset.
Though the program's designed so that Glenn's lectures are about 3-4 days behind whatever you're doing on the problem sets, so sometimes you can get a little bored :o
@TedShifrin, Well, the problem sets still give you a lot of guidance and ask clever questions so that you "self-discover." It's really difficult, but you're collaborating with other people, so it's doable.
@Huy Constants are in $W^{k,\infty}$, but not in $W_0^{k,\infty}$, since uniform limits of $C_c$ functions vanish at infinity (where "infinity" may be the boundary of $\Omega$).
Well, @nsanger, it's never been my favorite, but it's common for people in your situation to feel that way. And some go on to make it permanent, but not all :)
Yeah, I don't know if it'll stay my favorite either. I really want to learn some algebraic geometry, and also some algebraic topology. I know next to nothing about them right now, but talking to some people this summer I'd like to eventually
But, I'm thrilled to note that almost every undergrad at UGA who's taken that class from me who's gone on to grad school has ended up in some geometric area. Not all, but most.
@DanielF: I can see you've been an evil influence on me. One of my students just emailed me asking if I could give incite on one of his homework problems. I told him it was a good thing he wasn't an English major insighting a riot. :D
Do Artin instead of Herstein. Much better global understanding and far more interesting.
@TedShifrin, yeah I don't mean to give off the impression I'm freaking out and trying to learn everything before college. It's actually more because my top college choice doesn't really offer it, as far as I know :O
@TedShifrin: We just have one standard curriculum for the first two years. Not a special one for advanced students or a special one for not so advanced students.
A few kids who've taken our Spivak course here while in high school have gone to Chicago and taken that course. One continued to finish a math degree, but is now doing a Ph.D. in philosophy. The other gave up on math and is now doing ecology/biology.
@TedShifrin: I have mixed feelings about non-proof based math courses. Even in high school we proved everything properly, except maybe the fundamental theorem of algebra.
Well, if you're seriously going to work through my book and learn both theory and computation, you'll do fine in that Chicago course. But expect to spend 15-20 hours a week on math. They'll tell you that.
@TedShifrin Ahh, okay, we have mathematica on an institute license here, so I guess I'll try that out. I've been using matplotlib for a lot of stuff, but contour integrals, I'm having a hard time searching for it. :-)
Hatcher (also at Cornell, along with Hubbard) made a big deal when he published his Algebraic TOpology text that Cambridge Press would allow him to continue to freely distribute the .pdf.
