He's right as far as it would be nice if incoming math students were given a sheet saying, "by the way, here is the Greek alphabet with the names of the letters. We'll be using them a lot to stand for various things, so be prepared to seem them here and there". Using actual lecture time on them, though, is probably overkill.
@tb: I read both already. I was hoping that Riesz-Markov can be tweaked to work for c_0(N). I need a way to write a thing in (c_0(N))* as <., g > for some g in l^1.
@Matt Also the measure version for C_0(X) gives exactly what you're looking for. Just think about what a finite measure on N really is: it assigns some weight to each point and this must be summable
@AsafKaragila If W is a subspace of V then W^\ast is a quotient of V^\ast
@AsafKaragila Every Banach space has algebraic dimension \geq continuum. I don't think you can give an algebraic basis explicitly for any infinite-dimensional Banach space. This looks much more intricate than giving a Hamel basis for the reals.
I am starting to feel how much doing non set theoretic math without the axiom of choice is the same as hooking your gonads to a car battery just to make walking around the street harder.
A propos of that: If we're in a world without choice and have a vector space without a basis, I'm not sure the algebraic dual necessarily has any nonzero elements.
@AsafKaragila Can you visualize a Hilbert sphere contracting to a point within itself?
Expanding on before: The Hilbert sphere is the subset of ordinary countable-dimension Hilbert space consisting of vectors with norm 1, and the subset topology. Unlike finite-dimensional spheres, the Hilbert sphere is a contractible space!
@HenningMakholm I haven't played with this space before. I probably can't think about it because it is only a syntactic object for me right now. Objects I work with become semantical objects and get interpreted...
@HenningMakholm Quite >:-] (You also missed the comma there)
Brief point of order: Now that the artist formerly known as rob has left the building, do you guys think it is counterproductive (and/or rude) for me to reply to him as I did?
Also, amusingly, my algebraic geometry lecturer started talking about sheaves today (at last! with 5 lectures to go...) and then suddenly I heard, "sheaves are made of more basic objects called stacks" (or something like that) and I was shocked
then I realised he was actually talking about stalks
This question is to everybody who can answer: what is the simple intuition and simple idea of those three subject (algebraic geometry, algebraic topology, differential algebra), what are those use for, do they have any connection? Do you have to learn the topoloy in order to get to or learn those subject? What is its application?
@HenningMakholm I don't think that it was rude (I found the various trolling allusions by others quite rude). You are entitled to say that you're not particularly interested in division by zero and the other topics he brought up.
@Victor No, we're quite far away from each other. But Jonas mentioned earlier that he was drinking Whisky (I foresee a major headache tomorrow...)
@tb I don't doubt my entitlement as much as I wonder whether there's any point in trying to explain that the way he comes across makes people not want to talk to him. (Which is hard: each time I attempt to formulate a rule of thumb, it ends up looking like a double standard that we regulars get to shoot unmotivated mathematical musings at each other but he doesn't).
@Victor That's simply not true. If the student has a problem with homework they may ask (that's what office hours are for) and we may (and should!) also help them with their theses. As long as the main contents of the thesis is the student's own work there is absolutely no problem with that, on the contrary!
@HenningMakholm I understand the uneasy feeling completely. It is very hard to pin down the distinction. Why don't we consider division by zero interesting but we may chatter on about bases of vector spaces at the same time? If we're honest, we don't really know what it is that makes a mathematician either. But we know them when we see them...
It's quite true really. If we hadn't arithmetised our geometry we probably would never have dreamt up those monstrous things like non-measurable sets or continuous-nowhere-differentiable functions or whatever.
So, what Lagavulin do you drink? I decided to go for the 1981 edition which a friend brought me some years ago. Deliciously smooth, much less scratchy than the ordinary 16yo one.
@robjohn It seems like the kind of thing that should be more well-known. Maybe everybody is just blinded by the inclusion-exclusion approach. :) Ever thought of writing it up? There are a few teaching and expository-oriented journals that will publish really short pieces. For instance, it might work as a "Classroom Capsule" in The College Mathematics Journal.
@robjohn I really think you should write it up, and I think "Classroom Capsules" is the right kind of place. The guidelines say, "Classroom Capsules consists primarily of short notes (1-3 pages) that convey new mathematical ideas and effective teaching strategies for college mathematics instruction." Recent articles include titles like "An intuitive proof of the singular value decomposition of a matrix" and "Derivative sign patterns."
A short piece explaining that the D_n formula is normally proved via inclusion-exclusion, giving a combinatorial proof of the recurrence, and then getting the formula from the recurrence would fit right in.
