@DylanMoreland "For Halloween I want to dress up as Arturo Magidin. I don't know much group theory. Any good group theory texts that are advanced. I want to act like a group theorist also."
@BillDubuque The motivation is that some users feel such blunt answers are either too blunt or, worse, represent site policy. Then they flag—sometimes a lot. Making things more clear never hurts.
@MarianoSuárezAlvarez That sounds oh so familiar: recall your prior remark "ok I'll just ignore thi thread" after not wanting to do an easy link resolution - just as above. Are you surprised that I don't take you as sincere after that?
I have ceased to be surprised by you a while ago :)
in any case, I suggest you make a meta post about these multifarious intentions that you perceive in some users, for it may be important to deal with them if they are in any way disruptive
if you plan to study (as opposed to browse) some "real" analysis, pick something covering measure theory, the usual functional spaces, metric spaces and their function spaces
that will provide examples — then pick a book on functional analysis
But my current course on analysis is really putting me off. One minute fractals, then the next minute ODEs, then whatever else next. It's like a salad mix
@BenjaminLim you characterize the thing as jumping around a lot. see it as an opportunity to see what's out there. Abstraction only is part of the whole story.
I think that's why I don't get the point of why learn fractals and solutions to DEs using integral operators when we don't even know what an integral is
@MarianoSuárezAlvarez I am worried that my analysis is not strong
@MarianoSuárezAlvarez Which apostol book has things like a continuous function is integrable on a closed bounded subset of $\Bbb{R}$ because continuous functions on closed bounded sets are uniformly continuous?
my comment about step-functions was designed to persuade you that the set of integrable functions (on some domain) is larger than subset of continuous functions on that domain.
One of the problems with the riemann-darboux integral is that it really doesn't work well with pointwise limits. Even pointwise limits that are very "nice" (such as pointwise monotone limits of non-negative continuous functions) --- as in this question on mine: math.stackexchange.com/questions/102482/…
Again: Riemann introduced the concept of null-sets in his take on integration theory. Admittedly, he didn't call them null sets but rather "small sets" or something like that. No measure is required at all.
@JM thanks for the tip, but I need no help in that department :) But if I ever run the risk of becoming a sane person, I'll remember to try to rationalize the voting patterns.
My department is very strange. I looked at the last 11 years of master's-level courses and there were: 11 years of Category Theory (no surprises)... but only 10 of Differential Geometry, and 4 of Commutative Algebra
I think there were more years with non-commutative algebra!
@anon: I went into a bit more detail than this answer including a reference to here before I read the problem and realized that it was asking about all the $0$s.
@ZhenLin No way, don't worry - I just have to read a paper related to my field which is written by a guy from CS. I see this notation for the first time, and I just seen some examples where this notation is indeed useful, like $\lambda t.P(t,0,x) = (x,x-1,x^2)$
@kahen: that's what I usually do. My comment about convenience raised from the underexpectation I had about this notation (like using $\forall, \exists$ without saying it in words)
I try to stay away from it as well, but it is pretty useful for things like "The group operation $G\times G \owns (x,y) \mapsto xy \in G$ is required to be continuous [...]"
I'd write $((x, y) \mapsto x y) : G \times G \to G$, but that's because I think of $((x, y) \mapsto x y)$ as a function literal of type $G \times G \to G$.
@tb: well, how would you write that if you write an introductory book on groups which covers topological groups? Although you might not want to do that, I still claim that my question is well-posed
@ZhenLin two more iterations and you'll be very close to L. Tolstoy with his W&P
It's a convenient thing to do. Once "internal group" is defined one can give the same definition for Lie groups, commutative Hopf algebras, 2-groups...
