One of the most terribly bad things is to do mathematics without being able to see the profound beauty of things. I mean if one only solves things, but don't feel that beauty, it's all in vain. I wouldn't do mathematics without being able to perceive that beauty of the amazing connections.
@TedShifrin I have a question: When $f$ is differentiable at $a$ i.e. $\lim_{h \to 0} \frac{f(a+h)-f(a)-T(h)}{\|h\|}=0$ where $T$ is a continuous linear application, we have the result that $f$ is also continuous. But does the fact that $T$ is a continuous linear application is crucial?
@Chris'ssis Do you know if I can prove that for all real $x>0$ we have $$ \displaystyle\int_0^\frac{\pi}{2} (\sin t)^x \mathrm{d}t=\frac{\sqrt{\pi}}{x}\frac{\Gamma(\frac{x+1}{2})}{\Gamma(\frac{x}{2})}$$ without beta function?
Can someone give me a hint on how to prove the subadditivity of the limit supremum? I asked a question about it, but I'm kind of frantic and frustrated because I've been staring at it literally for days and gotten nowhere.
It's incredibly disheartening, really. This is the question I asked about it, and I'm sure it's trivial to a lot of people, but I can't even figure out how to start the proof.
Here's something you might find less frustrating, @Michael. If you have two continuous functions on $[a,b]$, $f$ and $g$, can you prove $\max (f+g) \le \max f + \max g$?
@TedShifrin Uhm... I'm not sure of the rigorous definition of continuity yet (I'm working through some books on my own and haven't gotten there yet) and all I can see off the top of my head is that your statement is obviously true intuitively.
@Michael Not commenting on the problem, but: I used to feel that way re: taking forever to understand/prove something (often still do). Do not be disheartened. Overcome it :)
@TedShifrin It's not equal in general because one or both of the functions might be negative, which would "lower the overall max" of the sum, so to speak.
@TedShifrin Right, that makes sense. Part of the problem is that with the limit supremum, I can't just say that the sequence is bounded above by that limit, right? Since the limit supremum is only the upper bound on some tail of the sequence?
@Gato: I certainly know examples of unbounded operators, but I've never thought of a differentiable function in that context. I suppose we could construct an example.
I used the letter $\upsilon$ in my differential geometry class today, and most of the students had no idea it was a letter or how it was pronounced in English.
@Gato Il est probablement en MPX[insert number here]. C'est dur de dire sand recul si le niveau est très élevé, mais enn tt cas il est de toute évidence plus élevé que celui de ma prépa de sup (à Anthony)
"Note that while the sign depends on the orientation of $S^3$, it does not depend on the orientation of $S^1$. (Proof: Unscrew a nut from a bolt, then flip it over and screw it back on.)"
@TedShifrin Ah my professor talks about this (derivative polynomial), but I don't know how can we prove that the derivative operator is non-continuous..
@Mike: I assigned my diff geo students a problem about spirals $r(t)(\cos t,\sin t)$. And the point was to prove that they have finite length ($0\le r\le 1$) if and only if $r$ and $|r'|$ were both integrable on $[0,\infty)$. One student left off the absolute value. Cool question to give a counterexample.
@TedShifrin That's a complicate question actually :D There are indeed good maths universities in Italy, but there are better ones in germany! Also I'm interested particularly in set theory and logic, which aren't really considered a lot in Italy.
Still, @Alessandro, I would say that you should learn a lot of mathematics in college/university and then specialize as you go to get your graduate degree.
I decided to specialize in one topic after a few years of undergrad; in grad school I'm doing something quite different. Whence I agree with @TedShifrin's judgement above.
@TedShifrin: that's true, but I also had other reasons to move to germany, I know people who live here and so on, so far I think it wasn't a bad choice :D
@evinda yes, but I think it's particularly easy with chinese (or other non alphabetic languages) where most of the study it's strictly mnemonic (is that a word?)... I mean, the grammar is very easy, but you must know everything by hearth
@TedShifrin I didn't get it as a critique :) well, I have to study a variety of topics anyway in the university before i get to specialize, who knows, I might chance my mind by then, abstract algebra is also very interesting :p
Another question before going to sleep, if $F,G$ two complementary subspaces in $E$ a normed vector space. Let $A$ be a linear map of $E$ onto $\Bbb{R}$. If $A$ restrained to $F$ is continuous and $A$ restrained to $G$ is continuous. Does $A$ is continuous?
I'm a big fan of Bott/Tu, but the poor guy is confused even about partitions of unity. On the other hand, he was thinking in a non-standard way which led to confusion, and I can't completely resolve his confusion :P
It's a good book, but I agree with the warning you gave me when I first started looking at it: one first needs to have a firm understanding of basic smooth manifold theory.
