and since when did saying "Mr. X says blablabla" become equivalent to supporting "Mr. X"?

When they invented the internet.

There will always be asshats, in both the Internet and real life.

@tb They are continuous and $X$ is compact but otherwise no hypotheses.

It's a step in a proof that $C(X)$ with the sup norm is complete.

I see, so you assume that $f_n$ is a $\|\cdot\|_\infty$-Cauchy sequence, don't you?

Yes, exactly.

You then have already shown that $f_n$ converges pointwise to a function $f: X \to \mathbb{R}$, now you want to show that this convergence is uniform if I understand correctly.

Yes.

@JM "There will always be asshats, in both the Internet and real life." Without asshats the brains couldn't get rid of their waste products.

The proof is here.

The proof is all around us.

I would do this as follows (a bit less efficiently but more transparently): choose $\varepsilon \gt 0$ and let $N$ be such that for $n,m \geq N$ you have $\|f_n - f_m \|_\infty \leq \varepsilon$. But this means that for all $x$ you have $|f_N (x) - f_m(x)| \leq \varepsilon$ and thus $f(x) \in [f_N(x) - \varepsilon, f_N(x)+\varepsilon]$.

This shows that $\|f_N - f\|_\infty \leq \varepsilon$ and, as $\|f_N - f_n\| \leq \varepsilon$ we also have $$\|f_n - f\| \leq \|f_n - f_N\| + \|f_N - f\| \leq 2\varepsilon$$

Oh that's just the proof that C(X,Y) where X compact Polish and Y Polish is Polish.

This doesn't look like a good idea.

Well...

I guess if he included the exercises themselves it would be alright.

@tb Nice : ) Thank you!

It bothers me that I don't understand the other thing though.

Let me think about it for a moment.

(I'm currently trying to figure out what brunoh is talking about in his errata thread.)

@tb About possible errors in print or otherwise in a certain book. :-)

Possibly the longest answer on the SE network.

@tb An internet source for errata for mathematics textbooks ... sounds like a good idea to me.

Why do you tell me?

And if there were one I'd very much prefer more elaborate assessments than "hint is of no use"

True

a source to find out if there is a misprint in your textbook would be handy

in my opinion

(I'm currently trying to figure out what brunoh is talking about in his errata thread.)

Wow... talk about inferiority complex.

@AsafKaragila it's not quite clear what "bigproblem" is referring to...

I'm really wondering why the suggestion to move to math.SE seems like a deal breaker...

@tb No, but his first reply to Benjamin was way too strong.

@JM I think it's also the fact he said "if you accept choice" and Mr. Bigproblem had no idea what that might mean.

Hooray, now I am sure that I'll hit 20k today!

Time to shower and go to go to the university.

@AsafKaragila Are you sure you answered that question? I understood it to ask for a space in which no singleton is a $G_\delta$. I think $[0,1]^{[0,1]}$ ought to do the trick.

Good morning.

Morning Jonas

Hello Jonas : )

@tb I hate it when the question I answer is not the question that was asked.

@AsafKaragila So do I. Makes me feel infinitely stupid.

I'm trying to be productive. I'll check out The Pomodoro Technique®.

Now I have to trigger the recalc anyway... oh well.

@JonasTeuwen Sounds tasty. With spaghetti?

It is not edible.

@JM: I came up with a nice slogan that you might appreciate

@JM Said like someone with 20K already :-p

Enough sodium hydroxide (dis)solves any problem.

What if my problem is Aurum?

@AsafKaragila If you're not part of the solution, you're part of the precipitate.

@robjohn If you're not part of the solution, get your fingers out of the test tubes.

@JonasTeuwen silentium est aureum

@robjohn :).

@AsafKaragila If the test tubes have NaOH, I'll get my fingers out.

@AsafKaragila that's a rather caustic comment, btw :-)

The test tubes have an even stronger content: Your mother. (now that's a caustic comment for you...)

I have to go now if I want to get to the class in time. Ciao.

@tb Will you hate me if I rewrite this?

@Matt The question is: is that a vacuous question?

@Matt why would I?

@tb Because it would rip your comments out of context. Plus I might produce a mistake in the rewritten version which would make you think I'm a hopeless case.

@robjohn You speak in riddles.

@Matt No, I was simply stating that if you rewrote that question, and tb hated you, you would never know if tb would have hated you anyway.

@Matt In other words, it was a joke, not a riddle.

@robjohn Like I said, it's for possible bounties. ;)

@robjohn You can't know with him anyway because he's nice to anyone. So he might just as well be hating me already : )

I can't stand the look of it though, so I'm going to do something about it.

@Matt Just go ahead, my comments aren't worth gold, I guess. Just tell me which comments I should remove.

For the moment, I'll leave them, though.

@tb I think the comments should stay really, even if ripped out of context.

@Matt such violence!

@robjohn Just as well, you didn't get my central heating joke the other day either : P

@Matt I remember you concluding that we didn't have central heating here.

@Matt The other thing you could do: Just post another answer. You can then accept the new answer and delete the old one. It will be saved and it will only be visible to you and those with more than 10k.

Good idea. : )

@Matt when I mentioned going for firewood.

Afternoon All:)

Hello there.

@FreakEnum afternoon, there.

@FreakEnum Afternoon? What timezone are you in?

GMT 2+

@robjohn: I'm glad about your answer and comment on the distribution thread. For a moment I thought I was going crazy when seeing the other answer.

@tb Ah, the $L^1$ with compact support? yeah, someone else already commented that the previous answer was bunk.

@robjohn Well, but that comment was bunk as well. The last inequality is perfectly justified but it doesn't show what the answerer wanted to show.

@robjohn 2 AM there , right?

Anyways I should have said mawning then :)

@tb Oh, heh. I didn't read the comment carefully because I assumed it said something sensible about the error of the answer. My bad.

@FreakEnum Indeed 2:09

@FreakEnum well, you have a hard time covering everyone here.

@FreakEnum So you might as well just stick to your own timezone unless you are addressing someone in particular.

You guyz have some special plans on Christmas?

am I allowed to chat off-topics here from maths?

@FreakEnum Play around on SE all day.

Bbl.

@Matt another SE addict :)

@FreakEnum Why not? Have at it.

@tb Ah, other than $\partial^\beta$ instead of $\partial^\alpha$, he gets something that should be bounded by a constant growing like a polynomial.

(Anybody here who wants to help me check that I'm not doing something geometrically stupid?)

@FreakEnum people do all the time. If it is about wine or cooking, it seems to be especially tolerated :-)

@JM Ooh, where?

@robjohn lol

@rob: It is being claimed in the comments to this that I have given a bum formula. Could you check if it's actually my fault or not?

@JM On the job...

Any of you here is programmer too? :)

@FreakEnum In my case, no. I just program for fun.

@JM programming using which language?

I remember here someone is computer scientist :)

@FreakEnum Nowadays, I just deal with FORTRAN and C++. If Mathematica counts, then that one too.

@FreakEnum Well, one regular and one (now) semi-regular.

@JM you've good taste of languages (C++, hardest one) !

Not so much "taste" as "it does the job", actually. :)

Hey @Ilya.

I'm a newbie in C++ :)

@JM: hi, how are you? :)

@Ilya Still fine. You're in the Netherlands again?

@JM nice to hear:) yes, I am

So the presentation went smoothly?

@JM The parameterization for the line perpendicular to the line with m=0 through (5,0) would be (5,-u), but the line with m=0 is (5+u,0). I don't know if that is the confusion.

@JM that was ok, I had nice discussions thereafter

@robjohn Hmm, that might be the case indeed. Unfortunately OP is being skimpy with details... :(

@Ilya: nice to have you back. I can feel whole again $\stackrel{\circ\ \circ}{\smile}$.

@robjohn hi, you're too kind )

@Ilya It hurts us when you keep the precious away for so long!

@JM Isn't what the OP wants, given two points, to find one of the two points forming an equilateral triangle with the first two?

@robjohn That was certainly my understanding. Have I misread the OP?

I got pinged, but I can't find the message...

@JM perhaps I need to read your whole post again. The comments are confusing me.

@JonasTeuwen Perhaps it was deleted. that happens to me.

@JonasTeuwen Also, if someone changes the address, then you still see the ping, I think.

@robjohn Well, the first part is basically where I present plug-and-chug formulae. The second part is where I derive a generic point-slope parametrization. The end result there still has to be changed for the application, of course.

@robjohn That must be it :).

@JM I would think the answer would be $\frac{1}{2}\begin{bmatrix}(x_2+x_1)-\sqrt{3}(y_2-y_1)&(y_2+y_1)+\sqrt{3}(x_2-x_1)\end{bmatrix}$

or $\frac{1}{2}\begin{bmatrix}(x_2+x_1)+\sqrt{3}(y_2-y_1)&(y_2+y_1)-\sqrt{3}(x_2-x_1)\end{bmatrix}$

Hi Ilya. Long time no see. How are you?

@robjohn The OP's special case?

@FreakEnum If you think C++ is hard then check out APL : )

@robjohn It seems the OP has withdrawn his objections... :)

Whee!

@JM Well, given two points, $\begin{bmatrix}x_1&y_1\end{bmatrix}$ and $\begin{bmatrix}x_2&y_2\end{bmatrix}$, those are the points of the equilateral triangle.

@JM: I didn't notice until now that you posted two answers.

Ah, yes, that's the simplest form for my answer. I was thinking a bit too generally. Sorry for the slowness.

@robjohn I like multiple methods for questions. ;)

@JM then perhaps I should post the matrix solution :-)

@robjohn Sure, why not? :)

@JM Done :-)

If there's one thing you can do to lose my attention is to mention in the title that your question is very interesting

(even more so if there's a typo)

At least, to me, it's not as egregious an error as "hurry I need it in 10 minutes you guys" ;)

heh

@JM no, in that case I usually vote to close...

Hey @Sri. How'd your AM/GM/HM experiments go?

@JM writing the question... :)

guys, do you remember any sufficient conditions for $\dot x = f(x)$ on $\mathbb R^2$ to have a closed (periodic) orbit?

I think there is a kind of fact that if we are given two bounded simply connected domains $B\subset A$ and if $f|_{\partial A}$ points inside $A$, $f|_{\partial B}$ points outside $B$ then there is a closed orbit in $A\setminus B$

Hey Ilya

this seems to be realted to Poincare-Bendixson Theory, but I forgot the name of the Theorem

@Srivatsan good morning ) what's up?

JM and tb, Want some advice. I probably have showed you this comment earlier: math.stackexchange.com/a/57063/13425. In my new post, I am proposing a general conjecture inspired by this post. I now want to say that I think that the proposed solution is crap. Politely... =)

@Ilya The usual... =)

@Srivatsan you can say that it is crap almost sure ;)

Unfortunately, though the answer claims that the inequality follows easily from Lagrange multipliers, I am unable to obtain a complete solution. Does this sound ok? Too soft? Too harsh?

@Srivatsan "The proposed solution is a bit too sparse on the details for my taste..."

@Srivatsan too soft. you've already written it in the comment

@Ilya I thought so. JM's a bit sharper.

Still polite, but a bit cutting. ;)

@Srivatsan you may be even direct and comment that this method is highly unlikely to yield the solution.

)) Matt was very brief

I wouldn't want to say "unlikely", as rob presented a solution that was entirely equivalent to Lagrange, but a bit more transparent.

@JM I don't want to say taste. I want to emphasise that I cannot yet get any real answer out of the "hint". It's not like I don't like the sparseness; the answer has been unhelpful.

@JM No, he did not. rob stopped mid-calculation...

well said JM

@Srivatsan Ah, I misremember.

As far as I understand, the real trouble is yet to start... =)

@Srivatsan Okay, "...it seems to me that the proposed answer is rather too sparse on the details; I am not entirely sure on how fruitful this line of attack would be."

"It might well be useful, but it is hard to tell with the details that weren't presented."

@Matt Your functional analysis questions are quite... funny!

First you ask stuff about weak convergence and then about the completeness of our friends the bounded functions 8-).

@JM Done. Thanks, JM and @Ilya.

@all: I posted the question, please read and comment.

@JM I had the books brought to my desk! :-). I'll check them out.

@JonasTeuwen Great to hear. :)

The book is full of cute formulas!

@AsafKaragila Look, one for you. Now you just need to get it into the Klein bottle.

@JonasTeuwen Do you already have a favorite?

@JonasTeuwen Being funny wasn't my intention T_T The course makes me do things I wouldn't do otherwise. I don't understand why it's funny though.

@JM Yes. Andrews/Askey/Roy.

Or my favorite formula?

@Matt Well, I couldn't find the correct word, maybe peculiar is more in place.

Yeah, formula.

@JonasTeuwen That is a synonym.

@JM Anything with lots of symbols will do. I'll check out between those.

Is it? It thought it was a synonym to "strange"!

At the very least "peculiar" is somewhat more highbrow than "weird"...

I think peculiar, weird, strange and funny all mean the same thing.

@Matt More or less, yeah.

@JM Selberg's integral is nice!

$n$-fold integration, how cool is that?

Oh yes, Atle Selberg is a gutsy man.

You sure need the stamina to do special functions.

Did Selberg do special functions?

@Srivatsan Well, he has a zeta function named after him, so there's that...

@JM Well, did Riemann do special functions? =)

@JM: I added some bling to my answer, too :-)

+1 for the bling!

@Srivatsan Does a bear.... never mind.

:D

@robjohn Is this something only you and JM understand? =)

@rob: I think that guy who got confused with your answer isn't used to the abuse $x=kx$... ;)

@Srivatsan "Is the Pope Catholic?" is the more polite version... :)

@JM Well, I did put a smiley there, you see...

@JM Ack! I meant to reply to them, and got carried away with making triangles. Thanks for responding. I will see if there is anything I should add.

@Srivatsan Clearly my eyesight is on the fritz... :D

And it's not that obvious. The wikipedia page says analysis and differential geometry. From where do I deduce the special functions bit?

@robjohn On that note, I'm a bit surprised at the number of upvotes my comment to your answer got...

@Srivatsan As JM says, "Is the Pope Catholic?" is one of the polite "Duh!" statements, "Does a bear s**t in the woods?" is another.

@robjohn See my previous comment. I still don't feel the "Duh!" at all. :)

@Srivatsan Well, the theory of Selberg zeta involves geodesics...

By the way, I posted that question, @robjohn.

@JM Perhaps there are a lot of physicists here :-)

Anyway, at least when somebody asks me about Stefan-Boltzmann, I can point them to your answer for some of the details.

@JM "I'm not physicist, you must believe me. Please." The OP seems to be rather terrified by your comment.

@Srivatsan Since Riemann dealt closely with the zeta and Gamma functions, and probably others, I thought that implied that he worked with special functions.

@robjohn I didn't know that bit =)

@Srivatsan That I'm quite puzzled at, also.

@Srivatsan It is the Riemann $\zeta$ function :-) and he did use the formula I proved in my answer. I shouldn't assume that everybody knows that.

@robjohn Of course, I know that it's called Riemann $\zeta$; that is why, I asked what I asked in the first place. =)

Gosh, this is going in circles. Let's just say we drop this =)

@Srivatsan obviously in circles around my head :-D

@Srivatsan I do remember the function could be described as a contour integral... :D

Hey, all, can you tell me if my question needs some trimming? I get carried away, so independent opinion will be nice...

Apart from that typo I ironed out, your question's hunky-dory to me...

Oh, thanks for that.

@Jonas: Apparently I missed "Does anyone know if there exist general methods to improve the speed of convergence of a series? I.e., can we find a different series that converges faster to the same number?"

At least for alternating series, you have the Euler transformation. There's also the Kummer transformation, but that's trickier to apply.

@Srivatsan I've looked at your question, and I'm working on it. Sorry for the delay.

@robjohn No problem, robjohn. Some comment has trickled in right now btw..

@Srivatsan Ah, then I should refresh...

hi

is there a path to accept more than one answer

No there isn't, sorry.

@JM: hi, would you help me a bit?

@Ilya if it's within my abilities. What about?

here in the bottom of page 222 the matrix function $H(t,x)$ is defined

$H(t,x) = D\phi_t(x)$ where $\phi_t(x)$ is a solution of $$ \dot \phi_t = f(\phi_t)$$ with $\phi_0 = x$

the author claims that $$\partial_t H(t,x) = D f(\phi_t(x)) H(t,x) $$

Yeah. I'm trying to review their notation since it's not entirely what I'm used to...

and I cannot understand where is the second multilplier from.

from what I understand, $D\phi_t(x) = (\partial_j \phi_{t,i})_{i,j=1}^n$, so $H_{ij} = \partial_j \phi_{t,i}$

I'm guessing it's the chain rule in a different form.

@JM seems to be so, but I don't understand how did they obtain it

then $\partial_t H_{ij} = \partial_j (\partial_t \phi_{t,i}) = \partial_j f_i$

and so $$\partial_t H(t,x) = D f(\phi_t(x)) $$

I'm having a hard time scanning through, as it seems some pages are missing here for me. $D$ is what sort of derivative?

D is the Jacobian matrix of f, $$Df = \left(\frac{\partial f_i}{\partial x_j}\right)$$

Ah! (Again, not the notation I'm used to. :D )

me too

Okay, so that means $t$ is constant with respect to the Jacobian in the definition of $H$...

(assuming I understood how they obtained $\phi$ from $\mathbf f$...)

yes, the Jacobian means only w.r.t. $\mathbf x$

from what I understand, the book is not too concise (

Hmm, yeah, it doesn't quite square for me... sorry for not being helpful. :( Maybe you should ask on main...

I will, thanks a lot for help, @JM

Nothing to thank for, I think; I know as much as you... :D

One of the comments here calls the answer a magnum opus. I wonder if that will actually come across as rude to anyone. =) Because, a magnum opus is supposed to be the greatest work of a creator, no?

@Srivatsan In a sense, it's a bit of a damper to be told that you've hit the ceiling... :)

@JM I'm sure some clever people must have made some nice puns/quotes out of this. Nothing comes to mind though...

(Like a compliment with a hidden subtle meaning...)

@Srivatsan Hehe, there is a new word they've invented for that: "complisult".

Aw, backhanded compliment -- that's the word I was looking for.

"That's such a nice dress. It does wonders for your figure." Heh

(from urban dictionary)

@Srivatsan Yes, that's the classical term. :)

@JM But new words are nice too. I hadn't come across this "complisult" =)

Hmm. Damn new mortgage rules 8-).

They average your salary over three years first and I only worked two of those. That ends up that I have 1000€ too little 8-). Even though I do have that 1000€...

@JonasTeuwen they paid already?

@Ilya No. Thursday.

but how do you know how much you receive?

@JM Oh, thanks for those improving the speed of convergence information. Do you have a book suggestion where they write this?

@Ilya No, it is about the mortgage to buy a house.

Oh... quid ending a post with [this is meant for the sake of the argument] --- really?

@Jonas: you're buying the house? cool

@tb hi

Well. I'm 1000€ too short in my salary averaged over three years :').

No exceptions.

So I must figure out something else.

@JonasTeuwen I wonder what is the lowest percentage you have to pay for the house

hi Ilya, welcome back to the old world!

@Ilya What do you mean?

The Middle Earth.

@Jonas I mean, the house costs 200k, you pay 40k and take a mortgage for the rest 160k, isn't it like that?

Basically yes.

@tb thanks. I'm in mixed feelings about new and old worlds.

But you need a minimum in your salary.

@JonasTeuwen so I wonder what is the minimum 40k that you have to pay

And to calculate this if you have only a temporary appointment they average your salary over the last three years.

That can be 0.

@JonasTeuwen 0_o

ABN Amro?

No, just a mortgage thingie office.

This eurocrisis sucks 8-). That is the reason of these new rules.

may I ask you about details (not necessary in chat if it's too private for you)? I would better pay 20-30% more for the mortgage than to waste money on the rent