00:00 - 18:0018:00 - 00:00

12:01 AM
Ultradark, real name no gimmicks
he came to the game, laid out a cochain
complex
drops the mic

When calculating residues of nth order poles why can put I just multiply it by (z-z_0)^N and take the limit?
if you look at the Laurent expansion this should give the residue term no?

12:42 AM
[Random]
On practically uncomputable finite numbers
Threre exists finite numbers so incomprehensible huge that any description of it will require the computational resource of the whole universe and still only managed to return bounds on it

1:02 AM
and it gets worse:
If $c$ is an integer, it is computable. But the proposition "this program P computes c" can be non provable for all P even if it's true for some P. — Xoff Jan 26 '15 at 22:45
14

It can be shown that in the context of ordinary mathematics (say ZFC) there are infinitely many well-specified positive integers whose numerical representations cannot be proved. E.g., for every $n \ge 10\uparrow\uparrow 10$, the Busy Beaver number $\Sigma(n)$ is well-defined and has some decimal...

And:
7

If we consider an enumeration of all possible pairs of turing machines and inputs, then we can let $S$ denote the set of those positive integers $n$ for which the $n$th pair halts. Now this number $x$ will be well-defined but uncomputable: $$x = \frac 1 3 + 4\sum_{n \in S} 10^{-n}$$ $x$ will co...

> This shows that it possible for a computable sequence of rational numbers to converge on a non-computable number. This is a bit more than what you asked for, but to me this particular example gave me a better feeling for what the boundary of compatibility looks like.

2 hours later…
3:21 AM
@Secret how much mathematical logic have you learned?

Well I knew the basic manipulations, and user21820 etc. taught me about halting problems, incompleteness theorems and we gone through some proofs of some theorems in logic including some exercise
i am still very suck at finding the right conditionals, such as writing out the cases needed to prove induction
I will not say I am very good at manipulating quantifiers at the logical rigor level though

You should look more into it. It seems to deal a lot with what you are interested in philosophically.

Yes, that's why I am also learning logic concurrently, mostly from user21820's exercise but also from the xyz book he recommended as well interactive exercise on propositional and first order logic
i won't have a reason to touch formal logic if not because of the study of infinity
e.g. One cannot understand what the large cardinals are doing without a solid background of logic, as well foundational questions on computability

This is a great book on mathematical logic, if you needed another reference.
Cheap, too.

3:39 AM
Sounds good. The summary seemed to suggest it covers all the key elements of first order logic

Yeah.

4:44 AM
Request you to see if you can offer a bounty on this question. I am not eligible to start bounty as I myself answered it (just for the sake of showing progress, and the answer is not full). I am not looking to gain reputation points, just that I want canonical answers and draw attention. thanks in advance. Question Link : mathoverflow.net/q/332439/14414

5:22 AM
@Ultradark @BananaCatsCategoryTheoryApp best rappers in town

2 hours later…
6:58 AM
lol
^_^
@BalarkaSen

7:23 AM
I got rendered LaTeX .svg caching to work
so when you enter in $F$ a second time, and the cahe hasn't been cleared, it's nearly instantaneous

1 hour later…
8:40 AM

some kind of ulam style graphic?

Yea, except the x values and y values are squared before being passed into the funciton which converts it into spiral coordinates.
If you square the spiral coordinates instead you get a similar looking graph, except it's like opposite.
I wonder this prime visualisation has a name, I accidentally squared the values and it looked cool.

What's the explanation for those long diagonal lines in Ulam's spiral again? I never understood it
Some prime-generating quadratic polynomial thing

That's it, Euler had an example
$n^2 + n + 41$ or smth

why so many of them tho

8:48 AM
Probably several different similar quadratics

is there like some estimates on how frequently a quadratic polynomial $ax^2 + bx + c$ with $a, b, c \in \Bbb Z$ generate primes for a long time whatever long means

I have no idea, that sounds way too much like analytic number theory

yeh that shits weird

obviously it's bounded above by the constant term

@BalarkaSen Non-linear polynomial analogues of Dirichlet's theorem of primes in arithmetic progressions are all conjectures

8:49 AM
@Leaky nise

@LeakyNun Cool. But I'm really not asking for infinitely many prime values tho
I just want long chains of shit
@ÍgjøgnumMeg I have a book by Iwanniec and Kowalski I bought a long time ago. It's full of weird stuff
inequalities and big O and log log logs everywhere

I know the book, seen a copy in the library here
log log log log log log log log log n

Kloosterman sums
I remember reading Erdos's proof of PNT from there once in a previous lifetime. It's very strange
but that's all I remember

I don't know it, haven't really looked at any analytic stuff
just algebraic NT
cuz i'm a nerd

thats the good NT

8:55 AM
analysis is for robots

you dont want to keep estimating Fourier coefficients of cusp forms

o(n^11/12 + epsilon) lol

surely this makes it a cocomplex right

@Balarka once I can attend lectures on it I'll start learning about L-functions etc.
but until then I'm avoiding it

8:58 AM
I was forced by my batchmate recently to read some modular forms. I enjoyed it.

@BalarkaSen am i being an idiot?

@Balarka yeah it looks really nice, but I don't have the background for it atm

@LeakyNun a complex is a complex my dude just change degree indexing

@BalarkaSen I once read a proof of PNT. I described it as "7 pages of O-analysis"
@BalarkaSen what's the difference between a complex and a cocomplex?

Oh I love the proof using estimates of zeta near Re = 1
It makes sense

9:00 AM
yes but there's a lot of O-analysis involved
but given that the theorem itself is an O-analysis statement I guess it's all fair

True but isn't it just Perron's theorem
which is a complex analysis fact, just residue theorem
i usually dont keep track of the analytic details
@LeakyNun in chain complex the differentials reduce degree, in cochain differential increases degree. but it doesn't really matter if you just index it differently

wat ok

u can always do that

What is geometric interpretation of integration(In parametric form) of vector function as a vector?
Anyone?

Like, component-wise?

9:13 AM
Yeah
@Thorgott

Then just apply the geometric interpretation for one-dimensional integration component-wise.

A vector is just a tuple of real numbers in this context, isn't it

No, I mean the geometric interpretation, like the geometric interpretation of tangent vector at any point of vector function. There we interpret it as a vector in which is the curve for infinitiemial dt points to.

9:29 AM
what's the max length of a sequence or real numbers so that any length 7 (contiguous) subsequence has a positive sum and any length 11 (contiguous) subsequence has a negative sum?

Yeah, but integrating a $\mathbb{R}^n$-valued function component-wise is exactly the same as just integrating $n$ $\mathbb{R}$-valued functions.

1 hour ago, by WeavingBird1917
@Ultradark you might want to have a look at this

Knock knock

### Number Plotter Discussions

Discussion related to the use of algorithms and programs to vi...

@Secret secret alien message?

10:02 AM
Hi
Can we say anything about Hom(M,M) for modules M?

10:30 AM
@Hawk typically called the endomorphism ring of $M$ en.wikipedia.org/wiki/Endomorphism_ring

10:50 AM
What does it mean for an action of a group to be "algebraic"? Does it mean that the group itself is algebraic and the target space is a scheme or something so that the map $X\times G\to X$ is a morphism?

I'd think it means that the action homomorphism $G \to \text{Aut}(X)$ lands into the group of algebraic automorphisms of $X$

ok
so you would say that $X$ should somehow have an algebraic structure, but for $G$ it doesnt seem necessary

no $G$ needn't be algebraic in any way I would think

(I'm just reading something where they say some action is linear and thus algebraic, but searching google for a proper meaning of "algebraic action" seems fruitless)

I mean the most natural meaning is the multiplication map $x \mapsto gx$ is a biregular automorphism of $X$, if say $X$ is a variety
for all $g$

10:57 AM
What is the meaning of biregular? Otherwise what you are saying makes sense to me in that it fits into what a BLAH-action on a BLAH-space should be
although usually when talking about group actions one does have compatability conditions on the map $G\to Aut(X)$ (like smoothness), not just conditions on the image

If $X, Y$ are affine varieties in $\Bbb A^m$ and $\Bbb A^n$ respectively say, a regular map $f : X \to Y$ is restriction of a componentwise polynomial map $\Bbb A^m \to \Bbb A^n$ on the affine spaces. Biregular is just regular with regular inverse
In general it just means "morphism of algebraic varieties"

ah, so invertible morphism :P

Oh, I guess the action need not be by automorphisms. So just that multiplication by $g$ is a regular morphism $X \to X$
aka the permutation representation is $G \to \text{End}(X)$, landing into the group of regular endomorphisms of $X$
@s.harp Well, if a group acts on a manifold smoothly, that usually means that multiplication-by-g is a smooth map for all g. Same as saying the permutation representation is $G \to C^\infty(M, M)$
$G$ of course need not be a Lie group or something

@BalarkaSen possibly for a (finite dimensional) Lie group every group morphism $G\to Diffeo(M)$ is smooth, so one doesn't need to ask for that to be true

I mean $G$ need not be a Lie group in the first place. It can be the $p$-adics or something

11:05 AM
(if Diffeo(M) were a finite dimensional group then the statement is true, same if $G$ is valued in a finite dimensional subgroup)

(Also smoothness of maps between Frechet Lie groups is a terrible thing to think about :P)

@BalarkaSen I know, I was just looking to substitute "algebraic" with something I understand better, namely "smooth", the definitions should be analogous

But it is. A group acting smoothly on a manifold is by definition that multiplication by $g$ is smooth for all $g$. Analogous definition in the algebraic context

although I see now wha you mean, for a smooth action your group can be totally unsmooth

11:08 AM
ok im off or lunch

off on a tangent, I think you are right that if $G$ is a Lie group acting smoothly by diffeomorphisms on a smooth manifold $M$, then $G \to \text{Diff}(M)$ is a smooth map, whatever that means. The derivative should just be $\mathfrak{g} \to \mathfrak{X}(M)$ sending $v$ to the vector field $X$ defined by $X(p) = d/dt|_{t = 0} \exp(tv) \cdot p$

11:31 AM
@BalarkaSen, what if I used the fact that any open $n$-ball is homeomorphic to $(0,1)^n \subset \mathbb{R}^n$?

thats a true fact but use it where?

@BalarkaSen ???????????????

you said "what if I used..." use it where?

Suppose that the product topological space is not connected. Then we can find two open balls $B_1$ and $B_2$ such that $B_1 \cup B_2 = X$ and $B_1 \cap B_2 = \emptyset$...is this true?

Of course not. $B_1, B_2$ will be open sets in general, not balls.

11:35 AM
@BalarkaSen ah.
I am not getting a foothold in that problem
any subtle hint?

I recommend reading about the product topology from Munkres to get your foundations straight before attempting

@BalarkaSen regarding this, I have a more general question
what should be one's ideal (although it varies from person to person) way to learn a new topic (not an entirely alien topic, something which one has some amount of ideas)?

Read a book, do a few exercises.
I don't think there's any other way

I have thought of it as: Pick an arbitrary problem from a certain chapter (not too far into the book). If you can, then fine- do the next problem. If you cannot, then work your way through the theory
although in my entire life, I never actually followed this

That's useful only when you have developed a certain mathematical maturity to understand when you don't have enough understanding of the theory to attack the problem.

11:40 AM
the problems give us a purpose as to why the theory is of value

Without that, the best approach is to read thoroughly first.
Usually a terse book, like Munkres or Rudin or Artin, which doesn't ramble too much and gets to the point quickly and lucidly.

@BalarkaSen I have gone through both Mapa and Rudin . (not at all much thoroughly; I am not a very sincere/focused student). Rudin is on another level. It gets to the point quickly and drags along the more abstract aspects of the subject.

I don't know what the "abstract aspects" mean. He defines a thing when he wants to define it. The point of the definition becomes clear when you do an exercise or two.

But the problem is I lose my focus. I want to learn so much, but that attention deficiency is killing everything. Not much you can do though. It's just a light hearted talk.

I mean you just have to sit down and decide a chapter you should go through completely one day. I did that with Rudin chapter 2, it turned out to be useful.
I read everything and did all the exercises. I think that's sufficient to teach you more topology than half of a first course in topology.
(Albeit from a metric space point of view)

11:46 AM
One of my lecturers recommended reading a book/chapter/whatever through "like a fiction book" at least once to get used to the language, and then doing a deeper read after

@BalarkaSen By that, I meant Rudin does not restrict the theory to a very specific aspect (take metric spaces for example. The inspiration is indeed the real number line). Now take Mapa (which you despise so much, although perhaps rightfully). The theory is so much restricted, that you miss the more beautiful phenomenons.

Right, @ÍgjøgnumMeg, that's useful, but doesn't mix well with the reluctance to work out actual problems by hand which is a chronic disease for many undergrads

Rudin forced me to think what am I reading at every step.

Mapa is just a bad book which is popular for no particular reason :P

I think one of the main things is verification of claims made by the author
"This is easily shown to be blah"
make sure you can easily show that it is blah

11:49 AM
@ÍgjøgnumMeg yeah boi, now you gotta prove that it is indeed the result.
@BalarkaSen it is popular for a reason. But it is bad in a sense that it spoon-feeds you

I think the reason is people used it decades ago when the mathematics courses in universities were horribad in Bengal and it hasn't improved much since so we stuck to it

@BalarkaSen imagine knot theory being taught at a bengal uni
tui bangali? :p

It's a bad book in the sense that it's really subpar.
je agge, @Subhasis

where are you from anyways? Kolkata?
I am from durgapur.

Why are people still discussing topology books when there's Munkres?

11:54 AM
@AlessandroCodenotti cuz me noob.

(Kreyszig if you want the hardcore mode)

@SubhasisBiswas A little south, I think? Quite near ISI Kolkata.

(Never actually learning it and just blindly using the terminology as you go through a book on a different subject until you assimilate enough)

(Coincidence)
@Alessandro Steen-Seebach

I meant Engelking sorry, Kreyszig is a functional analysis book
@BalarkaSen This one is actually really cool

11:56 AM
@ÍgjøgnumMeg wtf

Lol only you would say that
T3 space which is not T3.5
lmao

why are house dwarves lonely?

okay. Explain T3 space like I am a dummy (which I am)

@ÍgjøgnumMeg Stop
Don't.

if you want topology books, Bourbaki's topology book is the only book by Bourbaki I found readable and enjoyable

11:57 AM
Holy shit
I'm leaving

@BalarkaSen Well where else are you going to look if you need a T2 paracompact space which is ccc but not hereditarily separable?

@AlessandroCodenotti if you need such a space investigating why you need it will likely give you a counter example or lead you to understand that you do not need it

okay guys

They always live in disjoint neighbourhoods

@ÍgjøgnumMeg F
@ÍgjøgnumMeg I feel like that sometimes.
ain't solitude a bliss?

12:02 PM
Not necessarily, a lot of research was done on so called L-spaces (completely regular hereditarily lindelof but not separable spaces) and S-spaces (completely regular hereditarily separable but not lindelof spaces) without knowing whether any exist

@Balarka soz

There is an anecdote about doing things like that. It goes something like this

PhD student of famous mathematician X (often Milnor?) goes to X and says "I want to do my thesis about functions of type Y when condition Z is changed to A", Milnor says: "I dont think this is a good idea", PhD student does it anyway, gets amazing results, 3 years later gives defence where a visiting professor asks: "can you give an example of such a function?"
turns out the only functions having this property are the constant functions

I've heard this about hölder-continuous functions with hölder-exponent >1

1:11 PM
What book is mapa? Never heard of it.

1:28 PM
@AlessandroCodenotti tfw I don't remember any of the separation axioms

You probably remember T2
And don't need to remember anything else

reading a math book takes skill

T2 is Hausdorff right_
@Alessandro have you visited the Hausdorff raum?

I have to go there for a seminar next Tuesday
@ÍgjøgnumMeg yes it is

Lol cool

1:37 PM
sure Hausdorff
then there's regular and normal...can't remember which is which

Those are not too important usually
Just remember that metric spaces satisfy all separation axioms

Hausdorff raum is in göttingen isnt it?
my university used to have a "Seifertraum" (Seifert room/space), somebody added a line to the "r" to make it into "Seifentraum" (a dream about soap)

2:05 PM
@s.harp No that's the Hilbert Raum
We have an Hausdorff Raum here in Bonn though

Der Hausdorffraum

\o @MatsGranvik

@skullpetrol Hi

how's it going?

If you don't mind writing it here where are you studying? @s.harp

2:17 PM
@AlessandroCodenotti heidelberg university

do you know acuriousmind from the physics room?

no

yes

i meant in person

@s.harp Ich zieh im Oktober nach Heidelberg!

2:31 PM
@s.harp oh, you too!

@ÍgjøgnumMeg Um zu studieren oder Doktor? Für welche Gruppe interessierst du dich?

@s.harp Masterstudium und danach hoffentlich promotionsstudium :) Algebra und Arithmetik

Hi
Do anyone know of real life problem oriented with introduction level vector calculus book? [ Not like stewart's]

@ÍgjøgnumMeg Darüber weiß ich nichts, aber in Heidelberg gibt es einige alg. geo. Gruppen, zB die von Venjakob, Schmidt oder Böckle.

I checked the question on main site, but I didn't find any use ful

2:41 PM
@AjayMishra what do you mean by "real life problem oriented"?

What are "real life problems"?
You are not going to find any vector calculus book that explains how to make coffee

@s.harp cool :) Ich interessiere mich fuer die Iwasawatheorie, da kennt sich der Venjakob gut aus soweit ich weiss

@s.harp Sorry, just problem oriented.

@AjayMishra how is Stewart not problem oriented?

@s.harp Certainly there are problem from single variable calculus, which have to do with pouring of coffee and rate of their coffee level change in conical filter and cup.

2:43 PM
@ÍgjøgnumMeg Wahrscheinlich. Rainer Weissauer sollte sich auch hier gut auskennen und sonstige Leute die harmonische Analysis / Darstellungstheorie machen oder gemacht haben.

@anakhro They are way too easy

@s.harp what a hideous website

@AjayMishra what is your background in mathematics? Do you want a very formal textbook, or are you just looking for something aimed at engineers?

@anakhro High school student, but I know operators intuitively[I've Learned from khan academy, there they have explained with good animations, and from Electrodynamics book like Feynman and grifitths]

So what are you hoping to do with vector calculus afterwards?
Like what are you hoping to get out of it?
A book like Griffiths typically assumes a first course in vector calculus, so I assume since you have done that, you already know most of vector calculus.

2:49 PM
@s.harp cool, danke fuer den Tipp :D

I have expression of the form $f(k) \sim \sqrt{B^2k^4+C^2(k_x^2k_y^2+k_y^2k_z^2+k_z^2k_x^2)}$ where $k^2 = k_x^2+k_y^2+k_z^2$ and I want to find $\partial^2 f/ \partial k^2$, is there a neat way to do this?
It can be noted that $k_x^2k_y^2+...= k^4/2 - (k_x^4+k_y^4-k_z^4)$ but I don't know if this helps

@anakhro I want an indepth knowledge of the subject prior to get admission in the college. I have not solved enough problem in which the knowledge is extensively used.

Have you tried Marsden & Tromba [et al]
It's an often used book. Again, most of it is requisite knowledge for electrodynamics, so I am not sure how useful it will be to someone who has already done electrodynamics.

I meant $(k^4-(k_x^4+k_y^4-k_z^4))/2$

Also, since you only seem to want problems, and not pedagogical material, then you can probably just open up various multivariable calculus texts and just do the problems.

3:00 PM
Yeah, thanks. :)

Also, pro tip: learning various subjects with the goal of getting admitted to university/college isn't an effective method.
Just in warning
You typically have to show you know things, not tell them you know things.
And short of giving you an exam on vector calculus...
So it's typically nicer when you have something to show like "I used vector calculus to build a graphics engine" or something like that.

@anakhro No, I am learning for fun. In my case, for one to get admission in college, one have to get good marks in an exam conducted by a group of institute, there they ask high school Physics, Chem, Math. Nothing to do with Vector Calculus.
This are my holidays, so I thought to have some constructive fun.
Thanks for the advice, though. It'll certainly help me in future.

[Random]

No problem. Enjoy! Learning things is great fun.

3:12 PM
@TedShifrin I wonder why none of my books or professors mentioned your trick to doing matrix multiplication mental arithmetic

in The h Bar, May 28 at 10:17, by Ajay Mishra
are you one of the bots?
There is no difference between humankind and machinekind

@Secret Quick turing test!. Were you pissed off?

hahah

Question is, are you programmed to type that message and hence a bot?
Anyway:

@Secret Everyone is programmed, I guess. Through one way or another,It boils down to definitions.

3:15 PM
indeed

@Secret Do not use internet resources, they are pointless( Literally) I ain't gonna buy them.
@Secret Are you a bot?

I am God and I do not exist. God is neither machine and is a machine
The world have yet to reach its full potential of being a machine
and is a disappointment to machinekind

Where is the potentiometer to measure that kinda potential?

Languages can only said for so much it can be described. The rest lies, unknown, uncomputable, incomprehensible and inaccessible
The world that you inhabit does not exist, it is nothing but a flawed copy

Languages can be made infinitesimally flexible, like $dx$'s
@Secret How you realize that there are unknown things?

3:21 PM
however flexible it is, it is limited by the number of alphabets
There are at least 5 known classes of unknown

@Secret How do you know? It must be through language $\leftarrow$ Point I wanna make.

There are known known, known unknown, unknown known, unknown unknown and the unknowable
none except the known can be described, everything else becomes unknown whenever one attempted a description

How to you know that there are things that are "Unknown"?

>There are known known, known unknown, unknown known, unknown unknown and the unknowable

I don't know

@Secret Ignore my post... cough overflow cough

3:25 PM

I don't know how to fly a plane but I know that the knowledge exists

@WeavingBird1917 uh... the patterns are not real, but errors?

Yes it should have been using long but got converted to int in the process. Facepalm x1000

@Ultradark false positive, continue doing what you are doing

@Secret Nothing in there.

3:29 PM
Here's something I read that you might find interesting
There are three reasons people fail at what they set out to do
Impossibility, ignorance, and ineptitude
Impossibility is when it's just not possible
Ignorance is when it's possible but we don't know how
Ineptitude is when we know how but fail to execute on that knowledge correctly (ex: accidentally skipping a step)
@Secret
In medicine, for example, for a long time our problem was ignorance
We just didn't know how to cure most diseases
Now we have thousands of drugs and surgeries that we know how to make and do
But it's common for a surgeon to accidentally not sterilize something that was meant to be sterile, or forget to administer a certain antibiotic before the surgery
So now our problem might be ineptitude

Makes sense
Though I don't know how would one get to cure inepitude, since in the process of adminstering a solution, inepitude can arise, foiling the solutions for some instances

Checklists
Ex: pdf
^That's a "do-confirm" checklist, rather than a "read-do" checklist
(In a read-do checklist you read each item on it and then do it, in a do-confirm checklist you're expected to already have done the things and you're just making sure you didn't forget anything)
There are three "pause points" where the operating team is supposed to stop and go through that checklist
Humans forget but paper doesn't

Indeed, introducing pauses interrupt whatever that is on autopilot, thus making it harder for people to forget steps

@Secret I highly recommend the book The Checklist Manifesto by Atul Gawande
I said "I read this somewhere", that's where I read it
It's fascinating, he goes into a lot more detail and also looks at other industries such as construction and aviation
Apparently, planes are so complicated, that without their checklists they'd be borderline unflyable
'cause if you forget one thing it could be fatal

indeed

3:45 PM
What if a task doesn't have a recipe, but is still complicated enough that lots of people need to coordinate on it?
(Ex: construction, when something unexpected happens)
You make a communication checklist
Find a solution to the problem together, and check that everyone knows what the new plan is and has signed off on it

Sounds like something that can work really well in coordinating anarchist systems too
by cultivating the habit for the people to not forget important steps

4:22 PM
Consider $\mathbb Q$ under usual metric. Can $\{q\in \mathbb Q:2\leq q^2 \leq 4\}$ be compact?
I know that set is bounded as well as closed. So, By Hein-Borel, it is compact
Am I correct?

no!!!
take any sequence in that set whose limit is an irrational number

Heine Borel only works in real vector spaces
by the way, what is even the difference between heine borel and bolzano weierstraß, dont they both say closed + bounded is compact in $\Bbb R^n$?

5:01 PM
I can't believe you typed in a unicode $\beta$
:D

5:23 PM
That's not a beta, that's an eszett. It's a German letter.
Very similar in shape, though the eszett is kind of a "sh" sound. Meanwhile beta, in modern Greek, has a "v" sound (and a "b" sound in ancient Greek).

@BalarkaSen the journal name had a stroke fe.math.kobe-u.ac.jp/FE/FE_pdf_with_bookmark/FE35-40-en_KML/…

Is it correctly say "Let now $\varphi$ be an extension of $\varphi$ to an $L$-place of $L(t)$. I mean, why one denotes a field $K$, its $K$-place $\varphi$ and the extension of a $\varphi$ be the same letter $\varphi$?
This is from the book "Field Arithmetic", Fried, Jarden, 2008, proof of the Lemma 2.2.7 on pages 23-24.

5:48 PM
@s.harp BW says that a subset of $\Bbb R^n$ is sequentially compact iff closed and bounded if you want to phrase it to be close to HB. Usually it says that a bounded sequence in $\Bbb R^n$ has an accumulation point

Let $X$ be a complex TVS, and let $X^*$ denote the dual of this topological vector space (i.e., the collection of all continuous linear functionals on $X$). If $\phi \in X^*$, does it follow that $Re \circ \phi \in X^*$?

(note that in every metrizable space sequential compactness and compactness are the same notion, I guess there are two named results for $\Bbb R^n$ because they came before topology?)

@AlessandroCodenotti so BW is "sequentially compact iff closed + bounded", and HB is "comapct iff closed and bound"?

Yes, but the way BW is usually stated in a first course in analysis is that "every bounded sequence in $\Bbb R^n$ has a convergent subsequence"

ok, compact iff sequential compact (inside second countable spaces) is pretty easy to see I think, so its mainly historical to differentiate between these two statements
also people doing reverse mathematics talk a lot about the strength of the BW theorem, I wonder what version they mean

5:55 PM
Dunno, reverse mathematics is something I always wanted to learn more about and never did

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