The h Bar

0celouvsky

12:01 AM

@heather wait, do you know the binomial theorem yet?

Bernardo Meurer

I'm pretty sure I did something wrong because it LOVES sending things to the down-right end of the board

skill patrol

Now who's talking like an old woman?

dmckee

I see one bug right away. If the board is full unsigned int selected_row = 0, selected_col = 0; will cause it to return an incorrect answer.

heather

@0celouvsky, I don't think so - let me look it up.

dmckee

Initialize them to -1 or some other impossible value, or include a found one flag of some kind.

heather

12:02 AM

oh, it's like Pascal's triangle in equation form

dmckee

Yes, you could insist that a precondition to the routine be that there board is not full, but that is exactly the kind of thinking that lets bugs pop up out of well tested code far into the programs life.

0celouvsky

@heather You need to know what binomial coefficients are

Bernardo Meurer

@dmckee True, fixing, one second

@dmckee Can I initialize them to NULL?

and then verify before doing setCell

dmckee

Seecond, I see what you are doing with if (randRange(0, 100) > (int) (1 / counter * 100)) , but why not just use if (randRange(0,c) == 0)? Or does randRange not actually guarantee correct behavior on randRange(0,0)?

@BernardoMeurer They aren't pointers.

The reason some high-level language can have NUL or unset values on integer variables is that the variable is actually implemented as something akin to a c struct and has some meta data coming along with it.

Bernardo Meurer

@dmckee This is my randrange:

/**
 * randRange Generates random integers in range
 * @param lower Lower bound
 * @param upper Upper bound
 * @return random integer between lower and upper (inclusive)
 */
int randRange(int lower, int upper) {
    return rand() % (++upper - lower) + lower;
}

I guess (0,0) works?

0celouvsky

12:08 AM

@heather well I might as well tell you the algorithm and you can tell me if you understand

heather

okay

Bernardo Meurer

@dmckee Oh yeah!

Now it works :)

0celouvsky

@heather Ok, we take a continuous function $f$ on $[0,1]$. You recall what that means, right?

dmckee

I think I see the main problem. Doesn't if (randRange(0, 100) > (int) (1 / counter * 100)) accept with probability (c-1)/c?

Yashas Samaga

@heather Do you have any plan to write a research paper before (or during) high school?

heather

12:11 AM

@0celouvsky a function that has inputs and outputs between 0 and 1?

Bernardo Meurer

@dmckee Hm

0celouvsky

@heather no, I mean $f:[0,1]\to\Bbb R$ and is continuous

Bernardo Meurer

I guess so

heather

@YashasSamaga that would be cool - I'm working on two projects right now that possibly could lead to one, but I doubt I have the ability to bring them to fruition

0celouvsky

Sorry, 99% of time "function" without qualification means $\to\Bbb R$

dmckee

12:12 AM

And just for background knowledge, using a simple modulus operation as you have done with randRange won't give you exactly the desired probabilities. That's not an issue in this case where you don't need exactitude, but you wouldn't want to do it in a Monte Carlo.

heather

@0celouvsky a function that has inputs between 0 and 1 and outputs in $\mathbb{R}$?

0celouvsky

@heather Yes, but do you remember what continuous is

dmckee

On the other hand most scientific analysis tool-kits provide a well debugged version, so you don't need to implement it yourself.

Yashas Samaga

@heather Just do something whether it is good or bad; the college admission officers get excited when they see applicants with research papers

Bernardo Meurer

@dmckee I know, it won't be evenly distributed

@dmckee Can you remind me on how this works? Why do we change the probability as we go along?

heather

12:14 AM

@0celouvsky a function with no inputs where it is undefined? i.e., none of the outputs are outside $\mathbb{R}$?

0celouvsky

ahhh

what?

heather

like a function f(x) = 1/x is undefined at x=0; f(x) is not continuous

dmckee

@BernardoMeurer This method is used in cases where you don't know in advance how many items you have, but want to select one with uniform probability. The usual case is something like "Randomly and fairly pick one item from a linked list."

You can of course, walk the list counting and throw, then walk again to find the one you wanted.

Bernardo Meurer

But then you have O(2n)

dmckee

Yeah, and it's so inelegant.

So you start by provisionally selecting the first.

Then is there is a second you provisionally select it with probability 1/2.

0celouvsky

12:17 AM

@heather that's not what that means

dmckee

At that point there are even odd that your selected item is the first or the second.

Then you provisionally select the third with prob. 1/3.

Bernardo Meurer

Ah

Cool!

dmckee

Now check the odds for all three items: 1/3 for each. Cool beans.

Continue until you run out of stuff

It's so niffy it makes me grin every time I remember it.

Bernardo Meurer

@dmckee Very neat

Will keep close forever

0celouvsky

@heather I thought you knew calc 1...

Bernardo Meurer

12:20 AM

@dmckee If I have two allocated pointers to unsigned integers of same size, how do I check if they are exactly the same? Is memcmp the function for that?

heather

@0celouvsky i've read about calculus - i haven't taken the class

i'm sorry, i know i probably sound like an idiot

dmckee

@BernardoMeurer Do you want to check if the pointers are the same (that is point to the same memory), or if the integers pointed to are the same?

Bernardo Meurer

@heather Don't degradate yourself like that

dmckee

The first case is ptr1 == ptr2.

Bernardo Meurer

If you want someone to talk you down ask me

@dmckee The latter

dmckee

12:22 AM

The second is if (ptr1 & ptr2 & (*ptr1) == (*ptr2))

Bernardo Meurer

Wait, that checks if every element of my pointer has the same value?

dmckee

Gotta check that it is safe to defrerence the pointer before you actual do so.

Bernardo Meurer

I have these two

unsigned int *buf = calloc(self->size, sizeof(unsigned int));
unsigned int *rowcol = calloc(self->size, sizeof(unsigned int));

dmckee

In the first case you might also want a special result for both pointers NULL.

@BernardoMeurer Yeah?

Bernardo Meurer

And I do a bunch of operation on buf based on rowcol. I want to detect if the resulting buf is the same as rowcol, i.e. buf[0]=rowcol[0] && buf[1] = rowcol[1] && ...

"Gotta check that it is safe to defrerence the pointer before you actual do so." What do you mean by that?

dmckee

12:25 AM

@BernardoMeurer Doing *ptr1 when ptr1 is NULL makes the sate of your program undefined. Nasal demons are admitted from then on.

Also doing ptr[0].

So ... You should check that both pointers are non-NULL

Then loop over their contents

Bernardo Meurer

Oh, damn, I forgot to check for successful allocation of these pointers, good call

dmckee

flag = 1; for (i=0; i<ROWSIZE; ++i) { if (buf[i] != rowcol[i]) flag = 0; } if (flag) ...

Bernardo Meurer

@dmckee Can't I just do this:

unsigned int *buf = calloc(self->size, sizeof(unsigned int));
if(buf == NULL) catchError("Buffer allocation failed","");
unsigned int *rowcol = calloc(self->size, sizeof(unsigned int));
if(rowcol == NULL) catchError("Row/Col allocation failed","");

0celouvsky

@heather You don't sound like an idiot, but in your AMA you said you understood a good bit of calculus. Dan and Bernardo will probably kill me, but you don't.

dmckee

@BernardoMeurer In this case you probably can. But you should be in the habit of checking pointers before using them in c. And in c++ if you are using raw pointers.

Bernardo Meurer

12:29 AM

@dmckee Got it, I will review the code and add the check now :)

Okay, back to the issue, do I just do if (ptr1 & ptr2 & (*ptr1) == (*ptr2)) ?

0celouvsky

I'm not berating you, but if you want to get serious about learning calculus, I'm here for help.

Bernardo Meurer

How does this even work?

@heather He's not kidding, he held my hand throughout my entire freshman course

dmckee

@BernardoMeurer That works if you want to compare a single integer at the end of each pointer. (Which is what I thought you'd asked, but I should have known better.)

But what you want to do is to compare a buffer full of integers, and you have to do that one at a time.

Bernardo Meurer

@dmckee I was really worried that it was some bizarre pointer hack

dmckee

At least you don't have to do the NULL check before every comparison.

heather

12:31 AM

@0celouvsky okay.

Bernardo Meurer

@dmckee Why won't memcmp work?

heather

you have the Bernardo recommendation =)

Bernardo Meurer

I have it, but it doesn't work

dmckee

@BernardoMeurer It will.

Bernardo Meurer

@heather He does, he's pretty much the sole reason I passed Analysis

dmckee

12:32 AM

I just have some kind of blind spot for memcmp. Don't always remember that it is there.

Bernardo Meurer

@dmckee So, the objective is to only call that randCell function if the users move actually made a difference

i.e. if buf and rowcol are different

So I did this

if (memcmp(rowcol, buf, self->size * sizeof(*rowcol))) {
    self->setRandCell(self);
}

OH

typo

missed a !

There we go :)

Oh wait

dmckee

Looks good at first glance. But I"m nursing the remains of a migraine so details are not my strong suit right now.

Bernardo Meurer

@dmckee Ouch, migraines are the worse :/

I caught it, nevermind

Okay, it works :)

@dmckee Thanks for the help again!

Going to go to bed, tomorrow it's time to implement the history functionality!

That one will be fun

Yashas Samaga

sizeof(*rowcol) o0? @BernardoMeurer

Bernardo Meurer

@YashasSamaga What?

Yashas Samaga

12:45 AM

The other day you had some code like that, I told u why such code is bad.

nvm

Bernardo Meurer

@YashasSamaga But they are explicitly and necessarily allocated with the same size

Sir Cumference

1:00 AM

@0celouvsky You still have time

Learn it in your free time, there are plenty of websites

Plus programming can be useful in physics

0celouvsky

Huh?

I can program

Count Iblis

arxiv.org/abs/math/0411418

Sir Cumference

@0celouvsky Oh, what language

Slereah

h e l p

I can't sleep

The young people make too much noise

Physics Meta

-1

Q: What happened to CuriousOne?

He stopped posting here quite suddenly. I remember CuriousOne to be very critical of the MWI, so I'm going to poke in his eye here by saying that the Universe is in a superposition of all possibilities consistent with the record about his postings, so anything that could have happened to explain ...

0celouvsky

1:20 AM

@SirCumference matlab, formerly Java

learning fortran soon

Sir Cumference

neat

Slereah

lol FORTRAN

Emilio Pisanty

1:55 AM

@DanielSank yes

@Slereah it's a legitimate programming language

particularly for people really into that retro 70s computing scene like @KyleKanos

Slereah

Has anyone ever made a non-mathematical simulation program in FORTRAN

Emilio Pisanty

@Slereah plenty of hello worlds if nothing else

Slereah

Here's a math function

$f() = \mathfrak{hello world}$

DanielSank

2:34 AM

@EmilioPisanty ha! I still know things.

DanielSank

4:03 AM

@Slereah What's with all this mathfrak nonsense?

0celouvsky

@DanielSank it's the best font

$\mathfrak f:\mathfrak X\to\mathfrak Y$

honestly how can you not love it

Slereah

$\mathscr{HELLO WORLD}$

Secret

4:39 AM

Theorem: The set of truths are not closed under concatanation

Proof: Let $\mathscr{C}$ be the collection of true statements. Pick $S,T \in \mathscr{C}$. Define concatanation $\dots : \mathscr{C} \times \mathscr {C} \to \mathscr{K}$ where $\mathscr{K}$ is an ordered collection.

Suppose $S=\{\textrm{I was drunk}\}$, $T=\{\textrm{yesterday}\}$. Then $S \cdot T = \{\textrm{I was drunk yesterday}\}$. However for $\mathscr{C}|_{\textrm{my life}}$ there is no $R \in \mathscr{C}$ such that $R=S\cdot T$. Therefore $S \cdot T \not \in \mathscr{C}$ Q.E.D.

Sir Cumference

@Secret I've gotta ask: who are you posting these walls of text to?

Slereah

Also why do they always have to be nonsense

3

Sir Cumference

^That too

Slereah

"yesterday" isn't even a statement

And if you allow arbitrary concatenation then yesterday yesterday yesterday yesterday yesterday is a statement

Secret

4:56 AM

Well technically yes, thus that will be a proposition that is false or had undefined truth value. Meanwhile we also had a well defined proposition called "Buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo" that can be true or false depending on what really happened.

John Rennie

@BernardoMeurer you say you love C, but what you've done is to turn it into C++ :-)

Secret

But I agree, mistake, a single word should not be a statement

Slereah

Stop

Secret

I am just trying to formulate the observation that one can stitch together many truth statement and in general get a logical story yet it is completely false. From you guys reactions above, it seems I still get the maths concepts wrong...?

DanielSank

@JohnRennie Thank you.

Secret

5:09 AM

@SirCumference general, cause they are no specific users that I remember has relevant background to help enrich the analysis by pointing out directions on where to go next

Sir Cumference

@DanielSank I'm in some alternate universe where people prefer C over JS

Everyone I know is the opposite .-.

John Rennie

@DanielSank It is at least partially my fault, since it was me who told Bernardo how to implement member functions in a struct.

Sir Cumference

Then again, I don't have any friends who really use C...

0celouvsky

you probably only know dumb teenagers

John Rennie

@SirCumference Bernardo is only using C because his course requires it. No-one sane would use C these days.

Sir Cumference

5:12 AM

@0celouvsky Who know that all browsers can run JS

Wait, that's moronic

0celouvsky

oh don't give me that crap

I deleted my JS long ago

John Rennie

In effect what Bernardo has done is work round the limitations in C to turn it into a quasi OOL.

0celouvsky

@SirCumference of course it can run it, I just deleted it

Sir Cumference

That doesn't...

I surrender

Slereah

don't people still get taught C in school

C is usually the first language u learn

0celouvsky

5:13 AM

I thought it was Java or C++

Secret

Perhaps, for some of these wall of text, the best way to think about them is that they are questions, but I am not sure what the question is except from many years of observation you guys seemed to somehow can give relevant answers

Slereah

Gross

Sir Cumference

I was first taught JS

it has a special place in my heart

Slereah

Disgusting

John Rennie

C is a piss poor language to teach a beginner. C++, Java or C# would be a much better start.

Sir Cumference

5:14 AM

Then I learned some C++

@Slereah What languages do you prefer?

Slereah

Depends for what application

Generally I use C#

Sir Cumference

@Slereah You're a rare one

Slereah

Well what else can I use

Java is pig disgusting

John Rennie

C# is a very nice language. If it wasn't tainted by association with MS it would be more widely admired.

Slereah

python has weak types

John Rennie

5:16 AM

@SirCumference C# is widely used in industry.

Slereah

C++ is really poorly implemented

Sir Cumference

@JohnRennie Dang, never met anyone who really uses it

And I'm majoring in CS (as well as physics)

0celouvsky

> We assume that there exists a sequence $(\lambda_n)\subset D(J)$ with $|\lambda_n|\to\infty$ and that the family of operators $\{\lambda J_{\lambda_n}\}$ is equicontinuous. Then we have $$\mathrm{cl}\, R(J)=\{x\in X:\lim \lambda_n J_{\lambda_n}x=x\}$$ and hence $$N(J)\cap\mathrm{cl}\, R(J)=\{0\}.$$

I need some damn alcohol for this

Slereah

I have some Salmiakki in the pantry

0celouvsky

I don't know what's going on any more

John Rennie

5:17 AM

@SirCumference another example of how well academic CS courses train students for the real world.

0celouvsky

pseudoresolvents

who has even heard of these things?

Slereah

If I need some fast program I usually go with C

Or C++ that I use mostly like C

0celouvsky

@JohnRennie C hashtag?

C cardinality?

Slereah

It is pronounced C sharp

0celouvsky

o

Slereah

5:18 AM

$C \sharp$

John Rennie

C Sharp (programming language)

C# (pronounced as see sharp) is a multi-paradigm programming language encompassing strong typing, imperative, declarative, functional, generic, object-oriented (class-based), and component-oriented programming disciplines. It was developed by Microsoft within its .NET initiative and later approved as a standard by Ecma (ECMA-334) and ISO (ISO/IEC 23270:2006). C# is one of the programming languages designed for the Common Language Infrastructure. C# is a general-purpose, object-oriented programming language. Its development team is led by Anders Hejlsberg. The most recent version is C# 7.0 which...

Sir Cumference

"see sharp"

That's like pronouncing "CD" as "see dee"

0celouvsky

@JohnRennie so what's covariant C?

just plain C?

John Rennie

Covariant C :-)

I've seen deviant C ...

Sir Cumference

Most non-CS people are confused on the difference between C, C++, and C#

0celouvsky

5:19 AM

Musical isomorphism

In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle T*M of a Riemannian manifold given by its metric. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols ♭ and ♯. It is also known as raising and lowering indices. == Discussion == Let (M, g) be a Riemannian manifold. Suppose {∂i} is a local frame for the tangent bundle TM with dual coframe {dxi}. Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and...

Sir Cumference

Just like most are confused on the difference between Java and JavaScript

0celouvsky

all I know about Java Script is that I deleted it

John Rennie

@0celouvsky it's interesting that tis chat site still works without a JS interpreteter on you computer ...

Slereah

Javascript is all garbage

Sir Cumference

@Slereah no u

Slereah

5:20 AM

It was made by people too lazy to have well defined types

Sir Cumference

@Slereah Big deal

You and your types

0celouvsky

why is the chat so CS infested lately?

We haven't had a good point set topology discussion in weeks!

Sir Cumference

@0celouvsky JEE discussions had to be replaced

John Rennie

Most physicists have to do some coding.

Slereah

Well this isn't a math chat, either!

John Rennie

5:21 AM

Both experimentalists and theorists.

So there's a large intersection between physicists and programmers.

Slereah

I do have a mathematica program to derive all GR tensors from a metric

0celouvsky

JEE should be over soon

Slereah

It is quite nice

Doing GR calculations by hand is a pain

Sir Cumference

The nice thing about knowing one programming language is that learning another is way easier

Slereah

Is it

Sir Cumference

5:23 AM

Yeah, in my experience

Slereah

Here's a language I made :

Sir Cumference

You get a mindset

John Rennie

@Slereah did you write that yourself, or is it one of the various Mathematica GR programs floating about the web?

0celouvsky

@JohnRennie Most physicists know the definition of a manifold!!!!!

Slereah

esolangs.org/wiki/Lazy_Bird

@JohnRennie Wrote it myself

It's not very efficient

John Rennie

5:23 AM

@0celouvsky I strongly doubt that :-)

Slereah

But I don't mind too much

Sir Cumference

@Slereah Neat

0celouvsky

@Slereah lol what is that

Slereah

mbs``c``vk.1r``clbsbvt0``s``vksb`t0``v0i

Now that's programming

And it's Turing complete

So you can write anything with it

Sir Cumference

I think we can all agree that HQ9+ is the best language

Slereah

5:25 AM

please

It's not even Turing complete

0celouvsky

Lol. 218 pages into a functional analysis text we have the definition of a group

random af

Sir Cumference

@Slereah I'm not gonna bother to learn what that means

All I know is it doesn't apply to my use of JS

Slereah

Turing complete means that there is an isomorphism between that language and a Turing machine

0celouvsky

Daily reminder that I deleted the heck out of it

@Slereah they're probably not teaching him anything in CS worhtwhile

Slereah

Yeah I mean

Sir Cumference

5:27 AM

@0celouvsky Screw you!

Slereah

Turing completeness is pretty basic CS

Sir Cumference

You think double majoring is easy

Slereah

If you're not learning Turing completeness, you're not learning CS

You're just learning software development

0celouvsky

@SirCumference I'm a double major you child

Calm your nips

Secret

@0celouvsky do you wanna continue on our discussion about the proof or you want to do it later after the CS discussions?

Sir Cumference

5:28 AM

@Slereah T-thanks

This is a very well known school for CS, so I'd hesitate to say I'm not learning anything

0celouvsky

@SirCumference Carnegie-Mellon

Sir Cumference

@0celouvsky I refuse to say anything

0celouvsky

@OBE Confirmed.

Sir Cumference

Not necessarily

0celouvsky

@SirCumference Now I've got all I need

Sir Cumference

5:29 AM

All I'll say is that Carnegie Melon is a very tough school to double major in astrophysics and CS

Nothing more :P

0celouvsky

Damn

Not Carnegie Mellon

You misspelled it

Sir Cumference

It's 1:30am

I'm not at my A game... :/

0celouvsky

So it is Carnegie Mellon

Sir Cumference

What

0celouvsky

@SirCumference How's Stephanie?

Sir Cumference

5:31 AM

I don't know what that means

0celouvsky

O'Neil

Sir Cumference

Who?

0celouvsky

Lol

I'm on to you

Sir Cumference

@0celouvsky Uh...?

Well first, if you were aware of someone I knew, that'd be very creepy

So please don't try that

0celouvsky

oh it's too late for that

Going to search for the Yarmulke now

Sir Cumference

5:33 AM

I don't wear a yarmulke :P

0celouvsky

damn

John Rennie

Amazingly none of my servers are dead this morning - or even coughing up blood. It could be a relaxed morning.

Slereah

Possibly @SirCumference 's degree

Sir Cumference

Screw you...

John Rennie

Settle down chaps

Remember I have the power to ban users for, err, 60 seconds :-)

0celouvsky

5:37 AM

@Secret So we need some preparations for the proof...number 1: do you know that closed and bounded subsets of $\Bbb R$ are compact?

Secret

Uh, that sounds very different from what I and DHMO went through cause we can show that in the standard topology of reals, [0,1] is closed and bounded, but noncompact?

imgur.com/a/lGOvs

wat

My sides

he lied to you

[0,1] is most certainly compact

Slereah

5:40 AM

It is the compactest

0celouvsky

@Slereah no, that's the empty set :P

Slereah

Well you can cover it with $0$ sets

It's pretty compact

Secret

Ah sorry guys, [0,1] is not compact in the left order topology, not the standard one, I just realised...

in Mathematics, Feb 26 at 7:41, by Secret

@DHMO How about the countable union of {[1-1/n,1-1/(n+1))} with n=1,2,3,4,... and the union of [1,3/2) ?

Ok in that case, I should be fine for preparation 1, @0celouvsky

0celouvsky

@Secret Ok next, 2: Do you know what the product topology is?

3: Do you know Tychonov's theorem for two sets?

Secret

2. I think I need to revise this a little bit cause the only thing I remember is that the open sets in that topology are of the form $\Pi^{-1}(U) \cup \Pi^{-1}(V)$ where $\Pi^{-1}$ are projections onto the first and respectively second coordinates of the tuple formed by the cartesian product of the underlying sets

3. I think I had not reached that section in munkres yet, thus I am kinda unfamilar with the theorem

0celouvsky

5:51 AM

@Secret For 2, the basic open sets are of the form $A\times B$, that is, the rectangles form a basis for the topology

For 3: If $A,B$ are compact, then so is $A\times B$ in the product topology.

Secret

Ok (I like the way you phrase 2, it's a lot more intuitive than phrasing it as the intersections of $p^{-1}(U_i)$ that most text used)

but anyway, continue

0celouvsky

@Secret The relationship $c||x||_1\le ||x||_2\le C ||x||_1$ is call an "equivalence" of norms

Secret

yup it reminds of a similar looking relationship in metrics

0celouvsky

It's easy to check that this is an equivalence relation on the set of all norms on $X$

We want to check that all norms are equivalent. But since this relation is transitive, we can verify it for one of the norms some fixed norm

We'll use the $\ell^\infty$ norm for this since it has a very nice unit ball

Since $X$ is finite dimensional, it has a basis $e_1,\dotsc, e_n$

We define $||x||_\infty =\max |x_i|$, where $x=\sum x_ie_i$

make sense so far?

Secret

yup, so we have the unit n ball as a n cube under this norm

0celouvsky

6:00 AM

@Secret Yes, and now comes the nice part. It's easy to see in this norm that $X=\times^n\Bbb R$ as a topological space, where $\Bbb R$ carries the usual topology

Now this means that $B=\{x\in X:||x||_\infty\le 1\}$ is just $[0,1]^n$, and hence compact in $X$ by Tychonov's theorem.

Secret

ok

0celouvsky

@Secret We now have to show that $B$ is compact with respect to $||\cdot ||$ as well

This is...not so easy

First note that $||x||=||\sum x_ie_i||\le \sum |x_i|||e_i||\le C||x||_\infty$ where $C=\sum ||e_i||$

This implies that the $\infty$-topology is stronger than the first norm topology. Using this you can show that $B$ is compact with respect to $||\cdot ||$

Let $A=\{x\in X:||x||_\infty <1\}$. This is easily seen to be open with respect to either topology

$0\in A$ so we can fit an open ball $U=\{x:||x||<r\}$ inside of $A$ for some $r>0$

Now $||x||<r$ and $||x||_\infty\le 1$ implies $||x||_\infty<1$, that is, $U$ does not touch the boundary of $A$

Claim: $||x||<r$ implies $||x||_\infty<1$. To see this, let $||x||<r$ and put $x=\sum x_i e_i, \alpha=||x||_\infty$. So $||x/\alpha||_\infty=1$ and $x/a\in B$. Now if $\alpha\ge 1$, then $||x/\alpha||<r/\alpha\le r$, hence $||x/\alpha||_\infty <1$, which contradicts $||x/\alpha||_\infty=1$.

Thus $||x||_\infty =\alpha<1$, so we have the claim.

The claim then gives $||x||_\infty <r^{-1}||x||$ in $U$, so by scaling for all $x\in X$.

To sum it all up: for each $x\in X$, $r||x||_\infty <||x||\le C||x||_\infty$, which completes the proof.

@Secret I think this proof has an issue. I never used compactness, which is necessary. I'm thinking the construction of $U$ needs to be done more carefully and requires compactness. I'll write a better proof tomorrow

It's late, I must sleep

Secret

6:29 AM

g'night

Secret

7:05 AM

Some clarification on what I mean in the previous wall of text:

Consider the following sequence of events:
Truth

A: The book on the top of the bookcase fell
B: The dominos were knocked down
C: The ferris wheel is set into motion
D: A steel bead slid down the track
E: A splashing sound is made as it fell into a basin
F: A detector register the sound

What is being told:

D: A steel bead slid down the track
E: A splashing sound is made as it fell into a basin
F: A detector register the sound
C: The ferris wheel is set into motion
B: The dominos were knocked down
A: The book on the top of the bookcase fell

Without actually seeing the chain reaction setup in person, there is no way to deduce which version is true because it is internally consistent logically speaking

So the gist is that. You can have a video, and via clever cutting and rearrangment and the way each scene is arranged, create another video that depict the events differently, but still be perceived as unbroken and continuous

skill patrol

7:55 AM

Pro tip: never try to clarify a wall of text with another wall of text.

:-)

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