@ACuriousMind my brain is farting a bit, I'm confused by something I took for granted earlier. I stated that $N$ could be a single equivalence class, and that dividing $V/E = V/N$, but I think it's for the wrong reason. You can't have an equivalence relation only apply to a part of a space right? It would have to separate the whole space into partitions, it can't just be undefined for a section of it afaik. So the real reason is that it's $V/E$ which is pinching EACH class to a point, which...

Reduces the dimension by the dimensions of the classes that are being pinched, which is equal to the dimension you'd get after doing V/N, as dim(N) equals the dim of the classes

So V/E is isomorphic to, given a field K, a vector space of dimension dim(V) - dim(E) over that space

?

@Phase Slow down the horses there, where'd you get that $K$ is a field? In any case, the observation that $N$ is an equivalence class is correct - the quotient is supposed to be a vector space, what "role" does the class $N$ play in the quotient?

No no I'm being an idiot

Slow down the horses there?

I didn't mean that K is a subspace and a field

Are you being ironic or just don’t know the idiom?

I'm just changing how I've defined K because Idc about it anymore

I could've just said a field Q or something I suppose

@0celo7 What does it mean to you?

@ACuriousMind he's taking the mickey because it's "hold your horses" in England and US

I took it as a translation of Immer langsam mit den jungen Pferden

@ACuriousMind es heißt “Hold your horses”

You got the gist but it was wrong

@Phase I'm not taking the mickey, I just occasionally butcher some English idioms :D

I meant @0celo7 was

aaaaaaaaa i cant communicate anything today

But no in the thing above I didnt mean to imply K was both a subspace and a field, I was redefining it to be a field

I forgot to write it above too.. Smh.. I meant to state that if V is defined over K, so is the space it's isomorphic to

@Phase Okay. V and N are already vector spaces over some field, why did you feel the need to bring up the field there again?

I mean, it's not wrong, but if we're talking vector spaces, they're always implicitly over some field

Idk I never know how much detail is enough or too much

We don’t we do groups first

I’m an enemy of abstract nonsense but groups are easier than modules

But, on the other hand, aside from the superfluous mention of the field, that $V/E$ is a vector space of dimension $\dim(V)-\dim(E)$ is correct.

quotient spaces still seem a little..

Arcane though.

@Phase I assure you, I had the same feeling in my first semester of linear algebra

ok we are in bussiness. It works now

Me too

@ACuriousMind taught me quotients too

Like in Wikipedia they give an example as R^2 / Y where Y is a straight line, and that would separate it into lines parallel to Y, but wouldnt that be R^2 / E, where E is an equivalence relation of dim(1), giving you a resultant space isomorphic to R^1? @ACuriousMind halp

It didn't help that the lecturer insisted on calling the elements "left equivalence classes" by their even more confusing German name, i.e. Linksnebenklassen

He did a better job back then

How can you have a 1 dimensional space with different lines?

gosh...I try so hard to steer this discussion towards physics, and you guys go off on math again

lost causes

@0celo7 dont be rude : p

@enumaris im not lost, just in the process of getting lost : P

@0celo7 Phase did say they wanted to figure it out on their own

@Phase either he did a better job back then it you’re doing a worse job understanding 🤔

So I'm not teaching as much as I am giving hints

@ACuriousMind figure out what?

Quotient spaces

@0celo7 What these spaces are. Or, more precisely, they asked me for something to study and I suggested quotients

If I now just launched into a detailed explanation of quotients, it would kinda defeat the purpose

ACM halp

bare with me have never latexed on the fly before

will start soft

Given some vector space , a Hilbert space $H$

crap

@ACuriousMind is that what you did to me?

ok it rendered

hehe

@0celo7 your moms a quotient space lole

I feel like you just told me “space of equivalence classes with the natural linear structure” until it clicked

lol

Hilbert spaces...yummy...

I eat them for breakfast

@Phase please delete this.

@Phase Calm down, everything is fine :) I think the first thing to realize is that $N$ is the equivalence class of 0, i.e. in the quotient, it is literally the zero vector. "collapsing $N$ to a point" means that the whole of $N$ becomes 0.

@0celo7 Maybe? I don't recall exactly, it's been a long time

wait what's N here

Is N the straight line?

It has been a long time

@Phase You started writing $N$, what do you think it is?

i am nervous but I will say silly things anyways

ok

Im confused

D: what are you referring to

so $H$ $\rightarrow$ $H$

@Phase Okay, let's start from the beginning.

the thing that was confusin me was this

We have a vector space $V$, and an equivalence relation $\sim$, right?

Yes, that's correct,

I dont get how a space isomorphic to R^1 could contain distinct lines

@Phase Ahhhhh

uh oh

idk if thats a good or bad noise

The elements of $\mathbb{R}^2/Y$ are lines.

and you can add them like vectors.

If you have a vector space $V$ and a subspace $W$, you can declare the equivalence relation $x\sim y \iff x-y\in W$.

But.. If the space in the end is the space of lines parallel to Y

confus

Then you take the quotient $V/{\sim}$. The elements of the quotient are equivalence classes w.r.t. $\sim$.

Yeah, each equivalence class is a po-

oh

That doesn't seem particularly useful though..

for getting an idea of parallel lines

if each line is just represented by a point in R

In particular, $0\sim w \iff w\in W$, so the equivalence class of the zero is precisely $W$ itself.

ok so let $L$ ~ $frac{H}{\gamma}$

@0celo7 a l have never latexed on the spot lolz

@Phase Well, there is a bijection between $\mathbb{R}$ and lines parallel to $Y$ in $\mathbb{R}^2$. No one said it is particularly useful ;)

ahaha fair point x)

Is there anything I'm significantly missing with Quotient space stuff now?

@Cows do you not know how to use latex?

I suppose I should find some questions online

\frac{H}{\gamma}

@0celo7 I have never latexed on the fly lol

@Phase The point is that the ide ahas to become second nature. When someone writes something like "Let $W\subset V$ be a subvectorspace, and $x \in V/W$", do you know what $x$ "is"?

Wtf does that mean

Your TeX looks very strange

@ACuriousMind a representative of a coset

Or a coset

@ACuriousMind hm

Depends on the amount of pedantry

an equivalence class

?

@0celo7 I'm fine with either answer, as long as you can explain it

oh :(

idek what a coset is

@Phase Yeah. So, compared to a true vector $y\in V$, what's $x$?

a family of vectors?

Hm, maybe that's not a good quesiton

@Phase yeah, and how are they related?

rip

Oh

$H$ $\rightarrow$ $H$/ $\Gamma$

Idk, does y necessarily lie within x? I didn't get the impression from the question

That is, if it's a family of vectors, what is the relation between two vectors $z,z'\in x$?

oh

they're either in the same family or members of disjoint families

?

so

wait

didn't register the "in x"

are they in the same family? i.e. z E z'

@Cows You don't need to enclose every symbol in its own dollar sign, just put dollar signs around the entire LaTeX expression.

ok i will do that now

@Phase Well, you said that $x$ is a family, so yes

Idk what else to say other than they're in the same family [/ z~z']

I'm looking for $z-z'\in W$.

oh

The difference between two vectors in the same eqivalence class has to lie in the space you have quotiented out

OOOO

Yeah

@Phase this is fundamental in analysis too btw

You can’t avoid this stuff

now so $P( H/ \Gamma ) $ is the space of interest

All weakly defines spaces are quotients

As I said, it's a notion that's oddly hard to grasp at first, but becomes second nature once you're comfortable with it

Since $~$ is defined such that $x\sim y$ iff $x - y \in W$, if $x,y \in X \in V/W$, then that means that the difference between them lies within the original space, as otherwise they'd not exist in the same eq class to begin with

right?

Yep!

@Phase can I ask you to not curse please

Use \sim for the tilde

had to change x to X, I keep accidentally reusing variables

@0celo7 I normally would go "meh" but sure, you're being nice : P

Backslash, not slash :)

Yeah D: theres a 5 sec delay on the second edit :(

You're a good teacher :D

$\rightarrow$ are some ops acting on the space

come on dude > <

@Phase I'm actually feeling I'm doing abysmal right now, but thanks :D

$\rightarrow$ $\to$

just use \to, it's quicker

@ACuriousMind oh.. Why's that...? :(

say $\delta_{i}$

@Cows That's not how function notation works. You denote an operation by its own name, i.e. $f: X\to Y$ is a function that maps elements of $X$ to elements of $Y$, and $f(x)\in Y$ for $x\in X$.

The operation is $f$, not $\to$.

@Phase I feel I've done much better explaining this stuff to freshmen, but then again that was in front of a board where I could jump around and scribble down stuff much better than in chat...

oh sorry

so $f$

@ACuriousMind They also probably studied the prereqs formally and weren't massive dips

so don't be harsh with yourself : P

@Phase Nope, they were clueless freshmen right out of school :P

There might also have been a bar tour the evening before :D

so then there must be an $f_{k}$ from $P(H/ \Gamma)$ to $H$

Well, I'm probably just a bit of a dunce even compared to freshmen : P especially if its a top tier uni innit

I think we’re all stupid compared to German freshmen

I just looked at the side bar

"There are a few things ACM doesn't like: Nazis...[cont]", tbh who doesnt dislike nazis who isn't also a total smeghead

Well, personally I think ACM is a bit overzealous wrt. nazi jokes.

so I suppose what I've been thinking is that perhaps we are looking at a power sets of some subsets of a hilbertspace?

my mathy lang sucks

@Phase I think the point is that I'm pretty lenient and willing to grant second (and third, and fourth...) chances to most people but the last Nazi sympathizer that showed up here found himself banned from chat for a year rather quickly :P

Ouch

What counts as "Nazi" sympathiser tho

Like, do you mean literal nazis

Because the term is so overused these days

@Phase Approvingly posting Nazi screeds and using their symbolism as profile pictures.

Pretty literal.

Dang

@Phase thank you

The swastika is a cool symbol that should be reclaimed but if you're adding a white circle and red background you're asking for trouble

I am a firm believer in reclaiming symbols/phrases that were once misused but others (strongly) disagree.

The charlie chaplin moustache too

For example, the literal meaning of “separate but equal” is applicable to a wide range of situations but it’s suicide to use it unironically.

I dont really get why we stigmatise stuff like that still, it just gives the idea of these horrific assholes more poignance

d a n g e r z o n e

@Phase Chaplin himself did a pretty great job with The Great Dictator on that one

That speech is great

Ok phase let’s now discuss quotient spaces in analysis.

uh oh

ACM save me

I haven't even read what density is yet, I've barely read any analysis still, beyond trivial supremum and infimum stuff

We define L^p spaces to be the space of functions such that |f|^p is integrable, modulo the space of functions that are zero almost everywhere.

ok [redacted] off : P

@0celo7 is that marginally related to say a compact support?

You're not even trying xP

@Cows no

hehe

@Phase :(

so you mean a space except where it falls to zero

@Phase It is! Don't let yourself be ruled by machine men with machine hearts...

@Cows what?

hang tight let me fire up wikipedia

boy, it's not easy to find my way back here

ACM can confirm I’m not talking nonsense

@0celo7 if you're happy to spoonfeed the knowledge to a talking poo go for it

but don't expect me to know what $L^p$ is in any meaningful sense, or any sense at all really

One also can take quotients of the space of metrics by the diff group, and then study its homotopy type. Good stuff.

@0celo7 let us take a specific case, say L^2

oh dang

@0celo7 I dont doubt you're writing legit stuff, I just doubt that you're writing it in an approachable way for me, or even think you are : P

@ACuriousMind what about it

now that I switched to firefox...no more latex...

Most mathematicians get that one wrong btw

@0celo7 Just confirming you're not talking nonsense :)

this is a space of square integrable functions with an associated norm

we still on Hilbert spaces?

@ACuriousMind I was just saying that we set an a.e 0 function to zero in L^p

And that’s formally a quotient

@enumaris I'm not sure I know what the people in this chat are on

XD

::crickets while everyone evaluates their life decisions::

hehe

I’m avoiding talking to company

@0celo7 If you don't suggest a darn game

Also despairing over not understanding PDE

I will spend my last 20 quideronis

Shit boi what about PUBG

@Phase You finished Tyranny yet? :P

@ACuriousMind will u play that with us?

@ACuriousMind nah I haven't, I played a bit though it's really fun

@0celo7 Ew, no. I would play Dota, though.

The magic feels way better than the Melee

so basically I have a question

God the steam storefront is so laggy

@0celo7 PUBG is like 4 quid out my wallet count : P

i've been reading about strange things, basically what does it mean for different Lagrangian to describe the same theory? I mean I know it goes by the name of duality and so forth, but to a math person, what does this mean? in pde speak?

how do I actually install this chatjax++ ...

@enumaris i was struggling with that a short while ago. hang on i will give you a link

It's on the top right of the screen

@ACuriousMind what do you mean, “ew”?

like I got to the github but I'm not sure what the command to install it is

git clone URL

then what

lol

chrome.google.com/webstore/detail/se-chat-mathjax/…

@0celo7 I don't like deathmatch concepts

I'm a github noob

ah but that's the chrome one

just use chatjax

I'm on firefox right now

it adds a bookmark you click

@enumaris oh I got ur back hang on

addons.mozilla.org/en-US/firefox/tag/mathjax

great :D

time to test it $F$

@ACuriousMind so what is a duality? in soft (baby level) differential equations speak

hmmm how do I get it start up

I installed it lol

@ACuriousMind is it a reflection of your political leanings?

0celo7 lmao

It should be automatic shouldn't it...

@enumaris yes it should

hmm