My knee hurts. And my hip. And my shoulder, which is new. :/

xander: if your shoulder is new, it should still be under warranty

@leslietownes ah, yes. My language was unclear, and the referent was obscure. The pain is new, not the shoulder.

define new

sorry I'm being facetious

Say $W \leq V$ be a vector space. Say $\{ u_a: a\in A\}$ be a basis of U. Does the basis extension theorem still hold here, i.e, this U is part of a bigger basis for V?

This basis for U*

The general trick is that, if some vector falls outside the span of s given linearly independent set, appending this vector with this linearly independent set yields a bigger linearly independent set. All of this makes sense in FDVS but not sure generally

nick is that first W supposed to be a U, or what? U comes out of the blue there.

in complete generality there is no reason to expect every vector space to have a basis, unless you include something like the axiom of choice or something in your foundations, which is maybe surprising. but if you have that, yes, you can extend a basis of a subspace to a basis of the space irrespective of any assumption on dimension. but at the same time, the abstract linear algebra concept of basis is often not the one that one wants to use in the infinite dimensional setting.

May I ask how one should approach studying a math book through self-study? What should one focus on, and what should they do? If you are a professor, what do you expect students to do? I know I’ve brought this up countless times already sorry for the repetition but I still haven’t found a clear answer. I made a lot of mistakes that consumed my time while studying real analysis, and I don’t want to repeat them, but I really don’t know what to do.

@leslietownes oh yes, i hallucinated.

general arguments that proceed from a choice or even a use of a basis (whether an abstract linear algebra basis or a more specialized notion of basis) are way less common in non finite dimensional linear algebra. you are more likely to use them for purposes that are specific to particular operators or groups of operators, than you are to prove general theorems about your space.

@leslietownes yeah, assuming I have AC. How would I show this? Also I understand infinite dim VS theory is not useful without additional structure like convergence / distance, question is purely for curiosity purposes

just tossing that into the background. a lot of abstract linear algebra that isn't obviously false outside of the finite dimensional setting is still generally true in the infinite dimensional setting (assuming AC), but it's just not useful, so the statement that it is true can be kind of misleading if you assume there is anything of real substance behind it.

nick it is a standard zorn's lemma argument, same idea as what you do to prove the existence of a basis for an arbitrary space in general (which you should do first, it is basically a special case). math.stackexchange.com/questions/1044388/… for spoilers. it is probably a massive dupe on MSE, that being just one example,

@nickbros123 yes and the thing i'd add there is once you are adding in notions like convergence, the "things like a basis" that you begin to want to use are not abstract linear algebra bases anymore, e.g. because you allow for some notion of convergence or approximation, and don't require everything in the space to be a finite linear combination of vectors from your "thing like a basis"

and once you lose finiteness of linear combinations you lose a lot of the usefulness of a lot of arguments from the finite dimensional setting that start with "just pick a basis" and then announcing some kind of stepwise procedure that terminates because you will eventually run out of inputs to the procedure.

the analogy to the finite dimensional setting is maybe strongest with inner product spaces ("hilbert spaces") where a lot of the geometrical "pictures" people have in mind from finite dimensional examples of low dimension in R^n or C^n remain meaningful and instructive.

outside of that, good luck

@pie For what purpose are you studying it

@leslietownes makes sense.

@leslietownes thanks, I'll try to first prove the precursor theorem 👍

even there you have to pay attention to topological considerations more often than you might be used to

pie it is harder to, like, define The Right Way To Self Study Mathematics than it is to self study mathematics. you should be more specific about your goals. like, way more specific.

If you are studying it for some higher purpose then ideally you should find professor to guide you so that you can study only the parts that matter

If you are studying it for the sake of studying it, then well... just go through it, you'll do it eventually. But its a very slow process

So basically, you want to study the parts that matter, and what matters is not for us to say

it might depend on your course programme if you are studying for a course, it might depend on what your professor wants from you, if you are trying to write under them or something

and it might depend on what you want to study in the future

that's the important part of studying... the secondary things are things like exercises, approach towards reading proofs etc.

we can give advice towards those secondary things, but there's high chance it won't help you in a satisfactory way

you probably want some simple solution to all of your problems, but it doesn't work that way

@Jakobian I just like it and it might help be to apply for master degree or something like that since I want to have a career in math

If you want to be anyhow successful then you should have a vague idea of what you want to study in the future already.

but sure, you can study it, I don't recommend to dwell on the concepts too much though

how I'd do it is I wouldn't do any exercises, definitely. Just have a (formal but) vague understanding of concepts, read, move on, read, move on. Don't waste your time on it too much

something i would not do is spend months looking for the perfect textbook [or other resource] for whatever you want to learn. perfect resources do not exist, at most you can avoid horribly bad ones. one of the biggest pitfalls of self study is people turning it into this spiral of looking for perfection, and inevitably switching books when they can identify a fault with something

there's value in someone else saying "this is the textbook" and just being stuck with it as something you can't fight against or switch away from. maybe at some places it is over half the value of formalized education

moving too slowly to match your later goals (maybe what jakobian partly has in mind in "dwell on the concepts too much") is another pitfall

if you look at some people on MSE, from the patterns of questions they post, they are clearly spending months circling around some body of material that a class at a decent school might spend like two or three hours of lecture on, maximum

Right, what I see with people studying real analysis is they take so much time just studying out of one book (like Rudin or Folland)

which maybe isn't a problem if you don't plan on ever studying formally or getting a degree, but, would absolutely be a huge problem if you are attempting to prepare for courses that do move that fast. you need to try to match not just the subject matter, but the level of focus (which includes not focusing on certain things), if not the pace of instruction

in a formalized course at a high enough level, an instructor could be expected to skip a whole lot of material as not relevant to whatever the course's goal is, and cover only what is relevant to whatever the goal of the course is, and it might lead to something like an expectation that you can do problems in section 7.3 of some randomly chosen textbook after hearing one lecture on concepts that are dealt with in that section

and if your past experience is "i can only understand section 7.3 after reading chapters 1-6 and sections 7.1 and 7.2 and asking MSE questions on every point that the author doesn't expressly deal with, as well as some points that they do expressly deal with" it is not good preparation for that environment

even if it might be fine for background knowledge

@Jakobian or "i didn't understand this problem about polynomial rings in one variable so now i'm reading the entirety of lang's algebra book and three wikis on category theory"

@leslietownes I never met anyone like that

in my experience on forums here, people usually dwell on real analysis

yeah i guess it is maybe less common among pure self studiers and more common among people getting formal education in something adjacent

at my undergrad and grad school physics students would audit courses in topology and algebra a lot, and seemed way more likely to catch the category theory bug than people in the math department

but they were usually getting some sense that that stuff was important from somewhere else and not just operating under their own steam

for all i know it could have been like one prof in the physics department tossing off a remark like "this has something to do with algebraic topology" once every semester

in relation to the thread mathoverflow.net/questions/484620/…

→ 11 messages moved to Trashcan

@PrashantGokaraju (I won't handle flags on the message I'm replying to due to a conflict of interest) - @Mithical ended this, and I don't think an extended discussion surrounding the flags is productive. With that said, non-diamond users meeting certain requirements can and do handle non-custom chat flags, regardless of whether or not they're a room regular.

If you feel my actions were inappropriate, feel free to raise an In need of moderator intervention flag or use the Contact Us form to request either a diamond mod or SE staff (respectively) to review my decision.

@Mithical I don't really plan on continuing this, but if you ask or send that message ^ to the Trashcan, I'll drop it.

@leslietownes damnn i was actually able to show this myself :)

i guess i was just afraid of it

the only thing we have to zorn is zorn itself

@cocomac In general, it's not really worth getting involved in discussions around chat flags. In this case, pretty much all of the messages I removed were rather rude or passive aggressive; getting involved in a discussion about the exact messages is probably not going to be very constructive. If you think that a message that's flagged is not a problem, I'd recommend just anti-flagging and moving on in most cases.

(Anti-flagging, counter-flagging... whatever the term is.)

@Mithical My messages were blunt at best, but not rude. It was constructive feedback. Mathoverflow is a site for research mathematics, if you're not aware.

@Jakobian I am indeed aware, thank you. Constructive feedback still needs to be phrased in ways that are not dismissive or patronizing. I'd prefer for this discussion to not continue.

@pie Personally I treat math books as a source of information or problems for a topic. I have a separate notebook for each topic that I am learning about and when I learn some theorem I just write it down on the notebook to make sure I have a reference for the future and I understand that theorem. I don't really take note of all theorems, just the one's that I feel are important for that time. Maybe in the future by experience I will realize which theorems are actually important.

I don't really care about finishing a textbook. If I need to learn something, I just get the textbook that has the knowledge.

Also I don't care about learning things that I don't have any interest in or I don't find useful in the future.

I am not sure if this would be helpful for you, but this is what I did and it proved to be a healthy & useful habit for me so I thought of sharing.

→ 4 messages moved to Trashcan

To make a couple things clear: 1.) The effect of your words is what matters. It's mostly irrelevant if you were trying to be rude or not; once you've been informed that that is the effect that your words had, please recognize that and take it into account for future conversations. 2.) This is a general chat for Math.SE (not MathOverflow, I'd note). Disparaging those who are here to discuss math at a less-than-research level is not maintaining the professional atmosphere we expect for chat.

Why are you @Mithical acting as a math mod?

math.stackexchange.com/users?tab=moderators

@think_meaning_buildß Chat is jointly moderated by mods (and users) from around the network. That's why moderators on every site have moderator powers across all of chat.SE. The math.SE mods know where to find me if they want to discuss how I've handled something.

ok, Merry Christmas :-)

@Mithical Again, you are claiming that I am "disparaging someone". This is not the case

@robjohn nice hat

Its not that I am mad at being reprimanded, but at you drawing conclusions like these, thats stepping the line

Did you get suspended, pal?

@think_meaning_buildß wouldn't it show on my profile if I were to be suspended

yup

@Jakobian I feel like the victim here, and this situation is stressful to me, and I feel in no way responsible about it, because I was just acting the usual way, and my behaviour wasn't an issue, and you can't tell me that it was. Even if my behaviour could be improved in that particular instance, it wasn't something I had control over

(it only shows while you're suspended)

so telling me that I should improve next time, that's not okay with me, and yes I get it that you'd rather this discussion to not happen, I would rather it not happen either, I feel dragged into it

not only was I risking suspension because of someone flagging my comments, but also I got unfairly called out

These drive-by flags are a pain in the *ss

and for someone like me who can't phrase themselves well most of the time, this is causing great distress

your phrasing has improved a lot :-)

I'm the one who got a link sent to me by a mod about Do I have to be nice

>8(

if a misunderstanding occurs, then that can be amended with discussion and understanding is what I believe in, but the actions that are happening, and the firm beliefs of some people about my actions in particular, I believe they create environment where such understanding is impossible, this wasn't a fight, but a misunderstanding that didn't have the chance of being amended

I should have the right to speak up when someone is treating me unfairly, and I believe this is what happened here, and it keeps on happening, with the moderator here bending the truth disproportionately to the situation that occurred, that no one else can verify, which effectively works as defamation.

That's why I was advocating in my comments for stop to such actions, but those comments were deleted, while the comments in which such bending of the truth occurs, were left, as if to mock me

how do I construct an injective function from $S \times S$ to $S$ (S is an arbitrary infinite set, suppose) [assume AC]

I tried giving S a group structure, but actually that turned out useless

if at all such exists (?)

ok such exists according to wikipedia

@nickbros123 you can biject any infinite cardinal $\kappa$ with $\kappa^2$

They're saying that you can well-order $\aleph_\alpha\times\aleph_\alpha$ so that its order isomorphic to $\aleph_\alpha$

this even had a name, let me search

I'm not sure

I forgot to mention that cardinal numbers here are defined as ordinals which aren't bijective to any smaller ordinal

so ordering on $\aleph_\alpha$ means ordering of it as a set of all ordinals of size strictly smaller than $\aleph_\alpha$ i.e. its the ordinal $\omega_\alpha$

@BenSteffan Do you know stuff about sober spaces and so on?

hi

@Jakobian gonna be honest, understand almost close to none of this

@nickbros123 context

@Jakobian nothing at all

I don't even know the definition

every irreducible closed subset has a unique generic point

this is the black magic of algebraic geometers

Hi

@Thorgott The Noetherian spaces have the property that there is no infinite decreasing sequences in the specialization preorder

Do you know a property which would imply that any of non-empty dually well-ordered subsets in the specialization preorder has infimum?

Assume T_0

Something weaker than being Noetherian

My first intuition was to maybe try to see if I can let every closed set be a closure of a singleton. But I think that might be too strong

And also not sure what that would be equivalent to

Hi

hi

$\begin{cases} \frac{1}{2} x + z = 0 \\ x + 2y = 0 \\ x + y + hz = 0 \end{cases}$

if h = 1, what should I do?

$\begin{cases} z = -x/2 \\ y = -x/2 \\ 0=0 \end{cases}$

solve the sistem?

Yes

If h ≠ 1, in theory the ker is 0

So the dimension Is 0

last eq is wrong

@Mithical Good to know, that seems reasonable. Thanks!

@SineoftheTime In what sense ?

how did you find it?

But which one is wrong

I misread -_-

sorry

you get $y=z=-x/2$

Yes

which is the equation of a line

Yes

I also get (1-h)x=0

what's the task? Find a basis for the ker?

h such that dim ker f = 1

seems that you found h then

@Jakobian well, I just don't know the meaning and theorems surrounding of most of the things you mention. Just havent studied those things

@SineoftheTime h=1?

if h=1, what's dim ker?

The dimension Is 1

@SineoftheTime

@Binky what's your doubt?

Hi

The dimension is 1 if I get only one vector ≠ from (0,0,0) right?

huh?

@Pizza hi

you've found that dim ker =1 if h=1

But if x=0 -> dim=0

if x=0 you get the origin

but x in R

@SineoftheTime How is it going ?

@Pizza I'm tired

what about you?

I'm trying to calculate a Fourier transform

In my case it is enough to use the definition

Wait

Im done

$x^2 - 10x + 25 = (x-5)^2$

And that implies a nice u-sub i can do

So I got this:

Mmm

If you know how to compute the FT of $x^2 e^{-|x|}$ you're done

$\hat{f}(k) = \int_{-\infty}^\infty u^2 e^{-|u|} e^{-ik(u+5)} du$

So from here the absolute value was a bit annoying :)

So i did so:

Looks difficult

the FT of $e^{-|x|}$ is standard

$\hat{f}(k) = e^{-5ik} \left( \int_{-\infty}^0 u^2 e^u e^{-iku} du + \int_0^\infty u^2 e^{-u} e^{-iku} du \right)$

@Pizza Where did you get this exercise from?

didn't you compute it previously?

@SineoftheTime I did what I wrote in the chat

So I evaluated these two integrals by parts

Oh wait

sounds good

I combined the exp terms

$I_1 = \int_{-\infty}^0 u^2 e^{(1-ik)u} du$

So i can do $t = 1-ik$

not necessary tho

And then proceed by parts

looks good

This ?

@Pizza

@SineoftheTime However, I still don't know how to check on wolfram if I'm doing it right, I get strange results

@Binky yes

let me see

@SineoftheTime Did you have another way to proceed in mind?

I'm in a hurry right now, but it seems to me that after one integration by part you can apply the property $D \hat f =-i\widehat{xf}$

This is how computing is done I think

wolframalpha.com/…

put normalization 1 and oscillatory fractor=2 pi

since that's the convention you're using

the result does not seem "strange"

I've to go now but I try to answer from phone

Thanks! It's ok don't worry if you're busy

@Pizza But can't you learn the table by heart?

No, the transform must be demonstrated

$\int_0^{+\infty} \sin^2(x) dx$

Can you do this?

@Pizza did you find the correct answer?

@XanderHenderson Hilarious!

@SineoftheTime I have -2π

Yeah but this is not a problem

But there's also the option -2pi

I'll check for a moment, because before I had only calculated $I_1$

I_2 is basically the same, you have just to change the sign

In fact I get $-\frac{2}{(1+ik)^3}$

$I_1 = \frac{2}{(1 - ik)^3}, \quad I_2 = -\frac{2}{(1 + ik)^3}$

$e^{-5ik}\left(\frac{2}{(1-ik)^3} - \frac{2}{(1+ik)^3}\right)$

@SineoftheTime do you have LaTeX on phone ?

No

But I can read it from there code

Oh okay

What you get at the end?

After common denominator

$\hat{f}(k) = e^{-5ik} \cdot \frac{4ik (3 - k^2)}{(1 + k^2)^3}$

It doesn't match with wa, does it?

On wolfram I get e^-10 etc??

How is it possible

I don't know

Maybe it's the same, wait

Maybe I need to replace $k \to 2\pi k$

hi sine of the time

@Jakobian I'm confused by which way the order and dualization goes? You want an infimum for any decreasing chain of closures of points?

A sequence $(f_n)\subset L^p$ is Cauchy if for all $\epsilon>0$, there exists an $N$ such that for $n,m>N$, we have $\|f_n-f_m\|_p<\epsilon$.

In my book they state that if $(f_n)\subset L^p$ is Cauchy, then we may find a sequence of strictly positive integers $(k_n)$ such that for every $n\geq 1$,$$\|f_{k_{n+1}}-f_{k_n}\|_p<2^{-n}.$$Sorry if this is silly, but how do I find such a sequence $(k_n)$ from the definition? I see how for each $i$, we can find two positive integers $n,m$ such that $\|f_n-f_m\|_p<2^{-i}$, but that's not getting us the desired sequence.

@Sahaj hi

@Pizza did you manage to get the correct result or do you need help?

psie do you see how there is N_1 so that d(f_{N_1}, f_n) <= 1/2 for all n >= N_1? do you separately see how there is N_2 so that d(f_{N_2}, f_n)) <= 1/4 for all n >= N_2, but then also how if N_1 had already been chosen, we could additionally choose N_2 so that N_2 > N_1? it is an iteration of that one idea. if a condition holds for all n sufficiently large, you can adjust "sufficiently large" to include other constraints, like "and also larger than N_1, N_2, ..., N_k,"

and still have an infinite amount of n to choose from

note this has nothing to do with what the L^p norm is, the argument could be framed in a general metric space

yeah, I was just about to write what you were writing :) thanks for confirming

Sorry to interrupt the conversation, but I was just wondering if anyone had any feedback about the answer I wrote here. As far as I can tell, it is correct, and yet it has three downvotes. I also don't see why the marked duplicate actually answers the question that the OP is asking...

joe it is impossible to figure out why people downvoted you by asking people other than the people who downvoted you, and directly addressing the people who downvoted you is the one thing the system doesn't let you do, so, i dunno

some people really don't like it when people answer closed questions, or questions that look like they are about to be closed, that's my speculation, for whatever it is worth (which is nothing)

Fair enough, I was just wondering if somebody with set theory knowledge would be able to point out an outright error in my answer. I don't think there is one, but that would at least explain the downvotes...

people apply varying degrees of care to which they assess whether a question is worthy of being closed, or "deserves to be answered in its current form" independently if it is going to be closed or not. i don't think it's possible to tease out why

"i was just wondering if somebody with set theory knowledge could point out an outright error in my answer" is IMVHO a very good question and not about your downvotes, and maybe could make its own question on main if you were generally curious

particularly if you can formulate it in a way that maybe doesn't refer to the original question, so that you can more cleanly separate the set theory question from whatever people seem to think (or misperceive about) about what has tainted the OP

looking at the comments on the OP i wonder if some folks were thinking that maybe the OP hasn't clearly articulated a starting point or otherwise justified how the question fits into some context about their background and what OP understands/assumes or would be willing to follow as an answer, and foundational questions are maybe more sensitive to this kind of context than most

it's a habit of some users to vote to close + downvote

not being a specialist in set theory, i'd get nervous if someone seemed to be asking a question like this without laying out carefully what they know or assume about logical inference, and exactly what they included in their setup other than "the axiom of infinity" and "separation," and what they do not

and i would be hesitant to answer a question for the same reasons. would specialists all understand what is "usually presented" in set theory (or in "ZFC," or whatever you want to call it) in the same way, so that an answer to this question wouldn't need to refer to those details? i don't know

to put it more in my frame of reference, if someone asked a question on main about L^p spaces, but it seemed to depend on the details of how these spaces were constructed, i would probably comment "this depends on how the spaces are constructed, e.g. via X or via Y the explanation would differ. can you provide more information." i wouldn't answer it assuming that they knew some specific set of definitions about what measure and integration were, even if they were common definitions

That's fair enough, but for this question specifically it doesn't matter much. Even if you are working over a weak subtheory of ZFC, like say Zermelo set theory (which doesn't have the axiom of replacement or regularity), you can deduce the existence of the empty set in both ways mentioned in the question.

but i don't have the hair trigger on downvoting or closing so my own thought process maybe isn't too informative here. it might have more to do with how you approach what makes MSE questions great, vs. how much of the relevant background you know

i've run into this in the past too in set theory, not even in an answer, but a comment. a set theorist basically started a fight with me in the comments about how he didn't think my comments were a good way of looking at things. it was, in my view, a completely inappropriate use of the system (comments exist to improve/focus the OP, not as a kind of review system of how other people are commenting)

but whatevs

foundations seem to bring out the worst in some people (no offense joe if that is what you do for a living)

I don't make a living, I'm a graduate student :)

i've never had someone fight me in comments about whether my example of an operator on a hilbert space was so far out of some supposed operator mainstream that it was worthy of persecution

Well who would pick a fight with a lawyer \_(0_0)_/

hopefully lots of people, that's how i make a living

Hehe

@leslietownes my brother filed a 53 page something or another with the court on Thursday. Is that a lot?

He cited Magna Carta.

Because the insane prosecutor is trying to do something so crazy that the last time anyone thought to say "maybe funny do that" was prior to the founding of the US.

xander: it's a lot, certainly more than many judges would allow for routine stuff, although maybe not something potentially case dispositive

maybe throw in some claims to a right to trial by combat under the ancient law

@Thorgott yeah

Although doesn't necessarily have to be any chain, just the ones thats dual is well-ordered

@nickbros123 well-order S and apply the same ordering to S x S

what do you mean by 'dual'?

Swap $\leq$ with $\geq$

ah ok, so you want subcollections to have maxima

Yeah. Subcollections of the chain

Hey @leslietownes

Hey @XanderHenderson