Mathematics - 2020-11-25 (page 1 of 3)

Ted Shifrin

00:00

@AttractorNotStrangeAtAll No.

Thorgott

geometrically evident is usually a much more fuzzy notion than one would like to believe anyhow

Ted Shifrin

And I can lecture very fast.

Oh, Thor makes a good point. 2 hours a day 5 days a week, it's feasible.

AttractorNotStrangeAtAll

We have classes of 6 hours a week, 3 days 2 hours each.

Thorgott

that's doable

anakhro

@user2103480 that would be problematic.

My homology lecture for the high school students was a bust because it was too much information for one lecture.

Thorgott

00:04

I believe the analysis 1 lecture I took covered the same topics in a similar timeframe

user2103480

@Thorgott I checked, mine too

Ted Shifrin

Oh, so 21 hours is indeed a lot of time. Can do a lot!

user2103480

about 4 weeks for differentiation and integration, 4 hours of lectures per week. but we did 10 weeks on real & complex numbers, sequences, series and continuity before

Ted Shifrin

Differentiation in one variable has very little to do.

Thorgott

we even covered power series on top of that in this timeframe, iirc

robjohn

00:06

@MikeMiller but random walks in sober spaces are far less interesting ;-)

anakhro

What's the usual convention for conjugation in S_n? Is it $ghg^{-1}$ because composition is written backwards, or do they not usually change it?

Ted Shifrin

That's how I usually conjugate.

Thorgott

you can probably find either convention in some places

user2103480

@Thorgott we did that before starting continuity. But only pointwise. We did uniform continuity then afterwards in the first 1, 2 weeks of analysis 2

Then the lecturer went into point-set topology and alienated basically everyone lmao. I found it great though

AttractorNotStrangeAtAll

Thanks, guys. I asked because my colleagues are having a discussion in our group about the conduction of the classes.

Thorgott

00:09

we only did metric and normed spaces at the start of analysis 2

robjohn

@AttractorNotStrangeAtAll any resistance to that?

AttractorNotStrangeAtAll

The discussion is that the professor spent 2 months of class talking about
introductory things like set theory and induction, when he could have started with real numbers.

Ted Shifrin

Professors are human .... They do lots of stooopid things.

Not that I would know.

AttractorNotStrangeAtAll

At the same time some people had never seen those topics before, so it was good for them as it was a gentle introduction to analysis

Thorgott

I remember, in analysis 1, we introduced the real numbers axiomatically and then defined the naturals within the reals as smallest inductive set to obtain the induction principle

robjohn

00:13

@TedShifrin: did you see the falling ladders I posted?

Thorgott

silly from a foundational pov, but surprisingly effective

AttractorNotStrangeAtAll

The truth is that it is impossible to please everyone, and as the professor himself said, you can advance the content of the discipline independently, with the textbook itself

copper.hat

Real analysis is tough. Charles Pugh liked to describe it as complex analysis' evil twin.

Ted Shifrin

Yes, @robjohn. I can write animations too! :)

robjohn

okay, no more animations.

user2103480

00:14

@Thorgott our analysis 2 and 3 were hell. The exams were both really easy, but the topics were harsh

Ted Shifrin

LOL, they're good for other folk here.

I animated my bicycle path question years ago. Given the path of the rear wheel, plot the front wheel too.

copper.hat

How do you do the animations?

Ted Shifrin

Mathematica

copper.hat

Ahh. I'm to cheap too spend the $ on it.

Ted Shifrin

I finally did and now they want more to run on the latest Mac OS.

copper.hat

00:18

I used to love tech, but find the constant marginal changes (& charges) tiresome.

Ted Shifrin

Agreed.

copper.hat

Age, I suppose (in my case).

Ted Shifrin

I'm older.

user2103480

@Thorgott in one week, we were introduced to metric, normed, banach and hilbert spaces, and in the next one there was a short excursion into banach algebras, and then general topological spaces were introduced with metric spaces as first examples

Thorgott

I think the analysis 1 exam I took in my first semester was probably the harshest exam I've ever written. Like 7 exercises, none of which were standard and all of which required a clever trick to solve that could potentially be really hard to spot for inexperienced first semesters. It seems harmless looking back to it, but it was hella scary back then.

user2103480

00:19

Oh that's harsh

It was all standard in our first exams, with old exams and practice exams at hand

Ted Shifrin

It taught you to be a star in here, @Thor.

copper.hat

The problem with a lot of real analysis exams is that some questions have a trick and if you have seen it before it is straightforward, otherwise it depends on how creative you can be.

Ted Shifrin

Undergrad real analysis isn’t as crazy as graduate.

Rithaniel

Difficult problems are unironically great to see (in hindsight)

Thorgott

lol I wish

there was some weird integral on the exam I had no clue how to solve and I just randomly tried the $f=1\cdot f$ trick and partially integrated wrt $1$, it happened to work out and I felt like the luckiest person alive

copper.hat

00:22

I like/hate analysis. It is relatively easy to ask a simple question or minor variation that is very difficult.

AttractorNotStrangeAtAll

To be sincere, I like to see some of you saying that intro analysis was not easy peasy. The feeling I get is that most professors/researchers would say that intro analysis was actually very easy (so in order for me to be a researcher I would have to find it easy as well)

user2103480

@Thorgott in a probability exam that was nothing like the exercise sheets, stackexchange saved me lmao

the prof made the exam, the assistant the sheets, that was the disparity

Thorgott

oh lol

user2103480

and the prof had like 3 exam problems for maxima of random variables, which seem impossible to compute if you don't know the trick - and I knew it by reading SE solutions

copper.hat

Probability can be difficult in the same way as real analysis. Look up Erdos and the Monte Hall problem.

user2103480

00:25

@TedShifrin what is graduate analysis though? we don't have such a clear term here

Ted Shifrin

Lebesgue theory, measure theory, $L^p$

Thorgott

Let me share with you one of the worst homework exercises I ever had to do (analysis 1):
"Show by differentiation that the following function is constant and determine its constant value: $2\arctan\left(\frac{1}{2}\tan\frac{x}{2}\right)-\arccos\frac{3+5\cos x}{5+3\cos x}$"

Ted Shifrin

That's calculus, not analysis .

Meh.

Thorgott

the lecture was called analysis, what can I do

(it also had a lot of good exercises, but that one was horrible)

I hate tedious computations

copper.hat

For example, a rectifiable curve has a Lipschitz parametrisation hence differentiable ae. But does it always have a differentiable parameterisation? If not, give an example. I have struggled unsuccessfully with this for years.

user2103480

00:30

Our worst exercise was about the cauchy definition of the reals

@Thorgott mi.uni-koeln.de/Vorlesung_Sweers/Analysis-I-2015/skript/… page 19-20

(pdf)

copper.hat

I can't speak my native language, German is outside my realm :-(.

Thorgott

yeah, that's technical, but at least it's useful

user2103480

we had to prove existence of the multiplicative inverse, which sounds super easy and is easy if you sweep a few obvious things under the rug, but I was still new to uni and proved that the "adapted" algorithm (additive inverse-> multiplicative inverse) converged via some epsilon.delta

Thorgott

@user2103480 they once made us prove CLT for poisson variables by hand

user2103480

because I thought we had to prove everything in super detail

took like 2 pages

@Thorgott rip

if we share bad exercises

we had to prove that submanifolds of R^n have lebesgue measure 0 in analysis III

and weren't even given the fact that uncountable covers have countable subcovers (I think the proof needed something like that)

Thorgott

00:35

that's fairly easy, though, isn't it?

copper.hat

@user2103480 That is not too bad? Just as a strict subspace has measure zero.

Thorgott

cover with countably many precompact carts, each of which is the image of a measure zero set under a Lipschitz ($C^1$ on compactum) mapping

oh, if you don't have Lindelöf...

yeah, that's more awkward

but you can just take coordinate charts around all rational points

so using separability instead

user2103480

mi.uni-koeln.de/geometrische_analysis/16-17WSAnalysis3/…

I think that solution was also pretty close to what I wrote

Thorgott

yeah ok, if you also had to prove that differentiable maps map zero sets to zero sets, that's a good amount of effort

@user2103480 btw, same probability lecture also had us prove the continuous mapping theorem, Slutsky's theorem and parts of the portmanteau theorem as homework

which is funny, because these theorems didn't even come up in that (undergraduate) lecture, but only came up in the graduate probability lectures

user2103480

these are all not that bad if one is used to the concepts

But the whole problem with probability lectures is the bombardement with different types of convergence so I would have had serious problems

copper.hat

00:46

I am not sure I see the value in proof regurgitation in an exam.

user2103480

@Thorgott they came up in our undergrad probability lectures but there's not much motivation so students dont get the point

Thorgott

I remember I struggled really hard to prove that $X_n\rightarrow X$ in distribution iff $\mathbb{E}[f(X_n)]\rightarrow\mathbb{E}[f(X)]$ for every bounded Lipschitz function $f$

user2103480

@copper.hat yeh, but tbh oral exams mostly aren't more than proof regurgitation

@Thorgott tbf, that was our definition

copper.hat

I guess I would rather than someone knows the idea rather than the gory details.

Thorgott

our definition was pointwise convergence of the cdfs at every continuity point

copper.hat

00:49

Far too few folks focus on motivation which is essential for understanding in my opinion.

Thorgott

just looked it up and that + Slutsky were like 1.6 pages on my hand-in

we did probability before measure theory, which is horrible anyhow

copper.hat

Uurg.

user2103480

@Thorgott yeah same

but we had one good prof which didnt continue rushing through martingales after that

he started by 0 again, with the construction of the lebesgue integral and caretheodory's extension theorem etc

Thorgott

man, I spent all this time learning differential forms and I still have no clue what physicists are doing when they play with differentials. what on earth is "$d\vec{r}=\frac{d\vec{r}(t)}{dt}dt=\vec{v}(t)dt=d\vec{r}(t)$"??

user2103480

Which was good since I was damaged by analysis III (my trauma regarding measure theory is gone now, the wound inflicted by differential forms is still healing)

trivial math is difficult

00:54

Let $f, g$ be maps $B_m \to B_n$. Define $f \sim g$ when we can find relabelings $\alpha, \beta$ of the codomain, domain such that $\alpha \circ g \circ \beta = f$. I am wondering if we we want $\alpha \circ g \circ \beta = f$ to be true, then does one of the two maps (alpha or beta) have to be identity maps?

user2103480

@Thorgott total differential?

Thorgott

$\vec{r}$ is the position of a particle moving as the time $t$ varies and $\vec{v}$ is the velocity, but the calculation is tautological and literally just conjures notational dependency on $t$ up out of nowhere and doesn't accomplish anything else

it's bizarre

user2103480

what course?

Thorgott

this is from a theoretical physics 1 lecture lol

@trivial what're $B_m,B_n$?

trivial math is difficult

$B_k = \{1,\dots,k\}$

Thorgott

00:59

No, for example you can take $m=n=2$, $f=g=\operatorname{id}_{B_2}$, but $\alpha=\beta$ the map that interchanges $1$ and $2$

user2103480

In what sense was weak convergence of measures stronger than some other functional analytic form of convergence again?

Thorgott

which functional-analytic form of convergence

user2103480

weak* probably

Thorgott

do measures on a space naturally form a Banach space?

Lukas Heger

I think so?

you can take the total variation as a norm

oh wait, that only works for measures of bounded variation

user2103480

01:13

en.wikipedia.org/wiki/Ba_space there's this space

Thorgott

when you cant pay for a full banach space and only get the ba space

user2103480

but one could also interpret this as a form of riesz-markov on the dual space of C([0,1])

hrmpf, there was a document that included exactly the statement I'm trying to find

Thorgott

yeah i dunno what the right statement here is

or rather, weak convergence of measures is just convergence in the weak* topology on ba(\Sigma), no?

user2103480

https://math.stackexchange.com/questions/2111245/what-is-the-weak-topology-on-a-set-of-probability-measures?rq=1
it seems both viewpoints are valid

math.stackexchange.com/questions/1622873/… also a good explanation

skullpatrol

01:32

@robjohn please don't do that, sir

user2103480

@Thorgott "But the weak-* topology on $C_0(X)^\ast$ doesn't give the probabilist's weak topology on $\mathcal{M}(X)$; in particular, $\mathcal{P}(X)$ is not weak-* closed in $C_0(X)^\ast$." Probably that was what was meant

Thorgott

ah, I see

Ted Shifrin

03:42

@skullpatrol I think I hurt his feelings. :(

Rithaniel

Animations?

skullpatrol

yup

:(

Rithaniel

Question: If a ring doesn't have identity, then it might also not have any prime ideals. Would this qualify as having Krull dimension 0? Or would it not have Krull dimension at all?

Leaky Nun

is this a word game

copper.hat

Where are the animations?

Rithaniel

03:51

@LeakyNun The definition of rings that I'm familiar with doesn't necessarily require identity. It's almost always assumed, even if not stated, though.

(Unless of course you weren't replying to me)

Leaky Nun

it just depends on your definition of Krull dimension

this is not a mathematical question

skullpatrol

19 hours ago, by robjohn

or if you are curious

Rithaniel

Fair point on the definitions

copper.hat

@skullpatrol much appreciated!

skullpatrol

np, pal

robjohn

04:24

@skullpatrol I would never stop making animations. They can be useful in getting ideas across. For example, I think this animation shows what the 3-D shape looks like more clearly than a 2-D image would.

@Rithaniel I used to see exercises that explicitly said, "Let $R$ be a ring with unity." That sticks in my head to remind me that they don't of necessity have a unit.

Rithaniel

If $R$ is a $0-$dimensional ring without identity, then given any $x\in R$, does there necessarily exist a $y\in R$ such that $yx=x$?

@robjohn Yeah, it isn't mentioned all the time, but it's easy to just say "ring (not necessarily with identity)" or "ring with identity" so I'm comfortable with them going either way

robjohn

Like the ring of even integers

skullpatrol

@robjohn thank you, sir :D

robjohn

@TedShifrin No. My absence was due to taking the dogs to the park and getting dinner.

skullpatrol

04:40

@robjohn How's your rabbit doing, sir?

Rithaniel

(When you're working in a ring without identity, everything goes out the window, it seems)

robjohn

@skullpatrol It is quite happy. It gets lots of good food.

Rithaniel

(I'm trying to prove that if $R$ is $0-$dimesional and reduced, then $R$ is von neumann regular, but all proofs I've found rely on the fact that $x\in xR$, which might not be true if $R$ lacks identity. I'm sure that $x$ is in that ideal, but how do you prove it?)

Rithaniel

04:59

Wait, $x\in(x)$ by definition

I like arriving at that "aha" moment, but sometimes they also make you feel dumb

geocalc33

anybody here good at understanding $g=\frac{dxdy}{xy}+\frac{dudv}{v-uv}?$

can you pullback a vector space?

or is this just called an invertible linear map

RewCie

05:22

2

Phroop of Infinite Primez

user2661923

05:57

@Khachatur Mirijanyan I think that we should chat about it.

SenZen

06:40

test

Balarka Sen

10:47

@Thorgott Bounded continuous is enough

There's a short proof by Skorokhod representation theorem, see here. Coupling for the win, once again.

user 6232128

11:05

you have a very good knack of finding a winning argument

Thorgott

11:40

@Balarka bounded measurable and $\mu$-a.e. continuous is enough, but who cares about technicalities

Balarka Sen

not a very significant generalization; lipschitz to continuous is

Thorgott

i never learned skorokhod representation

was it that thing about a convergence in distribution being realizable by an a.e. convergence or sth

Balarka Sen

yeah

Thorgott

what a theorem

Balarka Sen

the proof is a coupling trick

Thorgott

11:51

Say I have a continuous thingy $\varphi\colon B\times[0,1]\rightarrow\mathbb{R}^n$ such that $\varphi(-,0)$ is the identity on $B$ (which is a subset of $\mathbb{R}^n$ btw) and $\varphi(-,1)$ is a homeomorphism $B\rightarrow B^{\prime}$ and each $\varphi(-,t)$ is also a homeomorphism onto its image. Is this what they call an ambient isotopy?

Balarka Sen

just an isotopy through embeddings

Thorgott

ah ok, and ambient would be the same thing but when defined on $\mathbb{R}^n\times[0,1]$ and it just restricts to a homeomorphism on $B$?

Balarka Sen

yeah

Mike Miller

@Thorgott This is the wrong notion, all knots are isotopic by this definition

All tame knots at least or knots with a tame arc

Balarka Sen

already the correct notion in the smooth category though

Thorgott

11:56

who cares about knots

I just want the right language to say I can nicely move one ball into another inside R^n lol

Mike Miller

It's wrong for that too

This would make the horned ball equivalent to the standard ball

Obviously wrong

Balarka Sen

lol

(I agree with Mike to be clear)

@MikeMiller Giving a talk on conformal maps and Riemann mapping theorem in my complex analysis class

Mike Miller

Prove Schoenflies!

Balarka Sen

boundary extension of the Riemann map is pretty annoying isnt it

Thorgott

Ok, here's what I'm doing. I have an open subset $U\subseteq\mathbb{R}^n$ and two intersecting balls $B,B^{\prime}\subset U$. Take $p$ in their intersection. First, continuously shrink the radius of $B$ so that it's smaller than the distance of $p$ to the boundary of $B\cap B^{\prime}$, then continuously move the center of $B$ to $p$, then to the center of $q$, then continuously blow up the radius to that of $B^{\prime}$. This process doesn't leave $B\cup B^{\prime}$ and hence $U$ at any point and continuously deforms $B$ into $B^{\prime}$ inside $U$.

Mike Miller

12:01

take that one for granted lol

Balarka Sen

yeah i just never thought about it until you told me to prove schoenflies

i was gonna say "obvious corollary of Riemann ma---"

Mike Miller

Too much notation can't read

I am still not sure what your goal is

Thorgott

i move one thing to another thing inside a third thing

Balarka Sen

Ok cool boundary extension is an exercise in Stein-Shakarchi

I'll prove it

Thanks for the tip

Mike Miller

Who cares about moving things

2

@BalarkaSen It's a theorem of Caratheodary lol

Balarka Sen

12:05

oh actually

SS says "generalize from piecewise linear things. requires further ideas"

Mike Miller

Maybe the hard part is showing boundary objectivity

objectivity

injectivity

Balarka Sen

lol

Mike Miller

AKA showing Riemann maps extend so long as your region is bounded by a Jordan curve

Thorgott

whatever, ill just call it phi

user 6232128

\o @epic_math

Mike Miller

12:06

Call it whatever you want but tell me why you're moving disks

Balarka Sen

also dude Koebe's Riemann map is so genius

its also an ex in SS, solved it yesterday

Thorgott

you don't wanna know

Mike Miller

OK so it's for an exercise about perfectoid rings

Balarka Sen

Start with the red domain $\Omega \subset \Bbb D$ on the left

Find a point $\alpha$ such that if you bloat a disk from the origin it osculates $\partial \Omega$ at $\alpha$ first time

the blue disk

Find a hyperbolic reflection sending $\alpha$ to $0$, reflect about the green geodesic

The red domain gets sent to the orange domain, still on the left picture

A branch of $\sqrt{z}$ is defined on this orange domain, cuz its away from the purple stick, the branch cut

So take square root; that sends the orange domain on left to the orange domain on the right

Reflect about the green geodesic so $\sqrt{\alpha}$ gets sent to $0$ again

You get a new red domain $\Omega_1$

Iterate it forever, $\lim_{n \to \infty} \Omega_n = \Bbb D$

This is the Riemann map

Mike Miller

@Thorgott I can say something useful if you ever tell me what the real point is lol

@BalarkaSen This is wild

Balarka Sen

12:17

yeah its beyond genius

my picture above is not accurate but im pretty sure this is what it looks like:

@MikeMiller Apparently Riemann's idea was the modern PDE proof you like of uniformization

Mike Miller

Interesting

He thought of it in terms of curvature then not complex diff

Balarka Sen

hm oh maybe im misremembering what you like to do. he wrote down a map $\partial \Omega \to S^1$ (this homeo coming from being a Jordan domain) and then tried to extend to the interior conformally by Dirichlet boundary value problem i think

Mike Miller

Ah no I like to solve the conformal change equation to produce a metric of constant curvature

Balarka Sen

ahh got it

Mike Miller

It's more irritating when you are noncompact since you need to ensure a complete metric

Balarka Sen

12:25

that makes sense

Mike Miller

So you usually do it as a boundary value problem on a compact exhaustion and take a limit

Thorgott

12:39

theres no real point

im just showing two maps are homotopic

Mike Miller

stupid

Thorgott

as per usual

Daniel Adams

13:11

Hi all

Imago

14:51

I started reading papers on Neural networks and I often come across the term: "degenerative minima". But what does this term mean, what does it express? Does someone know/is able to guess?

Leaky Nun

@Imago math.upenn.edu/~kazdan/312F12/Notes/max-min-notesJan09/…

it's when the Hessian is not definite (i.e. has a 0 eigenvalue)

Imago

@LeakyNun ah thank you :) that helps a lot

RewCie

@RewCie Hiiii :-)

satan 29

15:16

May I ask a question on triple integrals?

user2103480

@BalarkaSen I still don't get why we did all the partition shite when, directly afterwards, the hamiltonian is suddenly continuous on R^6N again

related to the statistical mechanics stuff

guess I'm gonna have to look up some infinitesimal lattice crap to find something that I find satisfying

Hm one could make a consistency condition that the integrals are equal due to averaging over microstates

Astyx

16:06

@satan29 Just ask, don't ask to ask

satan 29

nevermind now, its settled

Clarinetist

Does anyone have any suggestions for learning the basics of functional analysis as quickly as possible starting from a background on Rudin's PoMA?

Mike Miller

What do you want to know about functional analysis

Clarinetist

This is what was presented in class today, and it's gibberish to me.

Krein–Milman theorem

In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane ℝ2. == Statement == Throughout...

Mike Miller

Lovely theorem

Clarinetist

16:16

and its applications to measure theory

Mike Miller

The two texts I prefer are Rudin's functional book (less terse than his previous books are) and Conway's functional book

Conway is a rather dry writer --- one of the driest I know --- but he is at least clear

Clarinetist

Thanks for those suggestions

Mike Miller

This is John B Conway, not the more famous H Conway, who it is hard to imagine being a dry writer

Alessandro Codenotti

@MikeMiller omg I used Conway's book plenty of times and never realized this until now

Mike Miller

It's way too dull to be H lol

Still though he is very clear

Alessandro Codenotti

16:21

I think Conway's book is beautiful, but my favourite for a first introduction remains Brezis (Conway covers also much more advanced stuff)

Mike Miller

Does Brezis get to KM?

Alessandro Codenotti

I don't remember tbh

Clarinetist

I get the impression from Amazon reviews that Rudin's would not be recommended if you're coming straight from a PoMA background; would you agree?

Alessandro Codenotti

Is KM important for PDEs? I would guess so and I would also guess that it is covered by Brezis in that case

Mike Miller

KM shows up in construction of Haar measure

@Clarinetist That was not my experience, but YMMV

I would not blame anybody for reading something other than Rudin

Clarinetist

16:23

I might just get Conway

Mike Miller

I dislike his writing

I think FA is still clearer than the rest but I do not like his writing lol

Conway's book is nice because in his chapter on weak topologies he (unlike every other author) gives like 6 applications

Alessandro Codenotti

@MikeMiller It is stated but not proved in the comments to chapter 1, just checked, the proof is left as a guided Problem (Problem 1 at the end of the book)

Mike Miller

RIP

I have someone doing a project on Banach-Alaoglu and he wants applications so I pointed him to KM and Stone-Weierstrass

Alessandro Codenotti

Some exercises in Brezis book are quite hardcore, this one has a lot of hints though, it seems doable

Lukas Heger

how do you prove Stone-Weierstrass from Banach-Alaoglu?

Alessandro Codenotti

16:34

@MikeMiller Does (local) compactness of the space of characters of a commutative C*-algebra count as an application or is Gelfand duality going too far from the intended topic?

Mike Miller

@AlessandroCodenotti Already gave him that ref, up to him if he decides to pursue that or not

It's more difficult since he'd have to do some basics on multiplicative functions and calculating that there is one C* field etc

Alessandro Codenotti

I see, but I think that Gelfand duality is pretty cool. What kind of project is this?

Mike Miller

@LukasHeger You need Krein-Milman as well

You have a closed C*-subalgebra of C(X) that separates points. You want to show that it is all of C(X) --- equivalently that the set of functionals on C(X) which vanish on A is trivial

Call that A'. The unit ball in A' is weak* compact, Krein-Milman guarantees that the unit ball has an extreme point, which gives a measure on X for which everything in A integrates to zero

I don't really remember how it goes from here lol

But you use that it's an extreme point to derive a contradiction

Lukas Heger

I see, thanks

Mike Miller

I think you maybe show that the whole space has mu-measure zero

@Lukas I checked. You set K = support of mu (the extreme point in this unit ball). Because this is not the zero measure by assumption (towards a contradiction), the support is nonempty. The goal is to show that it is a point-mass, which will contradict that 1 in A and int f dmu = 0 for f in A

copper.hat

16:49

@Clarinetist A book I liked for my first round was "Introductory Real Analysis" by Kolmogorov & Fomin. Not as terse as Rudin (which I find hard going) and some older conventions such as all Hilbert spaces being separable but readable. I would suggest getting a few books rather than relying on one if you can. Another one I liked for its treatment of fixed points was "Functional Analysis" by Kantorovich & Akilov.

Mike Miller

Pick a point x in the support and a point y other than x. You can use that A separates points to construct a function f vanishing at y and nonzero at x, and show that f mu and (1-f) mu are both in A'. You do some irritating crap --- finally using that mu is an extreme point --- to then show that y is not in the support

So the support is one point as desired

There is surely content to this stuff I called irritating but I don't see it and I need to write lecture noters

Lukas Heger

thanks

Mike Miller

I never learned a "good" proof of SW. Do you know one

Lukas Heger

There's the proof that uses the Taylor series of $\sqrt{x}$, that's pretty elementary

Mike Miller

Oh, to be clear this is the statement that an algebra of functions on a compact metric space which separates points, is conjugation-closed, and has 1, is in fact C(X)

Lukas Heger

17:00

don't you need that it is closed as well? or you just get that it is dense in C(X)

I learned a proof of the real version in my intro analysis course

I think if you have the real version, you can also deduce the complex version

geocalc33

is putting a norm on a family of functions trivial from a functional analysis point of view

all the examples I've seen deal with more complex function spaces

like the space of C^0 functions on [-1,1]

with supremum norm

user486313

17:16

is operator theory a dead field?

user486313

what about functional analysis?

Mike Miller

@LukasHeger Yeah, sorry

geocalc33

function analysis is not dead

operator theory is still operating

Lukas Heger

@MikeMiller the proof of the real version went like this: if we let $K$ be a compact metric space and $B$ is a lattice of functions on $K$ that separtes points (where a lattice means it's a $\Bbb R$-vector space and satisfies $f \in B \Rightarrow |f| \in B$), then $B$ is dense.
Note that a lattice is closed under taking minima and maxima due to formulas: $\max(f,0)=\frac{1}{2}(f + |f|)$, $\max(f,g)=g+\max(f-g,0)$ and $\min(f,0)=\frac{1}{2}(f-|f|)$, $\min(f,g)=g+\min(f-g,0)$
Now the proof that a point-separating sublattice of $C(K)$ is dense:

user2103480

@Clarinetist pro tip:

use lecture notes instead of books

Balarka Sen

17:25

@MikeMiller law of large numbers

geocalc33

anyone listen to non mainstream hip hop music??

Balarka Sen

oh by SW do you mean the general thing about algebra separating points being dense?

then i dont know

i know for polynomials only

copper.hat

just europop & electronica here

Balarka Sen

@geocalc33 a little

user2103480

@copper.hat what a great niche

(seriously)

copper.hat

17:27

@user2103480 drives my 17 & 19 yo kids mad

Balarka Sen

lol

user2103480

they have no taste

i'M 24 aNd I liStEN tO ThIs

copper.hat

was better when i could dance to it, need some replacement parts in the hip area unfortunately :-)

user2103480

Go a step further and play krautrock on maximum volume

Balarka Sen

does hooverphonic classify as europop

copper.hat

17:30

I was taught functional analysis by an entertaining fellow by the name of Shoshichi Kobayashi, he had a wonderful way of putting things, for example, "sets are not like doros, they can be open and closed at the same time".

user2103480

@BalarkaSen google says it qualifies as electronica, so it's still within the acceptable range

Balarka Sen

mmkay

geocalc33

soundcloud.com/faustix/crying-in-the-sun-le-boeuf

Lukas Heger

Now that we have the lattice-version of Stone-Weierstrass we can prove the version for algebras: if $X$ is compact and $A \subset C(X)$ is a $\Bbb R$-subalgebra that separtes points, then $A$ is dense in $C(X)$.
Proof: it suffices, by the lattice-version of Stone-Weierstrass to prove that $\overline{A}$ is a lattice. wlog $A$ is closed. For $f \in A$, we have $f^2 \in A$. Now because $\sqrt{f^2}=|f|$, it suffices to show that we can take square roots of positive elements in $A$. Let $h \in A$ with $h \geq 0$. Set $g=1-h$ so that $h=1-g$. By rescaling we can assume $\|g\|_{\infty} < 1$, but …

Balarka Sen

@copper.hat is this actually the same kobayashi as kobayashi-nomizu

copper.hat

17:31

Yup. Nice fellow.

Balarka Sen

damn

Lukas Heger

@MikeMiller I won't say that this proof is pretty, but it's pretty elementary and down-to-earth, you don't need any functional analysis

copper.hat

i just listen to what i enjoy, i like jazz in the satchmo, ella fitzgerald, etc mode.

i suppose bach is no surprise, early electronica in my opinion :-)

Lukas Heger

also the appearance of the power series expansion for $\sqrt{1-x}$ is an oddity which I find amusing

Balarka Sen

haha

nice description

geocalc33

17:34

Is electronica like Martin Garrix?

user2103480

@geocalc33 nope

geocalc33

Kygo?

user2103480

more like M83

geocalc33

aho k

soundcloud.com/illeniumofficial/…

this is what I like^

user2103480

but electronica is vast and cannot be defined by one band, so take that with a grain of salt

copper.hat

17:36

not really, but whatever rocks your boat. i just use genres to find other stuff i like.

i find the repetiveness nice when i am trying to focus on something.

Balarka Sen

@copper.hat yeah thats really what cataloguing is useful for

good takes

user2103480

@BalarkaSen wait, it's not just a reason to be elitist?

WILD

Balarka Sen

hah

copper.hat

@BalarkaSen i suppose there is less information content (in a shannon sense) in the music i listen to when focusing :-)

Balarka Sen

lmao

user486313

17:42

anyone here like Dave Matthews Band ...

user2103480

3KZ - Please, Stay [FIDES006]

►

user486313

@geocalc33 wild insight, thanks bro

geocalc33

you're welcome brother

copper.hat

some friends politely describe my music taste as eclectic, more direct friends use the term crap :-).

user486313

@copper.hat what bands / music?

copper.hat

17:55

i like stuff like ncs, capella, the blue man group for example, edward maya, stuff like that. i also like bailando, panama, etc :-)

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$Mathematics$ Mathematics