Norms for limits

Conversation started Nov 19, 2015 at 2:59.

Alec Teal

Nov 19, 2015 02:59

Thanks @TedShifrin I had made some really weird mistake. it's actually trivial because it's =0 as you mentioned and the point is in the domain. (I stand by the $0<|h|<\delta$ bit, because the quotient isn't defined for $t=0$)

L33ter

our professor will use apostol for analysis next semester

so that is is good

I want to continue studying it by myself during the summer

Ted Shifrin

Oh, I thought you were in the valley, @Julian. I will be dropping folks at LAX on Sunday, 11/29, and then heading home. So that's not far from you.

Yes, @Alec, that's correct. But when you get to the end of the proof, you merely observe there's no problem when $h=0$.

Alec Teal

Yeah there isn't. Because it's in the domain

Julian Rachman

@Ted Also, I literally just noticed that I will be multi next year because I will be taking the AP BC exam this year so should I learn ahead or maybe learn some complex analysis?

L33ter

what is that !?

Alec Teal

Nov 19, 2015 03:01

I wish there was a way to re-wind time so that I could stop myself from typing things like "$x/x$ is differentiable but not continuous at $0$"

L33ter

@JulianRachman your a high school student?

Julian Rachman

@Ted Oh! That is actually pretty great. Should we meet?

@L33ter Yes I am.

Ted Shifrin

@Julian: I still want you to learn single variable rigorously and then multivariable rigorously, as opposed to the engineering course, but still learning computations.

L33ter

cool I took calculus BC exam when I was in high-school wasn't bad

I got 5 in it and calculus based physics

Ted Shifrin

We can figure something out, @Julian, or wait for my next trip.

L33ter

Nov 19, 2015 03:03

but I got 4 in chem

Ted Shifrin

The Calculus BC exam is a total travesty.

Julian Rachman

@Ted engineering course?

Ted Shifrin

They give 5s basically for anything from 60% to 100%. Totally uninformative, but everyone skips everything with that.

L33ter

no @TedShifrin when I went to uni I actually took calculus to raise my average

Julian Rachman

Ok. Lets try for next time since it is a holiday coming up and I am quite busy @Ted

Ted Shifrin

Nov 19, 2015 03:03

A typical multivariable course is computational/rote, @Julian, and often doesn't stress the interesting geometric ideas going on.

L33ter

and ended up getting 99 in it

Ted Shifrin

OK, @Julian. I'll be back up in the winter/spring, I'm sure.

Julian Rachman

L33ter

anyway back to studying

need to do algebra assignment

Alec Teal

Okay last one, and I hope this isn't daft. What do we actually define limits on? When I encountered them the "space" on $\mathbb{R}$ was never mentioned. It can't be just a metric space because limits in discrete metric would make no sense.

Julian Rachman

Nov 19, 2015 03:05

@L33ter I am only a sophomore in HS and so getting ahead will let me take any course I want

and do you remember anything from the test?

Alec Teal

I guess that a limit means $0<|x-a|<\delta$ and stuff actually means metric induced by a norm, right? Norms I can believe.

Julian Rachman

I am going to be so nervous

Ted Shifrin

You can define limits in any first countable topological space, @Alec.

L33ter

do you think if I have canadian passport with my name people at airport will give me attitude when I go to US @TedShifrin

Ted Shifrin

I don't know, Karim. I have very little belief in my country these days.

L33ter

Nov 19, 2015 03:07

we will see its not like I hold any religious beliefs anyway I thought it would be nice to escape the cold for sometime

Ted Shifrin

@Alec: There are more general notions of convergence, too, with nets.

L33ter

I know like people tend to generalize

Ted Shifrin

well, Karim, I don't think I'm volunteering to drive up to LA (for the third time in a month), but keep me posted.

L33ter

alright I definitely want to come and visit you !

Alec Teal

@TedShifrin I'm not "convinced" at the first countable bit. I would totally believe that for any NORMED space you could define limits and differentiability. A metric lacks...the scalar part (discrete metric for example) (how can I find out without pestering you?)

Ted Shifrin

Nov 19, 2015 03:08

I didn't say a word about differentiability. Then you want a Banach (or at least Frechet) space.

But you don't need a metric to talk about convergence of sequences, nets, etc.

Look at standard books, like Munkres or Dieudonné's Treatise on Analysis.

Alec Teal

@TedShifrin I wouldn't say that limits where you have the discrete metric say make any sense. But in my mind if you have a (complete, as you said) normed space, I can totally believe you get limits (derivatives and limits of functions, you know the classic real analysis)

Ted Shifrin

When you have the discrete metric, only a constant sequence converges.

Alec Teal

I don't know what'd break exactly but I cannot see "today we'll use the discrete metric for everything" leading to anything interesting as we know it.

Ted Shifrin

It's not very interesting.

Alec Teal

Yeah

Which Munkres book BTW?

Ted Shifrin

Nov 19, 2015 03:13

Just Topology.

Alec Teal

@TedShifrin if you consider the directional derivative for a moment, it matters not if you use $\Vert\cdot\Vert_\infty$ or $\Vert\cdot\Vert$ in the denominator, because these are equivalent norms, unless every first countable topological space is norm-able(?) (if it's metricisable, this isn't that big of a jump)

Conversation ended Nov 19, 2015 at 3:16.

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