Theorem: V-E+F=2. Proof: Take a spanning tree, take a spanning tree of its dual, those hit all the edges. (V-1)+(F-1)=E

Christoffel symbols are relaxing.

One day I should learn general relativity

(which is why you're doing Christoffel symbols, if I had to guess?)

Not even close

You should check out Baez & Muniain for a good book that covers some of the physics from the more mathy side.

Does the following make sense? Suppose that $U:=\{x\in \mathbb N: x\ge n \text{ for all $n$}\}$. Then, since U is a subset of $\mathbb N$, U has a minimum element $s$.

But I think that U is empty.

Indeed.

@AlessandroCodenotti it seems so. I tried to think of an uncountable well ordered set but haven’t yet come up with any.

@AMDG Someone else also said a similar thing once. Actually the nick is taken from a tv show wherein one character has this name. I didn’t watch the tv show completely as I read about how it was going to end :).

@TedShifrin I think using this one can also get the existence of the first uncountable ordinal.

provided the existence of uncountable well ordered set is given.

Identify sets with their indicator functions

Then $[a,b]=[a,\infty)-(b,\infty)$ and $(a,b)=(a,\infty)-[b,\infty)$

Thus, $[a,b]=-(b,a)$

Let $T(a,b,c)$ be the closed triangle with vertices a, b, and c if they're oriented counterclockwise, and $T^\circ(a,b,c)$ be the open triangle. Use similar "logic" to show $T(a,b,c)=-T^\circ(a,c,b)$

Wait no…

@Jakobian Dieudonne measure on $\omega_1+1$ is the usual counterexample

But on Polish spaces Borel probability measures are automatically Radon

@AlessandroCodenotti here it's subsets of a locally convex vector space so I don't think we can just take any probability measure

@TedShifrin okay, thank you! :)

@Jakobian Ah I see, maybe it works in that case, but I'm not really familiar with functional analysis outside of Banach spaces. I would trust your book, what are you reading, Conway? (If not maybe check out Conway too, iirc there is a discussion of the dual of $C_b$ in terms of spaces of measure in good generality)

@AlessandroCodenotti no, but I'm pretty sure they meant regular Borel probability measures.

This section is about Choquet theorem, and this was a more general case when you consider probability measures on the closure of the set of extreme points of a compact convex sets

Turns out any point can be represented by such measure. We'd like to drop the closure but turns out the setting is too general for that because the set of extreme points might not be a Borel set

Reminds me of barycentric coordinates

But if you assume that the convex set is additionally metrizable, then this is content of Choquet representation theorem.

@AlessandroCodenotti books.google.pl/books/about/…

This is what I'm using, a very good book in my opinion

why it is impossible to integrate this equation

What does your question even mean, @CroCo? What does dot mean, for starters?

It looks like $\frac{dy}{dx} = \tan\theta$.

ask a silly question...

This room has turned into naptime at Munchkin’s daycare.

Hello. I have a doubt that has been asked repeatedly on the site and yet I can't understand something in my notes. Two vector spaces are canonically isomorphic if there exists an isomorphism between the two spaces that does not depend on a particular choice of the basis. I don't understand whether $V^*\otimes W\cong Hom(V,W)$ are canonically isomorphic or just isomorphic

My notes say they are canonically isomorphic. The proof I know is like the one here. The fact that during the proof a basis of $V*$ and a basis of $W$ are used puzzles me

@Feynman_00 but every basis gives the same map

we only take basis to show that the map is an isomorphism

@Jakobian Oh ok that's it. Thanks.

@Feynman_00 Why not just map $\phi\otimes w$ to the linear map that takes $v$ to \phi(v)w$? Maybe that’s the mapping they gave.

certainly the mapping you should use

Yes, that was the map used in my notes

Oh, I see. To prove iso, we need to interpret linear maps using bases.

I am studying differential geometry and I'm loving it so far but I'm having some difficulties not being a math student and canonical isomorphisms are one of those things that puzzle me.

It’s really not that crucial to understand most of the time. In all the times I taught graduate differential geometry, it only came up a few times.

naturality actually is relevant when you try to extend linear algebra constructions fiberwise to bundles

Yes, but some formalities one can just take on faith sometimes …

The only time I needed a non canonical isomorphim was while locally trivializing the tangent bundle

And also with the dual space but I haven't studied the metric tensor yet