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 $s$.
@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 :).
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 $T(a,b,c)=-T^\circ(a,c,b)$
@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.
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
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.