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Stream: learning: questions

Topic: Infinite-dimensional manifolds as smooth sets

Matteo Capucci (he/him) (Sep 18 2020 at 08:53):

On the nlab is stated, in various places, that infinite-dimensional manifolds are smooth sets, i.e. sheaves on the (essentially small) site of Cartesian spaces.
I have an infinite-dimensional, separable Banach space in my hands (call it $P$), and I want to check if it's 'smooth'. Do you have any idea on how to use it to get a sheaf on that site?
An idea is to simply take maps $\Bbb R^n \to P$, but which kind of maps? The smooth ones, one might say, but how do I define what a smooth map into $P$ is?

David Michael Roberts (Sep 19 2020 at 02:04):

Smooth maps to Banach spaces are unambiguously defined (unlike for general locally convex spaces). But you might be thinking of the sheaf of continuous functions valued in $P$? This is the diffeology that takes all continuous functions and declares them to be "smooth".

John Baez (Sep 19 2020 at 05:42):

Matteo Capucci said:

An idea is to simply take maps $\Bbb R^n \to P$, but which kind of maps? The smooth ones, one might say, but how do I define what a smooth map into $P$ is?

The usual concept of a smooth map between Banach spaces (e.g. a smooth map from $\mathbb{R}^n$ to some other Banach space) is one that has derivatives of arbitrary orders, where the derivative involved is the Frechet derivative.

Matteo Capucci (he/him) (Sep 19 2020 at 11:01):

Oh well :thinking: yeah it didn't occur to me to use Frechet derivatives, but it feels like the correct notion

David Michael Roberts (Sep 20 2020 at 03:00):

I was thinking that past Fréchet, there are inequivalent notions, the best one is https://ncatlab.org/nlab/show/Michal-Bastiani+smooth+map, but for Banach everything is peachy.

John Baez (Sep 20 2020 at 15:29):

Yes. There's a weaker concept of derivative even for maps between Banach spaces (or Banach manifolds), where you don't require uniform convergence of the difference quotient for the derivative: it's called the Gateaux derivative. But this doesn't even define a linear transformation between tangent spaces unless you demand it does (thereby eliminating some examples). So I've always used the Fréchet derivative.

David Michael Roberts (Sep 20 2020 at 21:01):

Here's a paper listing seven different types of derivatives in locally convex spaces: https://link.springer.com/article/10.1007/BF01585512 and they all agree with the Fréchet derivative in finite dimensions!

Morgan Rogers (he/him) (Sep 20 2020 at 21:08):

This is starting to feel like another entry in this topic :joy:

David Michael Roberts (Sep 20 2020 at 23:03):

Yeah, but I'm not sure we have classified (or even can!) how many definitions of derivative there are :-)

David Michael Roberts (Sep 20 2020 at 23:07):

The upshot is, if @Matteo Capucci is working with Hilbert, Banach or Fréchet manifolds, then after realising we need something like the Fréchet derivative, where it is continuous in two slots, not just at each point, or in each direction, then there are no more choices.

If one goes out past Fréchet spaces as models for charts, then there are choices. Convenient calculus is a popular one, and fits well with diffeological spaces. But from the point of view of global analysis, the Michal–Bastiani calculus is "better". Or at least, in the circles I move close to, that what people prove theorems with, since it is a stronger notion, I believe.

Matteo Capucci (he/him) (Sep 21 2020 at 07:22):

Wow I didn't know there were so many options!