Zulip Chat Archive

Stream: maths

Topic: Implicit function theorem

Yury G. Kudryashov (Mar 29 2020 at 18:31):

I've just realized that I don't remember a lot of simple facts about infinite-dimensional Banach spaces. I'm trying to formulate a geometric version of the implicit function theorem. Currently we have a version that takes f : E × F → G such that df/dy is invertible and returns E × G → F. I'd like to have a version that takes f : E → F such that df is surjective, and returns ker (df) × F → E.

Patrick Massot (Mar 29 2020 at 18:32):

You'll probably hit Fredholm operators pretty soon.

Yury G. Kudryashov (Mar 29 2020 at 18:33):

The only issue is to identify E with ker (df) × (E / ker df). Do I need extra assumptions?

Yury G. Kudryashov (Mar 29 2020 at 18:33):

@Patrick Massot I'm afraid I forgot almost all of functional analysis after passing the test.

Yury G. Kudryashov (Mar 29 2020 at 18:37):

I'd like to have a version that works for f : E → real with infinite dimensional E (hence infinite-dimensional ker df)

Yury G. Kudryashov (Mar 29 2020 at 18:54):

It should work if E / ker df is finite dimensional: lift it somehow into E, and this lift is continuous because E / ker df is finite dimensional.

Yury G. Kudryashov (Mar 29 2020 at 18:56):

Are there any examples of E, p : submodule E, is_closed p such that there is no right inverse to the projection E → p.quotient?

Patrick Massot (Mar 29 2020 at 20:23):

I think that having a continuous right-inverse to the quotient map is the same thing as asking for a closed complement.

Patrick Massot (Mar 29 2020 at 20:24):

But when the target has finite dimension then the kernel has finite codimension and there won't be any problem, right?

Kevin Buzzard (Mar 29 2020 at 20:49):

https://math.stackexchange.com/a/108289/1162 claims that in general you need hypotheses to find a closed complement of a closed subspace of a Banach space.

Patrick Massot (Mar 29 2020 at 20:49):

Of course!

Patrick Massot (Mar 29 2020 at 20:49):

Does that contradict anything I wrote?

Kevin Buzzard (Mar 29 2020 at 20:50):

No not at all Patrick! I was just answering Yury's question.

Patrick Massot (Mar 29 2020 at 20:50):

I think that bit was clear to Yury as well. I'm pretty he met dense subspaces of Banach spaces (that are not the whole space).

Yury G. Kudryashov (Mar 30 2020 at 01:56):

I definitely met dense subspaces. The question was about a closed complement to a closed subspace of a Banach space.

Yury G. Kudryashov (Mar 30 2020 at 01:59):

So I should ask for a continuous right inverse instead of surjectivity, OK.

Yury G. Kudryashov (Mar 30 2020 at 02:06):

Possibly I'll formalize the fact that c_0 has no closed complement in l^\infty.

Last updated: Dec 20 2023 at 11:08 UTC