## 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.

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: May 06 2021 at 17:38 UTC