[Merged by Bors] - feat(analysis/inner_product_space/of_norm): Create an inner product from a norm #4798

hrmacbeth · 2020-10-27T02:28:49Z

A normed space respecting the polarization identity is an inner product space.

Co-authored-by: Frédéric Dupuis dupuisf@iro.umontreal.ca
Co-authored-by: Ruben Van de Velde 65514131+Ruben-VandeVelde@users.noreply.github.com

The idea will be to show that with this definition the dual space is indeed an inner product space. But many purely algebraic calculations remain, so this is WIP until I, Frédéric or any other volunteer has time to try them.

Separated out from #4379 (but independent of it). Discussion at
https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there.20code.20for.20X.3F/topic/Hilbert.20space.20is.20isometric.20to.20its.20dual

dupuisf · 2020-11-06T23:26:43Z

@hrmacbeth For parallelogram_law_with_dual_norm, wouldn't it follow from the fact that the dual is isomorphic to the completion of E?

Ruben-VandeVelde · 2022-08-26T13:44:32Z

@hrmacbeth I finished inner_product_space.of_norm (moving it to a separate file), and dropped parallelogram_law_with_dual_norm for lack of a proof. Do you think it's ready for review now?

sgouezel

The title of the PR has nothing to do with its content, right?

src/analysis/inner_product_space/of_norm.lean

sgouezel · 2022-11-24T21:40:17Z

src/analysis/inner_product_space/of_norm.lean

+# Inner product space derived from a norm
+
+This file defines an `inner_product_space` instance from a norm that respects the
+parallellogram identity.


Could you recall here what the parallelogram identity is? And maybe sketch the proof you will be giving somewhere in the file-level docstring, with pointers to intermediate lemmas in the proof.

I've recalled what the parallelogram identity is, but I can't link to intermediate lemmas because we're making them private.

sgouezel · 2022-11-24T21:41:22Z

src/analysis/inner_product_space/of_norm.lean

@@ -0,0 +1,407 @@
+/-
+Copyright (c) 2020 Heather Macbeth. All rights reserved.


Zombie PR, revived by Ruben :)

…_dual

eric-wieser · 2023-05-05T16:35:42Z

src/analysis/inner_product_space/of_norm.lean

+4⁻¹ * ((𝓚 ‖x + y‖) * (𝓚 ‖x + y‖) - (𝓚 ‖x - y‖) * (𝓚 ‖x - y‖)
+          + (I:𝕜) * (𝓚 ‖(I:𝕜) • x + y‖) * (𝓚 ‖(I:𝕜) • x + y‖)
+          - (I:𝕜) * (𝓚 ‖(I:𝕜) • x - y‖) * (𝓚 ‖(I:𝕜) • x - y‖))


Does this generalize as an approach to producing a conjugate-linear map over any star ring? My guess would be that summing over the generators does the trick...

I don't know much about star rings. Can you detail? How are the generators denoted in mathlib?

I guess what I probably meant here is "over the quaternions".

Probably you're right, but I have no idea how to do it in Lean. You will need to give me more info.

I should be clear that this comment is not intended to block this PR

mcdoll · 2023-06-08T11:39:47Z

Apart from the merge-conflict this should be good, right? @Ruben-VandeVelde

…_dual

…nner_product_space_dual

src/data/is_R_or_C/basic.lean

Ruben-VandeVelde · 2023-06-15T14:24:06Z

Apart from the merge-conflict this should be good, right? @Ruben-VandeVelde

Certainly as far as I'm concerned

urkud · 2023-06-21T17:04:23Z

LGTM modulo adding a sketch of the proof to the module docstring. Thank you!
bors d+

bors · 2023-06-21T17:04:25Z

✌️ hrmacbeth can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

YaelDillies · 2023-06-21T21:43:19Z

Yury, did you mean to delegate to Heather? She's been inactive on that PR for 2 years and a half.

kmill · 2023-07-12T19:19:26Z

Wow, a PR with an id that's less than the current Mathlib4 PRs :-)

bors r+

…rom a norm (#4798) A normed space respecting the polarization identity is an inner product space. Co-authored-by: Frédéric Dupuis <dupuisf@iro.umontreal.ca> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

bors · 2023-07-12T23:18:30Z

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feat(analysis.normed_space.dual): dual inner prod

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hrmacbeth added the WIP Work in progress label Oct 27, 2020

hrmacbeth changed the title ~~feat(analysis.normed_space.dual): dual inner product space~~ feat(analysis/normed_space/dual): dual inner product space Oct 27, 2020

fix namespace issue, update undergrad.yaml

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hrmacbeth mentioned this pull request Oct 27, 2020

[Merged by Bors] - feat(analysis/normed_space/dual): Fréchet-Riesz representation for the dual of a Hilbert space #4379

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Fix a sorry.

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YaelDillies changed the title ~~feat(analysis/normed_space/dual): dual inner product space~~ feat(analysis/inner_product_space/of_norm): Create an inner product from a norm Feb 28, 2023

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bors bot changed the title ~~feat(analysis/inner_product_space/of_norm): Create an inner product from a norm~~ [Merged by Bors] - feat(analysis/inner_product_space/of_norm): Create an inner product from a norm Jul 12, 2023

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[Merged by Bors] - feat(analysis/inner_product_space/of_norm): Create an inner product from a norm #4798

[Merged by Bors] - feat(analysis/inner_product_space/of_norm): Create an inner product from a norm #4798

hrmacbeth commented Oct 27, 2020 •

edited by YaelDillies

dupuisf commented Nov 6, 2020

Ruben-VandeVelde commented Aug 26, 2022

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hrmacbeth Nov 25, 2022

eric-wieser May 5, 2023

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bors bot commented Jul 12, 2023

		@@ -0,0 +1,407 @@
		/-
		Copyright (c) 2020 Heather Macbeth. All rights reserved.

[Merged by Bors] - feat(analysis/inner_product_space/of_norm): Create an inner product from a norm #4798

[Merged by Bors] - feat(analysis/inner_product_space/of_norm): Create an inner product from a norm #4798

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hrmacbeth commented Oct 27, 2020 • edited by YaelDillies

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mcdoll commented Jun 8, 2023

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urkud commented Jun 21, 2023

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hrmacbeth commented Oct 27, 2020 •

edited by YaelDillies