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refactor(LinearAlgebra/SesquilinearForm): Hermitian sesquilinear maps#18406

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refactor(LinearAlgebra/SesquilinearForm): Hermitian sesquilinear maps#18406
mans0954 wants to merge 22 commits intomasterfrom
mans0954/ConjSymm

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@mans0954 mans0954 commented Oct 29, 2024

Definition of a conjugate symmetric (or Hermitian) sesquilinear map.

An example of such a map is the inner product on a Hilbert C*-module.

Zulip Discussion

Alternative attempts to address this issue: #20177 #19809.


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@mans0954 mans0954 added the WIP Work in progress label Oct 29, 2024
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github-actions bot commented Oct 29, 2024

PR summary c381040244

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference

Declarations diff

+ IsConjSymm
+ domRestrict
+ isConjSymm_iff_eq_flip
+ isConjSymm_zero
++- eq
++- isRefl
++- ortho_comm

You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.


No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary
  • The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
  • The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

@github-actions github-actions bot added the t-algebra Algebra (groups, rings, fields, etc) label Oct 29, 2024
@mans0954 mans0954 changed the title refactor(LinearAlgebra/SesquilinearForm): Attempt to define conjugate symmetry refactor(LinearAlgebra/SesquilinearForm): Hermitian sesquilinear maps Oct 29, 2024
@mans0954 mans0954 marked this pull request as ready for review October 29, 2024 20:08
@mans0954 mans0954 removed the WIP Work in progress label Oct 29, 2024
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I'm surprised we don't appear to need the condition:

J (r • a) = (I r) • (J a)

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CC: @eric-wieser


variable [CommSemiring R] [AddCommMonoid M] [Module R M] {I : R →+* R} {B : M →ₛₗ[I] M →ₗ[R] R}
/-- The proposition that a sesquilinear form is conjugate symmetric -/
def IsConjSymm (B : M₁ →ₛₗ[I] M₁ →ₗ[R] M) (J : M →+ M) : Prop :=
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I worry this is more generality than we really care about, and maybe we just want to allow I = J = star. Rather than changing anything here, it's probably best to discuss on Zulip.

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@leanprover-community-bot-assistant leanprover-community-bot-assistant added the merge-conflict The PR has a merge conflict with master, and needs manual merging. (this label is managed by a bot) label Nov 23, 2024
@leanprover-community-bot-assistant leanprover-community-bot-assistant removed the merge-conflict The PR has a merge conflict with master, and needs manual merging. (this label is managed by a bot) label Nov 29, 2024
@leanprover-community-bot-assistant leanprover-community-bot-assistant added the merge-conflict The PR has a merge conflict with master, and needs manual merging. (this label is managed by a bot) label Dec 12, 2024
@leanprover-community-bot-assistant leanprover-community-bot-assistant removed the merge-conflict The PR has a merge conflict with master, and needs manual merging. (this label is managed by a bot) label Dec 12, 2024
@j-loreaux j-loreaux added the awaiting-zulip There is a Zulip discussion; the author should await and report/implement the decision reached there label Dec 22, 2024
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ocfnash commented Jan 25, 2025

In principle we could make a generalisation along these lines but I think we would need to be:

  1. very confident that it does not degrade the experience in much, much more common situation of a sesquilinear form
  2. very thorough and systematic in terms of API development

I just spent ~ 30 minutes looking over our library of results about bilinear forms / maps, sesquilinear forms and things are looking a bit unhealthy (and we even have egregious things like both https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SesquilinearForm.html#LinearMap.Nondegenerate and https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/Properties.html#LinearMap.BilinForm.Nondegenerate).

I don't think I would be comfortable making a generalisation along these lines until we tidied up quite a bit.

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Closed in favour of #20177

@mans0954 mans0954 closed this Jan 25, 2025
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mans0954 commented Jan 26, 2025

I just spent ~ 30 minutes looking over our library of results about bilinear forms / maps, sesquilinear forms and things are looking a bit unhealthy

It's a while since I last looked at it, but my impression was that a lot of the results about bilinear forms, bilinear maps and sesquilinear forms were really results about sesquilinear maps. I suspect it might be hard to untangle it all.

@YaelDillies YaelDillies deleted the mans0954/ConjSymm branch August 15, 2025 16:26
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