refactor(LinearAlgebra/SesquilinearForm): Hermitian sesquilinear maps#18406
refactor(LinearAlgebra/SesquilinearForm): Hermitian sesquilinear maps#18406
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PR summary c381040244Import changes for modified filesNo significant changes to the import graph Import changes for all files
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I'm surprised we don't appear to need the condition: |
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CC: @eric-wieser |
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| 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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In principle we could make a generalisation along these lines but I think we would need to be:
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 |
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. |
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.