# The star operation, bundled as a continuous star-linear equiv #

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def «term_→L⋆[_]_» :
Lean.TrailingParserDescr

Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications `M` and `M₂` will be topological modules over the topological ring `R`.

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- One or more equations did not get rendered due to their size.

Instances For

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def «term_≃L⋆[_]_» :
Lean.TrailingParserDescr

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications `M` and `M₂` will be topological modules over the topological semiring `R`.

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- One or more equations did not get rendered due to their size.

Instances For

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def starL (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
A ≃L⋆[R] A

If `A` is a topological module over a commutative `R` with compatible actions, then `star` is a continuous semilinear equivalence.

Equations

- starL R = { toLinearEquiv := starLinearEquiv R, continuous_toFun := ⋯, continuous_invFun := ⋯ }

Instances For

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@[simp]
theorem starL_symm_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] (a✝ : A) :
(starL R).symm a✝ = starAddEquiv.symm a✝
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@[simp]
theorem starL_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] (a✝ : A) :
(starL R) a✝ = star a✝
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def starL' (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
A ≃L[R] A

If `A` is a topological module over a commutative `R` with trivial star and compatible actions, then `star` is a continuous linear equivalence.

Equations

- One or more equations did not get rendered due to their size.

Instances For

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@[simp]
theorem starL'_symm_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] (a✝ : A) :
(starL' R).symm a✝ = starAddEquiv.symm ({ toFun := id, map_add' := ⋯, map_smul' := ⋯, invFun := id, left_inv := ⋯, right_inv := ⋯ }.symm a✝)
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@[simp]
theorem starL'_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] (a✝ : A) :
(starL' R) a✝ = star a✝
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theorem continuous_selfAdjointPart (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
Continuous ⇑(selfAdjointPart R)
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theorem continuous_skewAdjointPart (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
Continuous ⇑(skewAdjointPart R)
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theorem continuous_decomposeProdAdjoint (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [IsTopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A] :
Continuous ⇑(StarModule.decomposeProdAdjoint R A)
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theorem continuous_decomposeProdAdjoint_symm (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] :
Continuous ⇑(StarModule.decomposeProdAdjoint R A).symm
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def selfAdjointPartL (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
A →L[R] ↥(selfAdjoint A)

The self-adjoint part of an element of a star module, as a continuous linear map.

Equations

- selfAdjointPartL R A = { toLinearMap := selfAdjointPart R, cont := ⋯ }

Instances For

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theorem selfAdjointPartL_apply_coe (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] (x : A) :
↑((selfAdjointPartL R A) x) = ⅟ 2 • x + ⅟ 2 • star x
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def skewAdjointPartL (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
A →L[R] ↥(skewAdjoint A)

The skew-adjoint part of an element of a star module, as a continuous linear map.

Equations

- skewAdjointPartL R A = { toLinearMap := skewAdjointPart R, cont := ⋯ }

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@[simp]
theorem skewAdjointPartL_apply_coe (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] (x : A) :
↑((skewAdjointPartL R A) x) = ⅟ 2 • (x - star x)
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def StarModule.decomposeProdAdjointL (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [IsTopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A] :
A ≃L[R] ↥(selfAdjoint A) × ↥(skewAdjoint A)

The decomposition of elements of a star module into their self- and skew-adjoint parts, as a continuous linear equivalence.

Equations

- StarModule.decomposeProdAdjointL R A = { toLinearEquiv := StarModule.decomposeProdAdjoint R A, continuous_toFun := ⋯, continuous_invFun := ⋯ }

Instances For

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@[simp]
theorem StarModule.decomposeProdAdjointL_apply (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [IsTopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A] (i : A) :
(decomposeProdAdjointL R A) i = ((selfAdjointPart R) i, (skewAdjointPart R) i)
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@[simp]
theorem StarModule.decomposeProdAdjointL_symm_apply (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [IsTopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A] (a : ↥(selfAdjoint A) × ↥(skewAdjoint A)) :
(decomposeProdAdjointL R A).symm a = (selfAdjoint.submodule R A).subtype a.1 + (skewAdjoint.submodule R A).subtype a.2