File: Mathlib/Topology/Algebra/Module/Star.lean

full_name: «_aux_Mathlib_Topology_Algebra_Module_Star___macroRules_term_→L⋆[_]__1»
code: @[inherit_doc]
notation:25 M " →L⋆[" R "] " M₂ => ContinuousLinearMap (starRingEnd R) M M₂
goal: Lean.Macro

full_name: _aux_Mathlib_Topology_Algebra_Module_Star___unexpand_ContinuousLinearMap_1
code: @[inherit_doc]
notation:25 M " →L⋆[" R "] " M₂ => ContinuousLinearMap (starRingEnd R) M M₂
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Topology_Algebra_Module_Star___unexpand_ContinuousLinearEquiv_1
code: @[inherit_doc]
notation:50 M " ≃L⋆[" R "] " M₂ => ContinuousLinearEquiv (starRingEnd R) M M₂
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Topology_Algebra_Module_Star___macroRules_term_≃L⋆[_]__1»
code: @[inherit_doc]
notation:50 M " ≃L⋆[" R "] " M₂ => ContinuousLinearEquiv (starRingEnd R) M M₂
goal: Lean.Macro

full_name: starL_apply
code: /-- If `A` is a topological module over a commutative `R` with compatible actions,
then `star` is a continuous semilinear equivalence. -/
@[simps!]
def starL (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L⋆[R] A 
goal: ∀ (a : A), (starL R) a = star a

full_name: starL_symm_apply
code: /-- If `A` is a topological module over a commutative `R` with compatible actions,
then `star` is a continuous semilinear equivalence. -/
@[simps!]
def starL (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L⋆[R] A 
goal: ∀ (a : A), (starL R).symm a = starAddEquiv.symm a

full_name: starL
code: /-- If `A` is a topological module over a commutative `R` with compatible actions,
then `star` is a continuous semilinear equivalence. -/
@[simps!]
def starL (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L⋆[R] A 
goal: A ≃L⋆[R] A

full_name: starL'_symm_apply
code: /-- If `A` is a topological module over a commutative `R` with trivial star and compatible actions,
then `star` is a continuous linear equivalence. -/
@[simps!]
def starL' (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L[R] A 
goal: ∀ (a : A),
  (starL' R).symm a =
    (starL R).symm ({ toFun := id, map_add' := ⋯, map_smul' := ⋯, invFun := id, left_inv := ⋯, right_inv := ⋯ }.symm a)

full_name: starL'_apply
code: /-- If `A` is a topological module over a commutative `R` with trivial star and compatible actions,
then `star` is a continuous linear equivalence. -/
@[simps!]
def starL' (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L[R] A 
goal: ∀ (a : A), (starL' R) a = star a

full_name: starL'
code: /-- If `A` is a topological module over a commutative `R` with trivial star and compatible actions,
then `star` is a continuous linear equivalence. -/
@[simps!]
def starL' (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A]
    [StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
    A ≃L[R] A 
goal: A ≃L[R] A

full_name: continuous_selfAdjointPart
code: theorem continuous_selfAdjointPart [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    Continuous (selfAdjointPart R (A := A)) 
goal: Continuous ⇑(selfAdjointPart R)

full_name: continuous_skewAdjointPart
code: theorem continuous_skewAdjointPart [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    Continuous (skewAdjointPart R (A := A)) 
goal: Continuous ⇑(skewAdjointPart R)

full_name: continuous_decomposeProdAdjoint
code: theorem continuous_decomposeProdAdjoint [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : Continuous (StarModule.decomposeProdAdjoint R A) 
goal: Continuous ⇑(StarModule.decomposeProdAdjoint R A)

full_name: continuous_decomposeProdAdjoint_symm
code: theorem continuous_decomposeProdAdjoint_symm [TopologicalAddGroup A] :
    Continuous (StarModule.decomposeProdAdjoint R A).symm 
goal: Continuous ⇑(StarModule.decomposeProdAdjoint R A).symm

full_name: selfAdjointPartL
code: /-- The self-adjoint part of an element of a star module, as a continuous linear map. -/
@[simps!]
def selfAdjointPartL [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] selfAdjoint A 
goal: A →L[R] ↥(selfAdjoint A)

full_name: selfAdjointPartL_toFun_coe
code: /-- The self-adjoint part of an element of a star module, as a continuous linear map. -/
@[simps!]
def selfAdjointPartL [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] selfAdjoint A 
goal: ↑((selfAdjointPartL R A) x) = ⅟2 • x + ⅟2 • star x

full_name: selfAdjointPartL_apply_coe
code: /-- The self-adjoint part of an element of a star module, as a continuous linear map. -/
@[simps!]
def selfAdjointPartL [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] selfAdjoint A 
goal: ↑((selfAdjointPartL R A) x) = ⅟2 • x + ⅟2 • star x

full_name: skewAdjointPartL_toFun_coe
code: /-- The skew-adjoint part of an element of a star module, as a continuous linear map. -/
@[simps!]
def skewAdjointPartL [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] skewAdjoint A 
goal: ↑((skewAdjointPartL R A) x) = ⅟2 • (x - star x)

full_name: skewAdjointPartL_apply_coe
code: /-- The skew-adjoint part of an element of a star module, as a continuous linear map. -/
@[simps!]
def skewAdjointPartL [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] skewAdjoint A 
goal: ↑((skewAdjointPartL R A) x) = ⅟2 • (x - star x)

full_name: skewAdjointPartL
code: /-- The skew-adjoint part of an element of a star module, as a continuous linear map. -/
@[simps!]
def skewAdjointPartL [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] skewAdjoint A 
goal: A →L[R] ↥(skewAdjoint A)

full_name: StarModule.decomposeProdAdjointL_symm_apply
code: /-- The decomposition of elements of a star module into their self- and skew-adjoint parts,
as a continuous linear equivalence. -/
@[simps!]
def StarModule.decomposeProdAdjointL [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : A ≃L[R] selfAdjoint A × skewAdjoint A 
goal: (StarModule.decomposeProdAdjointL R A).symm a =
  (selfAdjoint.submodule R A).subtype a.1 + (skewAdjoint.submodule R A).subtype a.2

full_name: StarModule.decomposeProdAdjointL
code: /-- The decomposition of elements of a star module into their self- and skew-adjoint parts,
as a continuous linear equivalence. -/
@[simps!]
def StarModule.decomposeProdAdjointL [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : A ≃L[R] selfAdjoint A × skewAdjoint A 
goal: A ≃L[R] ↥(selfAdjoint A) × ↥(skewAdjoint A)

full_name: StarModule.decomposeProdAdjointL_apply
code: /-- The decomposition of elements of a star module into their self- and skew-adjoint parts,
as a continuous linear equivalence. -/
@[simps!]
def StarModule.decomposeProdAdjointL [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : A ≃L[R] selfAdjoint A × skewAdjoint A 
goal: (StarModule.decomposeProdAdjointL R A) i = ((selfAdjointPart R) i, (skewAdjointPart R) i)