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

@[simp]
theorem starL_symm_apply (R : Type u_1) {A : Type u_2} [] [] [] [] [Module R A] [] [] [] :
@[simp]
theorem starL_apply (R : Type u_1) {A : Type u_2} [] [] [] [] [Module R A] [] [] [] :
∀ (a : A), () a = star a
def starL (R : Type u_1) {A : Type u_2} [] [] [] [] [Module R A] [] [] [] :

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

Equations
• = { toLinearEquiv := , continuous_toFun := , continuous_invFun := }
Instances For
@[simp]
theorem starL'_symm_apply (R : Type u_1) {A : Type u_2} [] [] [] [] [] [Module R A] [] [] [] :
∀ (a : A), a = (LinearEquiv.symm ().toLinearEquiv) ((LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := id, map_add' := }, map_smul' := }, invFun := id, left_inv := , right_inv := }) a)
@[simp]
theorem starL'_apply (R : Type u_1) {A : Type u_2} [] [] [] [] [] [Module R A] [] [] [] :
∀ (a : A), () a = star a
def starL' (R : Type u_1) {A : Type u_2} [] [] [] [] [] [Module R 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
theorem continuous_selfAdjointPart (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] :
theorem continuous_skewAdjointPart (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] :
theorem continuous_decomposeProdAdjoint (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] :
theorem continuous_decomposeProdAdjoint_symm (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] :
@[simp]
theorem selfAdjointPartL_apply_coe (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] (x : A) :
(() x) = 2 x + 2 star x
@[simp]
theorem selfAdjointPartL_toFun_coe (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] (x : A) :
(() x) = 2 x + 2 star x
def selfAdjointPartL (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] :
A →L[R] ()

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

Equations
• = { toLinearMap := , cont := }
Instances For
@[simp]
theorem skewAdjointPartL_apply_coe (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] (x : A) :
(() x) = 2 (x - star x)
@[simp]
theorem skewAdjointPartL_toFun_coe (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] (x : A) :
(() x) = 2 (x - star x)
def skewAdjointPartL (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] [] :
A →L[R] ()

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

Equations
• = { toLinearMap := , cont := }
Instances For
@[simp]
theorem StarModule.decomposeProdAdjointL_symm_apply (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] (a : () × ()) :
= a.1 + a.2
@[simp]
theorem StarModule.decomposeProdAdjointL_apply (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] (i : A) :
= (() i, () i)
def StarModule.decomposeProdAdjointL (R : Type u_1) (A : Type u_2) [] [] [] [] [Module R A] [] [] [] [] [] [] :
A ≃L[R] () × ()

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

Equations
• = { toLinearEquiv := , continuous_toFun := , continuous_invFun := }
Instances For