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Mathlib.Topology.Algebra.Module.Star

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

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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    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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      @[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
      @[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
      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] :

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

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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 = (starL R).symm ({ toFun := id, map_add' := , map_smul' := , invFun := id, left_inv := , right_inv := }.symm a)
        @[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
        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.

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

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            @[simp]
            @[simp]

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

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

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