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Mathlib.Topology.Algebra.ContinuousMonoidHom

Continuous Monoid Homs #

This file defines the space of continuous homomorphisms between two topological groups.

Main definitions #

structure ContinuousAddMonoidHom (A : Type u_7) (B : Type u_8) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] extends AddMonoidHom :
Type (max u_7 u_8)

The type of continuous additive monoid homomorphisms from A to B.

  • toFun : AB
  • map_zero' : (self.toAddMonoidHom).toFun 0 = 0
  • map_add' : ∀ (x y : A), (self.toAddMonoidHom).toFun (x + y) = (self.toAddMonoidHom).toFun x + (self.toAddMonoidHom).toFun y
  • continuous_toFun : Continuous (self.toAddMonoidHom).toFun

    Proof of continuity of the Hom.

Instances For
    theorem ContinuousAddMonoidHom.continuous_toFun {A : Type u_7} {B : Type u_8} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : ContinuousAddMonoidHom A B) :
    Continuous (self.toAddMonoidHom).toFun

    Proof of continuity of the Hom.

    structure ContinuousMonoidHom (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] extends MonoidHom :
    Type (max u_2 u_3)

    The type of continuous monoid homomorphisms from A to B.

    When possible, instead of parametrizing results over (f : ContinuousMonoidHom A B), you should parametrize over (F : Type*) [ContinuousMonoidHomClass F A B] (f : F).

    When you extend this structure, make sure to extend ContinuousAddMonoidHomClass.

    • toFun : AB
    • map_one' : (self.toMonoidHom).toFun 1 = 1
    • map_mul' : ∀ (x y : A), (self.toMonoidHom).toFun (x * y) = (self.toMonoidHom).toFun x * (self.toMonoidHom).toFun y
    • continuous_toFun : Continuous (self.toMonoidHom).toFun

      Proof of continuity of the Hom.

    Instances For
      theorem ContinuousMonoidHom.continuous_toFun {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : ContinuousMonoidHom A B) :
      Continuous (self.toMonoidHom).toFun

      Proof of continuity of the Hom.

      ContinuousAddMonoidHomClass F A B states that F is a type of continuous additive monoid homomorphisms.

      You should also extend this typeclass when you extend ContinuousAddMonoidHom.

      • map_add : ∀ (f : F) (x y : A), f (x + y) = f x + f y
      • map_zero : ∀ (f : F), f 0 = 0
      • map_continuous : ∀ (f : F), Continuous f

        Proof of the continuity of the map.

      Instances

        Proof of the continuity of the map.

        class ContinuousMonoidHomClass (F : Type u_1) (A : outParam (Type u_7)) (B : outParam (Type u_8)) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] extends MonoidHomClass :

        ContinuousMonoidHomClass F A B states that F is a type of continuous additive monoid homomorphisms.

        You should also extend this typeclass when you extend ContinuousMonoidHom.

        • map_mul : ∀ (f : F) (x y : A), f (x * y) = f x * f y
        • map_one : ∀ (f : F), f 1 = 1
        • map_continuous : ∀ (f : F), Continuous f

          Proof of the continuity of the map.

        Instances
          theorem ContinuousMonoidHomClass.map_continuous {F : Type u_1} {A : outParam (Type u_7)} {B : outParam (Type u_8)} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] [self : ContinuousMonoidHomClass F A B] (f : F) :

          Proof of the continuity of the map.

          @[instance 100]
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          @[instance 100]
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          • ContinuousAddMonoidHom.funLike = { coe := fun (f : ContinuousAddMonoidHom A B) => (f.toAddMonoidHom).toFun, coe_injective' := }
          theorem ContinuousAddMonoidHom.funLike.proof_1 {A : Type u_2} {B : Type u_1} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A B) (h : (fun (f : ContinuousAddMonoidHom A B) => (f.toAddMonoidHom).toFun) f = (fun (f : ContinuousAddMonoidHom A B) => (f.toAddMonoidHom).toFun) g) :
          f = g
          Equations
          • ContinuousMonoidHom.funLike = { coe := fun (f : ContinuousMonoidHom A B) => (f.toMonoidHom).toFun, coe_injective' := }
          theorem ContinuousAddMonoidHom.ext {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] {f : ContinuousAddMonoidHom A B} {g : ContinuousAddMonoidHom A B} (h : ∀ (x : A), f x = g x) :
          f = g
          theorem ContinuousMonoidHom.ext {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {f : ContinuousMonoidHom A B} {g : ContinuousMonoidHom A B} (h : ∀ (x : A), f x = g x) :
          f = g

          Reinterpret a ContinuousAddMonoidHom as a ContinuousMap.

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          • f.toContinuousMap = { toFun := (f.toAddMonoidHom).toFun, continuous_toFun := }
          Instances For

            Reinterpret a ContinuousMonoidHom as a ContinuousMap.

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            • f.toContinuousMap = { toFun := (f.toMonoidHom).toFun, continuous_toFun := }
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              theorem ContinuousAddMonoidHom.toContinuousMap_injective {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] :
              Function.Injective ContinuousAddMonoidHom.toContinuousMap
              theorem ContinuousMonoidHom.toContinuousMap_injective {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] :
              Function.Injective ContinuousMonoidHom.toContinuousMap

              Construct a ContinuousAddMonoidHom from a Continuous AddMonoidHom.

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                def ContinuousMonoidHom.mk' {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →* B) (hf : Continuous f) :

                Construct a ContinuousMonoidHom from a Continuous MonoidHom.

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                  theorem ContinuousAddMonoidHom.comp.proof_1 {A : Type u_1} {B : Type u_3} {C : Type u_2} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousAddMonoidHom B C) (f : ContinuousAddMonoidHom A B) :
                  Continuous ((g.toAddMonoidHom).toFun fun (x : A) => f.toAddMonoidHom x)

                  Composition of two continuous homomorphisms.

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                    @[simp]
                    theorem ContinuousAddMonoidHom.comp_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousAddMonoidHom B C) (f : ContinuousAddMonoidHom A B) :
                    ∀ (a : A), (g.comp f) a = g.toAddMonoidHom (f.toAddMonoidHom a)
                    @[simp]
                    theorem ContinuousMonoidHom.comp_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) :
                    ∀ (a : A), (g.comp f) a = g.toMonoidHom (f.toMonoidHom a)

                    Composition of two continuous homomorphisms.

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                      Product of two continuous homomorphisms on the same space.

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                        theorem ContinuousAddMonoidHom.sum.proof_1 {A : Type u_1} {B : Type u_3} {C : Type u_2} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A C) :
                        Continuous fun (x : A) => ((f.toAddMonoidHom).toFun x, g.toAddMonoidHom x)
                        @[simp]
                        theorem ContinuousAddMonoidHom.sum_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A C) (i : A) :
                        (f.sum g) i = (f.toAddMonoidHom i, g.toAddMonoidHom i)
                        @[simp]
                        theorem ContinuousMonoidHom.prod_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom A B) (g : ContinuousMonoidHom A C) (i : A) :
                        (f.prod g) i = (f.toMonoidHom i, g.toMonoidHom i)

                        Product of two continuous homomorphisms on the same space.

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                          Product of two continuous homomorphisms on different spaces.

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                            theorem ContinuousAddMonoidHom.sum_map.proof_1 {A : Type u_1} {B : Type u_2} {C : Type u_4} {D : Type u_3} [AddMonoid A] [AddMonoid B] [AddMonoid C] [AddMonoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousAddMonoidHom A C) (g : ContinuousAddMonoidHom B D) :
                            Continuous fun (p : A × B) => ((f.toAddMonoidHom).toFun p.1, (g.toAddMonoidHom).1 p.2)
                            @[simp]
                            theorem ContinuousAddMonoidHom.sum_map_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [AddMonoid A] [AddMonoid B] [AddMonoid C] [AddMonoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousAddMonoidHom A C) (g : ContinuousAddMonoidHom B D) (i : A × B) :
                            (f.sum_map g) i = (f.toAddMonoidHom i.1, g.toAddMonoidHom i.2)
                            @[simp]
                            theorem ContinuousMonoidHom.prod_map_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid A] [Monoid B] [Monoid C] [Monoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) (i : A × B) :
                            (f.prod_map g) i = (f.toMonoidHom i.1, g.toMonoidHom i.2)

                            Product of two continuous homomorphisms on different spaces.

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                              The trivial continuous homomorphism.

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                                theorem ContinuousAddMonoidHom.zero.proof_1 (A : Type u_1) (B : Type u_2) [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                Continuous fun (x : A) => AddZeroClass.toZero.1
                                @[simp]
                                theorem ContinuousMonoidHom.one_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                ∀ (x : A), (ContinuousMonoidHom.one A B) x = 1
                                @[simp]

                                The trivial continuous homomorphism.

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                                  The identity continuous homomorphism.

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                                    @[simp]

                                    The identity continuous homomorphism.

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                                      The continuous homomorphism given by projection onto the first factor.

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                                        @[simp]
                                        theorem ContinuousAddMonoidHom.fst_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) :
                                        (ContinuousAddMonoidHom.fst A B) self = self.1
                                        @[simp]
                                        theorem ContinuousMonoidHom.fst_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) :
                                        (ContinuousMonoidHom.fst A B) self = self.1

                                        The continuous homomorphism given by projection onto the first factor.

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                                          The continuous homomorphism given by projection onto the second factor.

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                                            @[simp]
                                            theorem ContinuousAddMonoidHom.snd_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) :
                                            (ContinuousAddMonoidHom.snd A B) self = self.2
                                            @[simp]
                                            theorem ContinuousMonoidHom.snd_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) :
                                            (ContinuousMonoidHom.snd A B) self = self.2

                                            The continuous homomorphism given by projection onto the second factor.

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                                              The continuous homomorphism given by inclusion of the first factor.

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                                                @[simp]
                                                theorem ContinuousMonoidHom.inl_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A) :
                                                (ContinuousMonoidHom.inl A B) i = ((ContinuousMonoidHom.id A).toMonoidHom i, (ContinuousMonoidHom.one A B).toMonoidHom i)
                                                @[simp]
                                                theorem ContinuousAddMonoidHom.inl_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A) :
                                                (ContinuousAddMonoidHom.inl A B) i = ((ContinuousAddMonoidHom.id A).toAddMonoidHom i, (ContinuousAddMonoidHom.zero A B).toAddMonoidHom i)

                                                The continuous homomorphism given by inclusion of the first factor.

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                                                  The continuous homomorphism given by inclusion of the second factor.

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                                                    @[simp]
                                                    theorem ContinuousMonoidHom.inr_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B) :
                                                    (ContinuousMonoidHom.inr A B) i = ((ContinuousMonoidHom.one B A).toMonoidHom i, (ContinuousMonoidHom.id B).toMonoidHom i)
                                                    @[simp]
                                                    theorem ContinuousAddMonoidHom.inr_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B) :
                                                    (ContinuousAddMonoidHom.inr A B) i = ((ContinuousAddMonoidHom.zero B A).toAddMonoidHom i, (ContinuousAddMonoidHom.id B).toAddMonoidHom i)

                                                    The continuous homomorphism given by inclusion of the second factor.

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                                                      The continuous homomorphism given by the diagonal embedding.

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                                                        @[simp]
                                                        theorem ContinuousMonoidHom.diag_toFun (A : Type u_2) [Monoid A] [TopologicalSpace A] (i : A) :
                                                        (ContinuousMonoidHom.diag A) i = ((ContinuousMonoidHom.id A).toMonoidHom i, (ContinuousMonoidHom.id A).toMonoidHom i)
                                                        @[simp]

                                                        The continuous homomorphism given by the diagonal embedding.

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                                                          The continuous homomorphism given by swapping components.

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                                                            @[simp]
                                                            theorem ContinuousAddMonoidHom.swap_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B) :
                                                            (ContinuousAddMonoidHom.swap A B) i = ((ContinuousAddMonoidHom.snd A B).toAddMonoidHom i, (ContinuousAddMonoidHom.fst A B).toAddMonoidHom i)
                                                            @[simp]
                                                            theorem ContinuousMonoidHom.swap_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B) :
                                                            (ContinuousMonoidHom.swap A B) i = ((ContinuousMonoidHom.snd A B).toMonoidHom i, (ContinuousMonoidHom.fst A B).toMonoidHom i)

                                                            The continuous homomorphism given by swapping components.

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                                                              The continuous homomorphism given by addition.

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                                                                @[simp]
                                                                theorem ContinuousMonoidHom.mul_toFun (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E] :
                                                                ∀ (a : E × E), (ContinuousMonoidHom.mul E) a = a.1 * a.2

                                                                The continuous homomorphism given by multiplication.

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                                                                  The continuous homomorphism given by negation.

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                                                                    The continuous homomorphism given by inversion.

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                                                                      Coproduct of two continuous homomorphisms to the same space.

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                                                                        @[simp]
                                                                        theorem ContinuousMonoidHom.coprod_toFun {A : Type u_2} {B : Type u_3} {E : Type u_6} [Monoid A] [Monoid B] [CommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalGroup E] (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) :
                                                                        ∀ (a : A × B), (f.coprod g) a = (ContinuousMonoidHom.mul E).toMonoidHom ((f.prod_map g).toMonoidHom a)
                                                                        @[simp]
                                                                        theorem ContinuousAddMonoidHom.coprod_toFun {A : Type u_2} {B : Type u_3} {E : Type u_6} [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom B E) :
                                                                        ∀ (a : A × B), (f.coprod g) a = (ContinuousAddMonoidHom.add E).toAddMonoidHom ((f.sum_map g).toAddMonoidHom a)

                                                                        Coproduct of two continuous homomorphisms to the same space.

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                                                                          theorem ContinuousAddMonoidHom.instAddCommGroup.proof_5 {A : Type u_2} {E : Type u_1} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] :
                                                                          ∀ (n : ) (x : ContinuousAddMonoidHom A E), nsmulRec (n + 1) x = nsmulRec (n + 1) x
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                                                                          theorem ContinuousAddMonoidHom.instAddCommGroup.proof_11 {A : Type u_2} {E : Type u_1} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] :
                                                                          ∀ (n : ) (a : ContinuousAddMonoidHom A E), zsmulRec (Int.ofNat n.succ) a = zsmulRec (Int.ofNat n.succ) a
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                                                                          • ContinuousAddMonoidHom.instTopologicalSpace = TopologicalSpace.induced ContinuousAddMonoidHom.toContinuousMap ContinuousMap.compactOpen
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                                                                          • ContinuousMonoidHom.instTopologicalSpace = TopologicalSpace.induced ContinuousMonoidHom.toContinuousMap ContinuousMap.compactOpen
                                                                          theorem ContinuousAddMonoidHom.inducing_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                                                          Inducing ContinuousAddMonoidHom.toContinuousMap
                                                                          theorem ContinuousMonoidHom.inducing_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                                                          Inducing ContinuousMonoidHom.toContinuousMap
                                                                          theorem ContinuousAddMonoidHom.embedding_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                                                          Embedding ContinuousAddMonoidHom.toContinuousMap
                                                                          theorem ContinuousMonoidHom.embedding_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                                                          Embedding ContinuousMonoidHom.toContinuousMap
                                                                          theorem ContinuousAddMonoidHom.range_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                                                          Set.range ContinuousAddMonoidHom.toContinuousMap = {f : C(A, B) | f 0 = 0 ∀ (x y : A), f (x + y) = f x + f y}
                                                                          theorem ContinuousMonoidHom.range_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] :
                                                                          Set.range ContinuousMonoidHom.toContinuousMap = {f : C(A, B) | f 1 = 1 ∀ (x y : A), f (x * y) = f x * f y}
                                                                          theorem ContinuousMonoidHom.closedEmbedding_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [ContinuousMul B] [T2Space B] :
                                                                          ClosedEmbedding ContinuousMonoidHom.toContinuousMap
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                                                                          theorem ContinuousMonoidHom.continuous_of_continuous_uncurry {B : Type u_3} {C : Type u_4} [Monoid B] [Monoid C] [TopologicalSpace B] [TopologicalSpace C] {A : Type u_7} [TopologicalSpace A] (f : AContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun (x : A) (y : B) => (f x) y)) :
                                                                          theorem ContinuousAddMonoidHom.compLeft.proof_2 {A : Type u_3} {B : Type u_2} (E : Type u_1) [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A B) (_g : ContinuousAddMonoidHom B E) (_h : ContinuousAddMonoidHom B E) :
                                                                          { toFun := fun (g : ContinuousAddMonoidHom B E) => g.comp f, map_zero' := }.toFun (_g + _h) = { toFun := fun (g : ContinuousAddMonoidHom B E) => g.comp f, map_zero' := }.toFun (_g + _h)
                                                                          theorem ContinuousAddMonoidHom.compLeft.proof_1 {A : Type u_1} {B : Type u_3} (E : Type u_2) [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A B) :
                                                                          (fun (g : ContinuousAddMonoidHom B E) => g.comp f) 0 = (fun (g : ContinuousAddMonoidHom B E) => g.comp f) 0

                                                                          ContinuousAddMonoidHom _ f is a functor.

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                                                                            ContinuousMonoidHom _ f is a functor.

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                                                                              ContinuousAddMonoidHom f _ is a functor.

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                                                                                theorem ContinuousAddMonoidHom.compRight.proof_2 (A : Type u_2) {E : Type u_3} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] {B : Type u_1} [AddCommGroup B] [TopologicalSpace B] [TopologicalAddGroup B] (f : ContinuousAddMonoidHom B E) (g : ContinuousAddMonoidHom A B) (h : ContinuousAddMonoidHom A B) :
                                                                                { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := }.toFun (g + h) = { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := }.toFun g + { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := }.toFun h

                                                                                ContinuousMonoidHom f _ is a functor.

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