Documentation

Mathlib.Topology.Bornology.Hom

Locally bounded maps #

This file defines locally bounded maps between bornologies.

We use the FunLike design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.

Types of morphisms #

Typeclasses #

structure LocallyBoundedMap (α : Type u_6) (β : Type u_7) [Bornology α] [Bornology β] :
Type (max u_6 u_7)

The type of bounded maps from α to β, the maps which send a bounded set to a bounded set.

Instances For
    class LocallyBoundedMapClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [Bornology α] [Bornology β] extends FunLike :
    Type (max (max u_6 u_7) u_8)

    LocallyBoundedMapClass F α β states that F is a type of bounded maps.

    You should extend this class when you extend LocallyBoundedMap.

    Instances
      theorem Bornology.IsBounded.image {F : Type u_1} {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F) {s : Set α} (hs : Bornology.IsBounded s) :
      def LocallyBoundedMapClass.toLocallyBoundedMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F) :

      Turn an element of a type F satisfying LocallyBoundedMapClass F α β into an actual LocallyBoundedMap. This is declared as the default coercion from F to LocallyBoundedMap α β.

      Instances For
        instance instCoeTCLocallyBoundedMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] :
        theorem LocallyBoundedMap.ext {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] {f : LocallyBoundedMap α β} {g : LocallyBoundedMap α β} (h : ∀ (a : α), f a = g a) :
        f = g
        def LocallyBoundedMap.copy {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : αβ) (h : f' = f) :

        Copy of a LocallyBoundedMap with a new toFun equal to the old one. Useful to fix definitional equalities.

        Instances For
          @[simp]
          theorem LocallyBoundedMap.coe_copy {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : αβ) (h : f' = f) :
          ↑(LocallyBoundedMap.copy f f' h) = f'
          theorem LocallyBoundedMap.copy_eq {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : αβ) (h : f' = f) :
          def LocallyBoundedMap.ofMapBounded {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : αβ) (h : ∀ ⦃s : Set α⦄, Bornology.IsBounded sBornology.IsBounded (f '' s)) :

          Construct a LocallyBoundedMap from the fact that the function maps bounded sets to bounded sets.

          Instances For
            @[simp]
            theorem LocallyBoundedMap.coe_ofMapBounded {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : αβ) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded sBornology.IsBounded (f '' s)} :
            @[simp]
            theorem LocallyBoundedMap.ofMapBounded_apply {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : αβ) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded sBornology.IsBounded (f '' s)} (a : α) :

            id as a LocallyBoundedMap.

            Instances For
              @[simp]
              theorem LocallyBoundedMap.coe_id (α : Type u_2) [Bornology α] :
              @[simp]
              theorem LocallyBoundedMap.id_apply {α : Type u_2} [Bornology α] (a : α) :
              def LocallyBoundedMap.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) :

              Composition of LocallyBoundedMaps as a LocallyBoundedMap.

              Instances For
                @[simp]
                theorem LocallyBoundedMap.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) :
                ↑(LocallyBoundedMap.comp f g) = f g
                @[simp]
                theorem LocallyBoundedMap.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) (a : α) :
                ↑(LocallyBoundedMap.comp f g) a = f (g a)
                @[simp]
                @[simp]
                @[simp]
                theorem LocallyBoundedMap.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] {g₁ : LocallyBoundedMap β γ} {g₂ : LocallyBoundedMap β γ} {f : LocallyBoundedMap α β} (hf : Function.Surjective f) :
                @[simp]
                theorem LocallyBoundedMap.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] {g : LocallyBoundedMap β γ} {f₁ : LocallyBoundedMap α β} {f₂ : LocallyBoundedMap α β} (hg : Function.Injective g) :