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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_1) (β : Type u_2) [inst : Bornology α] [inst : Bornology β] :
Type (maxu_1u_2)

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_1) (α : outParam (Type u_2)) (β : outParam (Type u_3)) [inst : Bornology α] [inst : Bornology β] extends FunLike :
    Type (max(maxu_1u_2)u_3)

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

    You should extend this class when you extend LocallyBoundedMap.

    Instances
      theorem IsBounded.image {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] [inst : 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} [inst : Bornology α] [inst : Bornology β] [inst : 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.

      Equations
      instance instCoeTCLocallyBoundedMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Bornology α] [inst : Bornology β] [inst : LocallyBoundedMapClass F α β] :
      Equations
      Equations
      • One or more equations did not get rendered due to their size.
      theorem LocallyBoundedMap.ext {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : 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_1} {β : Type u_2} [inst : Bornology α] [inst : 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.

      Equations
      @[simp]
      theorem LocallyBoundedMap.coe_copy {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] (f : LocallyBoundedMap α β) (f' : αβ) (h : f' = f) :
      ↑(LocallyBoundedMap.copy f f' h) = f'
      theorem LocallyBoundedMap.copy_eq {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] (f : LocallyBoundedMap α β) (f' : αβ) (h : f' = f) :
      def LocallyBoundedMap.ofMapBounded {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : 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.

      Equations
      @[simp]
      theorem LocallyBoundedMap.coe_ofMapBounded {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] (f : αβ) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded sBornology.IsBounded (f '' s)} :
      @[simp]
      theorem LocallyBoundedMap.ofMapBounded_apply {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] (f : αβ) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded sBornology.IsBounded (f '' s)} (a : α) :
      def LocallyBoundedMap.id (α : Type u_1) [inst : Bornology α] :

      id as a LocallyBoundedMap.

      Equations
      @[simp]
      theorem LocallyBoundedMap.coe_id (α : Type u_1) [inst : Bornology α] :
      @[simp]
      theorem LocallyBoundedMap.id_apply {α : Type u_1} [inst : Bornology α] (a : α) :
      def LocallyBoundedMap.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : Bornology α] [inst : Bornology β] [inst : Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) :

      Composition of LocallyBoundedMaps as a LocallyBoundedMap.

      Equations
      @[simp]
      theorem LocallyBoundedMap.coe_comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : Bornology α] [inst : Bornology β] [inst : Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) :
      ↑(LocallyBoundedMap.comp f g) = f g
      @[simp]
      theorem LocallyBoundedMap.comp_apply {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : Bornology α] [inst : Bornology β] [inst : Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) (a : α) :
      ↑(LocallyBoundedMap.comp f g) a = f (g a)
      @[simp]
      theorem LocallyBoundedMap.comp_assoc {α : Type u_4} {β : Type u_3} {γ : Type u_1} {δ : Type u_2} [inst : Bornology α] [inst : Bornology β] [inst : Bornology γ] [inst : Bornology δ] (f : LocallyBoundedMap γ δ) (g : LocallyBoundedMap β γ) (h : LocallyBoundedMap α β) :
      @[simp]
      theorem LocallyBoundedMap.comp_id {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] (f : LocallyBoundedMap α β) :
      @[simp]
      theorem LocallyBoundedMap.id_comp {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst : Bornology β] (f : LocallyBoundedMap α β) :
      theorem LocallyBoundedMap.cancel_right {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : Bornology α] [inst : Bornology β] [inst : Bornology γ] {g₁ : LocallyBoundedMap β γ} {g₂ : LocallyBoundedMap β γ} {f : LocallyBoundedMap α β} (hf : Function.Surjective f) :
      theorem LocallyBoundedMap.cancel_left {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : Bornology α] [inst : Bornology β] [inst : Bornology γ] {g : LocallyBoundedMap β γ} {f₁ : LocallyBoundedMap α β} {f₂ : LocallyBoundedMap α β} (hg : Function.Injective g) :