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Mathlib.Topology.ContinuousFunction.Bounded

Bounded continuous functions #

The type of bounded continuous functions taking values in a metric space, with the uniform distance.

structure BoundedContinuousFunction (α : Type u) (β : Type v) [TopologicalSpace α] [PseudoMetricSpace β] extends ContinuousMap :
Type (max u v)

α →ᵇ β is the type of bounded continuous functions α → β from a topological space to a metric space.

When possible, instead of parametrizing results over (f : α →ᵇ β), you should parametrize over (F : Type*) [BoundedContinuousMapClass F α β] (f : F).

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

  • toFun : αβ
  • continuous_toFun : Continuous self.toFun
  • map_bounded' : ∃ (C : ), ∀ (x y : α), dist (self.toFun x) (self.toFun y) C
Instances For
    theorem BoundedContinuousFunction.map_bounded' {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (self : BoundedContinuousFunction α β) :
    ∃ (C : ), ∀ (x y : α), dist (self.toFun x) (self.toFun y) C
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    • One or more equations did not get rendered due to their size.
    Instances For
      class BoundedContinuousMapClass (F : Type u_2) (α : outParam (Type u_3)) (β : outParam (Type u_4)) [TopologicalSpace α] [PseudoMetricSpace β] [FunLike F α β] extends ContinuousMapClass :

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

      You should also extend this typeclass when you extend BoundedContinuousFunction.

      • map_continuous : ∀ (f : F), Continuous f
      • map_bounded : ∀ (f : F), ∃ (C : ), ∀ (x y : α), dist (f x) (f y) C
      Instances
        theorem BoundedContinuousMapClass.map_bounded {F : Type u_2} {α : outParam (Type u_3)} {β : outParam (Type u_4)} [TopologicalSpace α] [PseudoMetricSpace β] [FunLike F α β] [self : BoundedContinuousMapClass F α β] (f : F) :
        ∃ (C : ), ∀ (x y : α), dist (f x) (f y) C
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        • BoundedContinuousFunction.instCoeTC = { coe := fun (f : F) => { toFun := f, continuous_toFun := , map_bounded' := } }
        @[simp]
        theorem BoundedContinuousFunction.coe_to_continuous_fun {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) :
        f.toContinuousMap = f

        See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

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          theorem BoundedContinuousFunction.bounded {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) :
          ∃ (C : ), ∀ (x y : α), dist (f x) (f y) C
          theorem BoundedContinuousFunction.ext_iff {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
          f = g ∀ (x : α), f x = g x
          theorem BoundedContinuousFunction.ext {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} (h : ∀ (x : α), f x = g x) :
          f = g
          def BoundedContinuousFunction.mkOfBound {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : C(α, β)) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) :

          A continuous function with an explicit bound is a bounded continuous function.

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            @[simp]
            theorem BoundedContinuousFunction.mkOfBound_coe {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : C(α, β)} {C : } {h : ∀ (x y : α), dist (f x) (f y) C} :

            A continuous function on a compact space is automatically a bounded continuous function.

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              @[simp]
              theorem BoundedContinuousFunction.mkOfDiscrete_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [DiscreteTopology α] (f : αβ) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) :
              ∀ (a : α), (BoundedContinuousFunction.mkOfDiscrete f C h) a = f a
              @[simp]
              theorem BoundedContinuousFunction.mkOfDiscrete_toFun {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [DiscreteTopology α] (f : αβ) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) :
              ∀ (a : α), (BoundedContinuousFunction.mkOfDiscrete f C h) a = f a
              def BoundedContinuousFunction.mkOfDiscrete {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [DiscreteTopology α] (f : αβ) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) :

              If a function is bounded on a discrete space, it is automatically continuous, and therefore gives rise to an element of the type of bounded continuous functions.

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                The uniform distance between two bounded continuous functions.

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                theorem BoundedContinuousFunction.dist_eq {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
                dist f g = sInf {C : | 0 C ∀ (x : α), dist (f x) (g x) C}
                theorem BoundedContinuousFunction.dist_set_exists {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
                ∃ (C : ), 0 C ∀ (x : α), dist (f x) (g x) C

                The pointwise distance is controlled by the distance between functions, by definition.

                theorem BoundedContinuousFunction.dist_le {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} {C : } (C0 : 0 C) :
                dist f g C ∀ (x : α), dist (f x) (g x) C

                The distance between two functions is controlled by the supremum of the pointwise distances.

                theorem BoundedContinuousFunction.dist_le_iff_of_nonempty {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} {C : } [Nonempty α] :
                dist f g C ∀ (x : α), dist (f x) (g x) C
                theorem BoundedContinuousFunction.dist_lt_of_nonempty_compact {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} {C : } [Nonempty α] [CompactSpace α] (w : ∀ (x : α), dist (f x) (g x) < C) :
                dist f g < C
                theorem BoundedContinuousFunction.dist_lt_iff_of_compact {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} {C : } [CompactSpace α] (C0 : 0 < C) :
                dist f g < C ∀ (x : α), dist (f x) (g x) < C

                The type of bounded continuous functions, with the uniform distance, is a pseudometric space.

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                • One or more equations did not get rendered due to their size.

                The type of bounded continuous functions, with the uniform distance, is a metric space.

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                theorem BoundedContinuousFunction.nndist_eq {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
                nndist f g = sInf {C : NNReal | ∀ (x : α), nndist (f x) (g x) C}
                theorem BoundedContinuousFunction.nndist_set_exists {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
                ∃ (C : NNReal), ∀ (x : α), nndist (f x) (g x) C

                On an empty space, bounded continuous functions are at distance 0.

                theorem BoundedContinuousFunction.dist_eq_iSup {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
                dist f g = ⨆ (x : α), dist (f x) (g x)
                theorem BoundedContinuousFunction.nndist_eq_iSup {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : BoundedContinuousFunction α β} {g : BoundedContinuousFunction α β} :
                nndist f g = ⨆ (x : α), nndist (f x) (g x)
                theorem BoundedContinuousFunction.tendsto_iff_tendstoUniformly {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {ι : Type u_2} {F : ιBoundedContinuousFunction α β} {f : BoundedContinuousFunction α β} {l : Filter ι} :
                Filter.Tendsto F l (nhds f) TendstoUniformly (fun (i : ι) => (F i)) (⇑f) l
                theorem BoundedContinuousFunction.inducing_coeFn {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] :
                Inducing (UniformFun.ofFun DFunLike.coe)

                The topology on α →ᵇ β is exactly the topology induced by the natural map to α →ᵤ β.

                theorem BoundedContinuousFunction.embedding_coeFn {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] :
                Embedding (UniformFun.ofFun DFunLike.coe)
                @[simp]
                theorem BoundedContinuousFunction.const_apply (α : Type u) {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (b : β) :
                (BoundedContinuousFunction.const α b) = fun (x : α) => b
                @[simp]
                theorem BoundedContinuousFunction.const_toFun (α : Type u) {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (b : β) :
                (BoundedContinuousFunction.const α b) = fun (x : α) => b

                Constant as a continuous bounded function.

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                  If the target space is inhabited, so is the space of bounded continuous functions.

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                  When x is fixed, (f : α →ᵇ β) ↦ f x is continuous.

                  The evaluation map is continuous, as a joint function of u and x.

                  Bounded continuous functions taking values in a complete space form a complete space.

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                  • =

                  Composition of a bounded continuous function and a continuous function.

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                  • f.compContinuous g = { toContinuousMap := f.comp g, map_bounded' := }
                  Instances For
                    @[simp]
                    theorem BoundedContinuousFunction.coe_compContinuous {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (f : BoundedContinuousFunction α β) (g : C(δ, α)) :
                    (f.compContinuous g) = f g
                    @[simp]
                    theorem BoundedContinuousFunction.compContinuous_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (f : BoundedContinuousFunction α β) (g : C(δ, α)) (x : δ) :
                    (f.compContinuous g) x = f (g x)
                    theorem BoundedContinuousFunction.lipschitz_compContinuous {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (g : C(δ, α)) :
                    LipschitzWith 1 fun (f : BoundedContinuousFunction α β) => f.compContinuous g
                    theorem BoundedContinuousFunction.continuous_compContinuous {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (g : C(δ, α)) :
                    Continuous fun (f : BoundedContinuousFunction α β) => f.compContinuous g

                    Restrict a bounded continuous function to a set.

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                      @[simp]
                      theorem BoundedContinuousFunction.coe_restrict {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) (s : Set α) :
                      (f.restrict s) = f Subtype.val
                      @[simp]
                      theorem BoundedContinuousFunction.restrict_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) (s : Set α) (x : s) :
                      (f.restrict s) x = f x

                      Composition (in the target) of a bounded continuous function with a Lipschitz map again gives a bounded continuous function.

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                        The composition operator (in the target) with a Lipschitz map is Lipschitz.

                        The composition operator (in the target) with a Lipschitz map is uniformly continuous.

                        The composition operator (in the target) with a Lipschitz map is continuous.

                        def BoundedContinuousFunction.codRestrict {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (s : Set β) (f : BoundedContinuousFunction α β) (H : ∀ (x : α), f x s) :

                        Restriction (in the target) of a bounded continuous function taking values in a subset.

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                          A version of Function.extend for bounded continuous maps. We assume that the domain has discrete topology, so we only need to verify boundedness.

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                            @[simp]
                            theorem BoundedContinuousFunction.extend_apply' {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] {f : α δ} {x : δ} (hx : xSet.range f) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β) :
                            @[simp]
                            theorem BoundedContinuousFunction.dist_extend_extend {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α δ) (g₁ : BoundedContinuousFunction α β) (g₂ : BoundedContinuousFunction α β) (h₁ : BoundedContinuousFunction δ β) (h₂ : BoundedContinuousFunction δ β) :
                            dist (BoundedContinuousFunction.extend f g₁ h₁) (BoundedContinuousFunction.extend f g₂ h₂) = max (dist g₁ g₂) (dist (h₁.restrict (Set.range f)) (h₂.restrict (Set.range f)))
                            theorem BoundedContinuousFunction.arzela_ascoli₁ {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [CompactSpace β] (A : Set (BoundedContinuousFunction α β)) (closed : IsClosed A) (H : Equicontinuous fun (x : A) => x) :

                            First version, with pointwise equicontinuity and range in a compact space.

                            theorem BoundedContinuousFunction.arzela_ascoli₂ {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] (s : Set β) (hs : IsCompact s) (A : Set (BoundedContinuousFunction α β)) (closed : IsClosed A) (in_s : ∀ (f : BoundedContinuousFunction α β) (x : α), f Af x s) (H : Equicontinuous fun (x : A) => x) :

                            Second version, with pointwise equicontinuity and range in a compact subset.

                            theorem BoundedContinuousFunction.arzela_ascoli {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [T2Space β] (s : Set β) (hs : IsCompact s) (A : Set (BoundedContinuousFunction α β)) (in_s : ∀ (f : BoundedContinuousFunction α β) (x : α), f Af x s) (H : Equicontinuous fun (x : A) => x) :

                            Third (main) version, with pointwise equicontinuity and range in a compact subset, but without closedness. The closure is then compact.

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                            @[simp]
                            theorem BoundedContinuousFunction.coe_zero {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] :
                            0 = 0
                            @[simp]
                            theorem BoundedContinuousFunction.coe_one {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [One β] :
                            1 = 1
                            theorem BoundedContinuousFunction.forall_coe_zero_iff_zero {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : BoundedContinuousFunction α β) :
                            (∀ (x : α), f x = 0) f = 0
                            theorem BoundedContinuousFunction.forall_coe_one_iff_one {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [One β] (f : BoundedContinuousFunction α β) :
                            (∀ (x : α), f x = 1) f = 1

                            The pointwise sum of two bounded continuous functions is again bounded continuous.

                            Equations
                            • BoundedContinuousFunction.instAdd = { add := fun (f g : BoundedContinuousFunction α β) => { toFun := fun (x : α) => f x + g x, continuous_toFun := , map_bounded' := } }
                            @[simp]
                            theorem BoundedContinuousFunction.coe_add {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f : BoundedContinuousFunction α β) (g : BoundedContinuousFunction α β) :
                            (f + g) = f + g
                            theorem BoundedContinuousFunction.add_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f : BoundedContinuousFunction α β) (g : BoundedContinuousFunction α β) {x : α} :
                            (f + g) x = f x + g x
                            theorem BoundedContinuousFunction.add_compContinuous {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f : BoundedContinuousFunction α β) (g : BoundedContinuousFunction α β) [TopologicalSpace γ] (h : C(γ, α)) :
                            (g + f).compContinuous h = g.compContinuous h + f.compContinuous h
                            @[simp]
                            Equations
                            • BoundedContinuousFunction.instSMulNat = { smul := fun (n : ) (f : BoundedContinuousFunction α β) => { toContinuousMap := n f.toContinuousMap, map_bounded' := } }
                            @[simp]
                            theorem BoundedContinuousFunction.coe_nsmul {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (r : ) (f : BoundedContinuousFunction α β) :
                            (r f) = r f
                            @[simp]
                            theorem BoundedContinuousFunction.nsmul_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (r : ) (f : BoundedContinuousFunction α β) (v : α) :
                            (r f) v = r f v
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                            @[simp]
                            theorem BoundedContinuousFunction.coeFnAddHom_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] :
                            ∀ (a : BoundedContinuousFunction α β) (a_1 : α), BoundedContinuousFunction.coeFnAddHom a a_1 = a a_1

                            Coercion of a NormedAddGroupHom is an AddMonoidHom. Similar to AddMonoidHom.coeFn.

                            Equations
                            • BoundedContinuousFunction.coeFnAddHom = { toFun := DFunLike.coe, map_zero' := , map_add' := }
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                              The additive map forgetting that a bounded continuous function is bounded.

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                                @[simp]
                                theorem BoundedContinuousFunction.coe_sum {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddCommMonoid β] [BoundedAdd β] [ContinuousAdd β] {ι : Type u_2} (s : Finset ι) (f : ιBoundedContinuousFunction α β) :
                                (∑ is, f i) = is, (f i)
                                theorem BoundedContinuousFunction.sum_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddCommMonoid β] [BoundedAdd β] [ContinuousAdd β] {ι : Type u_2} (s : Finset ι) (f : ιBoundedContinuousFunction α β) (a : α) :
                                (∑ is, f i) a = is, (f i) a

                                The pointwise difference of two bounded continuous functions is again bounded continuous.

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                                • BoundedContinuousFunction.instSub = { sub := fun (f g : BoundedContinuousFunction α R) => { toFun := fun (x : α) => f x - g x, continuous_toFun := , map_bounded' := } }
                                theorem BoundedContinuousFunction.sub_apply {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Sub R] [BoundedSub R] [ContinuousSub R] (f : BoundedContinuousFunction α R) (g : BoundedContinuousFunction α R) {x : α} :
                                (f - g) x = f x - g x
                                @[simp]
                                theorem BoundedContinuousFunction.coe_sub {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Sub R] [BoundedSub R] [ContinuousSub R] (f : BoundedContinuousFunction α R) (g : BoundedContinuousFunction α R) :
                                (f - g) = f - g
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                                @[simp]
                                theorem BoundedContinuousFunction.natCast_apply {α : Type u} [TopologicalSpace α] {β : Type u_2} [PseudoMetricSpace β] [NatCast β] (n : ) (x : α) :
                                n x = n
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                                @[simp]
                                theorem BoundedContinuousFunction.intCast_apply {α : Type u} [TopologicalSpace α] {β : Type u_2} [PseudoMetricSpace β] [IntCast β] (m : ) (x : α) :
                                m x = m
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                                • BoundedContinuousFunction.instMul = { mul := fun (f g : BoundedContinuousFunction α R) => { toFun := fun (x : α) => f x * g x, continuous_toFun := , map_bounded' := } }
                                @[simp]
                                theorem BoundedContinuousFunction.coe_mul {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Mul R] [BoundedMul R] [ContinuousMul R] (f : BoundedContinuousFunction α R) (g : BoundedContinuousFunction α R) :
                                (f * g) = f * g
                                theorem BoundedContinuousFunction.mul_apply {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Mul R] [BoundedMul R] [ContinuousMul R] (f : BoundedContinuousFunction α R) (g : BoundedContinuousFunction α R) (x : α) :
                                (f * g) x = f x * g x
                                Equations
                                • BoundedContinuousFunction.instPow = { pow := fun (f : BoundedContinuousFunction α R) (n : ) => { toFun := fun (x : α) => f x ^ n, continuous_toFun := , map_bounded' := } }
                                theorem BoundedContinuousFunction.coe_pow {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ) (f : BoundedContinuousFunction α R) :
                                (f ^ n) = f ^ n
                                @[simp]
                                theorem BoundedContinuousFunction.pow_apply {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ) (f : BoundedContinuousFunction α R) (x : α) :
                                (f ^ n) x = f x ^ n
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                                theorem BoundedContinuousFunction.norm_eq {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) :
                                f = sInf {C : | 0 C ∀ (x : α), f x C}

                                The norm of a bounded continuous function is the supremum of ‖f x‖. We use sInf to ensure that the definition works if α has no elements.

                                theorem BoundedContinuousFunction.norm_eq_of_nonempty {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) [h : Nonempty α] :
                                f = sInf {C : | ∀ (x : α), f x C}

                                When the domain is non-empty, we do not need the 0 ≤ C condition in the formula for ‖f‖ as a sInf.

                                theorem BoundedContinuousFunction.dist_le_two_norm' {β : Type v} {γ : Type w} [SeminormedAddCommGroup β] {f : γβ} {C : } (hC : ∀ (x : γ), f x C) (x : γ) (y : γ) :
                                dist (f x) (f y) 2 * C
                                theorem BoundedContinuousFunction.dist_le_two_norm {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (x : α) (y : α) :
                                dist (f x) (f y) 2 * f

                                Distance between the images of any two points is at most twice the norm of the function.

                                theorem BoundedContinuousFunction.norm_le {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] {f : BoundedContinuousFunction α β} {C : } (C0 : 0 C) :
                                f C ∀ (x : α), f x C

                                The norm of a function is controlled by the supremum of the pointwise norms.

                                theorem BoundedContinuousFunction.norm_lt_iff_of_compact {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [CompactSpace α] {f : BoundedContinuousFunction α β} {M : } (M0 : 0 < M) :
                                f < M ∀ (x : α), f x < M

                                Norm of const α b is less than or equal to ‖b‖. If α is nonempty, then it is equal to ‖b‖.

                                def BoundedContinuousFunction.ofNormedAddCommGroup {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : αβ) (Hf : Continuous f) (C : ) (H : ∀ (x : α), f x C) :

                                Constructing a bounded continuous function from a uniformly bounded continuous function taking values in a normed group.

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                                  @[simp]
                                  theorem BoundedContinuousFunction.coe_ofNormedAddCommGroup {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : αβ) (Hf : Continuous f) (C : ) (H : ∀ (x : α), f x C) :
                                  theorem BoundedContinuousFunction.norm_ofNormedAddCommGroup_le {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] {f : αβ} (hfc : Continuous f) {C : } (hC : 0 C) (hfC : ∀ (x : α), f x C) :

                                  Constructing a bounded continuous function from a uniformly bounded function on a discrete space, taking values in a normed group.

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                                    Taking the pointwise norm of a bounded continuous function with values in a SeminormedAddCommGroup yields a bounded continuous function with values in ℝ.

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                                      theorem BoundedContinuousFunction.coe_normComp {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) :
                                      f.normComp = norm f

                                      If ‖(1 : β)‖ = 1, then ‖(1 : α →ᵇ β)‖ = 1 if α is nonempty.

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                                      The pointwise opposite of a bounded continuous function is again bounded continuous.

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                                      theorem BoundedContinuousFunction.coe_zsmulRec {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (z : ) :
                                      (zsmulRec (fun (x1 : ) (x2 : BoundedContinuousFunction α β) => x1 x2) z f) = z f
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                                      • BoundedContinuousFunction.instSMulInt = { smul := fun (n : ) (f : BoundedContinuousFunction α β) => { toContinuousMap := n f.toContinuousMap, map_bounded' := } }
                                      @[simp]
                                      theorem BoundedContinuousFunction.coe_zsmul {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (r : ) (f : BoundedContinuousFunction α β) :
                                      (r f) = r f
                                      @[simp]
                                      theorem BoundedContinuousFunction.zsmul_apply {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (r : ) (f : BoundedContinuousFunction α β) (v : α) :
                                      (r f) v = r f v
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                                      The nnnorm of a function is controlled by the supremum of the pointwise nnnorms.

                                      BoundedSMul (in particular, topological module) structure #

                                      In this section, if β is a metric space and a 𝕜-module whose addition and scalar multiplication are compatible with the metric structure, then we show that the space of bounded continuous functions from α to β inherits a so-called BoundedSMul structure (in particular, a ContinuousMul structure, which is the mathlib formulation of being a topological module), by using pointwise operations and checking that they are compatible with the uniform distance.

                                      instance BoundedContinuousFunction.instSMul {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] :
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                                      • BoundedContinuousFunction.instSMul = { smul := fun (c : 𝕜) (f : BoundedContinuousFunction α β) => { toContinuousMap := c f.toContinuousMap, map_bounded' := } }
                                      @[simp]
                                      theorem BoundedContinuousFunction.coe_smul {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) (f : BoundedContinuousFunction α β) :
                                      (c f) = fun (x : α) => c f x
                                      theorem BoundedContinuousFunction.smul_apply {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) (f : BoundedContinuousFunction α β) (x : α) :
                                      (c f) x = c f x
                                      instance BoundedContinuousFunction.instIsCentralScalar {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] [SMul 𝕜ᵐᵒᵖ β] [IsCentralScalar 𝕜 β] :
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                                      instance BoundedContinuousFunction.instBoundedSMul {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [BoundedSMul 𝕜 β] :
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                                      instance BoundedContinuousFunction.instModule {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Semiring 𝕜] [AddCommMonoid β] [Module 𝕜 β] [BoundedSMul 𝕜 β] [BoundedAdd β] [ContinuousAdd β] :
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                                      • BoundedContinuousFunction.instModule = Function.Injective.module 𝕜 { toFun := DFunLike.coe, map_zero' := , map_add' := }
                                      def BoundedContinuousFunction.evalCLM {α : Type u} {β : Type v} (𝕜 : Type u_2) [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Semiring 𝕜] [AddCommMonoid β] [Module 𝕜 β] [BoundedSMul 𝕜 β] [BoundedAdd β] [ContinuousAdd β] (x : α) :

                                      The evaluation at a point, as a continuous linear map from α →ᵇ β to β.

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                                        @[simp]
                                        theorem BoundedContinuousFunction.evalCLM_apply {α : Type u} {β : Type v} (𝕜 : Type u_2) [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Semiring 𝕜] [AddCommMonoid β] [Module 𝕜 β] [BoundedSMul 𝕜 β] [BoundedAdd β] [ContinuousAdd β] (x : α) (f : BoundedContinuousFunction α β) :
                                        @[simp]

                                        The linear map forgetting that a bounded continuous function is bounded.

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                                          Normed space structure #

                                          In this section, if β is a normed space, then we show that the space of bounded continuous functions from α to β inherits a normed space structure, by using pointwise operations and checking that they are compatible with the uniform distance.

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                                          Postcomposition of bounded continuous functions into a normed module by a continuous linear map is a continuous linear map. Upgraded version of ContinuousLinearMap.compLeftContinuous, similar to LinearMap.compLeft.

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                                          • One or more equations did not get rendered due to their size.
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                                            @[simp]

                                            Normed ring structure #

                                            In this section, if R is a normed ring, then we show that the space of bounded continuous functions from α to R inherits a normed ring structure, by using pointwise operations and checking that they are compatible with the uniform distance.

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                                            theorem BoundedContinuousFunction.coe_npowRec {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (f : BoundedContinuousFunction α R) (n : ) :
                                            (npowRec n f) = f ^ n
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                                            • BoundedContinuousFunction.hasNatPow = { pow := fun (f : BoundedContinuousFunction α R) (n : ) => { toContinuousMap := f.toContinuousMap ^ n, map_bounded' := } }
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                                            theorem BoundedContinuousFunction.coe_natCast {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ) :
                                            n = n
                                            @[simp]
                                            theorem BoundedContinuousFunction.coe_ofNat {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ) [n.AtLeastTwo] :
                                            (OfNat.ofNat n) = OfNat.ofNat n
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                                            theorem BoundedContinuousFunction.coe_intCast {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ) :
                                            n = n
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                                            Normed commutative ring structure #

                                            In this section, if R is a normed commutative ring, then we show that the space of bounded continuous functions from α to R inherits a normed commutative ring structure, by using pointwise operations and checking that they are compatible with the uniform distance.

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                                            Normed algebra structure #

                                            In this section, if γ is a normed algebra, then we show that the space of bounded continuous functions from α to γ inherits a normed algebra structure, by using pointwise operations and checking that they are compatible with the uniform distance.

                                            def BoundedContinuousFunction.C {α : Type u} {γ : Type w} {𝕜 : Type u_2} [NormedField 𝕜] [TopologicalSpace α] [NormedRing γ] [NormedAlgebra 𝕜 γ] :

                                            BoundedContinuousFunction.const as a RingHom.

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                                              instance BoundedContinuousFunction.instAlgebra {α : Type u} {γ : Type w} {𝕜 : Type u_2} [NormedField 𝕜] [TopologicalSpace α] [NormedRing γ] [NormedAlgebra 𝕜 γ] :
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                                              • BoundedContinuousFunction.instAlgebra = Algebra.mk BoundedContinuousFunction.C
                                              @[simp]
                                              theorem BoundedContinuousFunction.algebraMap_apply {α : Type u} {γ : Type w} {𝕜 : Type u_2} [NormedField 𝕜] [TopologicalSpace α] [NormedRing γ] [NormedAlgebra 𝕜 γ] (k : 𝕜) (a : α) :
                                              ((algebraMap 𝕜 (BoundedContinuousFunction α γ)) k) a = k 1
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                                              Structure as normed module over scalar functions #

                                              If β is a normed 𝕜-space, then we show that the space of bounded continuous functions from α to β is naturally a module over the algebra of bounded continuous functions from α to 𝕜.

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                                              • One or more equations did not get rendered due to their size.
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                                              Star structures #

                                              In this section, if β is a normed ⋆-group, then so is the space of bounded continuous functions from α to β, by using the star operation pointwise.

                                              If 𝕜 is normed field and a ⋆-ring over which β is a normed algebra and a star module, then the space of bounded continuous functions from α to β is a star module.

                                              If β is a ⋆-ring in addition to being a normed ⋆-group, then α →ᵇ β inherits a ⋆-ring structure.

                                              In summary, if β is a C⋆-algebra over 𝕜, then so is α →ᵇ β; note that completeness is guaranteed when β is complete (see BoundedContinuousFunction.complete).

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                                              The right-hand side of this equality can be parsed star ∘ ⇑f because of the instance Pi.instStarForAll. Upon inspecting the goal, one sees ⊢ ↑(star f) = star ↑f.

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                                              • BoundedContinuousFunction.instSup = { sup := fun (f g : BoundedContinuousFunction α β) => { toFun := f g, continuous_toFun := , map_bounded' := } }
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                                              • BoundedContinuousFunction.instInf = { inf := fun (f g : BoundedContinuousFunction α β) => { toFun := f g, continuous_toFun := , map_bounded' := } }
                                              @[simp]
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                                              @[deprecated BoundedContinuousFunction.coe_sup]

                                              Alias of BoundedContinuousFunction.coe_sup.

                                              @[deprecated BoundedContinuousFunction.coe_abs]

                                              Alias of BoundedContinuousFunction.coe_abs.

                                              The nonnegative part of a bounded continuous -valued function as a bounded continuous ℝ≥0-valued function.

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                                                The absolute value of a bounded continuous -valued function as a bounded continuous ℝ≥0-valued function.

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                                                  @[simp]
                                                  theorem BoundedContinuousFunction.nnnorm_coeFn_eq {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ) :
                                                  f.nnnorm = nnnorm f

                                                  Decompose a bounded continuous function to its positive and negative parts.

                                                  Express the absolute value of a bounded continuous function in terms of its positive and negative parts.