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Mathlib.Topology.ContinuousMap.ZeroAtInfty

Continuous functions vanishing at infinity #

The type of continuous functions vanishing at infinity. When the domain is compact C(α, β) ≃ C₀(α, β) via the identity map. When the codomain is a metric space, every continuous map which vanishes at infinity is a bounded continuous function. When the domain is a locally compact space, this type has nice properties.

TODO #

structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β] [TopologicalSpace β] extends C(α, β) :
Type (max u v)

C₀(α, β) is the type of continuous functions α → β which vanish at infinity from a topological space to a metric space with a zero element.

When possible, instead of parametrizing results over (f : C₀(α, β)), you should parametrize over (F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F).

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

C₀(α, β) is the type of continuous functions α → β which vanish at infinity from a topological space to a metric space with a zero element.

When possible, instead of parametrizing results over (f : C₀(α, β)), you should parametrize over (F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F).

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

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

C₀(α, β) is the type of continuous functions α → β which vanish at infinity from a topological space to a metric space with a zero element.

When possible, instead of parametrizing results over (f : C₀(α, β)), you should parametrize over (F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F).

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

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class ZeroAtInftyContinuousMapClass (F : Type u_2) (α : outParam (Type u_3)) (β : outParam (Type u_4)) [TopologicalSpace α] [Zero β] [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass F α β :

ZeroAtInftyContinuousMapClass F α β states that F is a type of continuous maps which vanish at infinity.

You should also extend this typeclass when you extend ZeroAtInftyContinuousMap.

Instances
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    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_toContinuousMap {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : ZeroAtInftyContinuousMap α β) :
    f.toContinuousMap = f
    theorem ZeroAtInftyContinuousMap.ext {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {f g : ZeroAtInftyContinuousMap α β} (h : ∀ (x : α), f x = g x) :
    f = g
    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_mk {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {f : αβ} (hf : Continuous f) (hf' : Filter.Tendsto f (Filter.cocompact α) (nhds 0)) :
    { toFun := f, continuous_toFun := hf, zero_at_infty' := hf' } = f
    def ZeroAtInftyContinuousMap.copy {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : ZeroAtInftyContinuousMap α β) (f' : αβ) (h : f' = f) :

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

    Equations
    • f.copy f' h = { toFun := f', continuous_toFun := , zero_at_infty' := }
    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_copy {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : ZeroAtInftyContinuousMap α β) (f' : αβ) (h : f' = f) :
    (f.copy f' h) = f'
    theorem ZeroAtInftyContinuousMap.copy_eq {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : ZeroAtInftyContinuousMap α β) (f' : αβ) (h : f' = f) :
    f.copy f' h = f

    A continuous function on a compact space is automatically a continuous function vanishing at infinity.

    Equations
    • One or more equations did not get rendered due to their size.
    @[simp]
    theorem ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty_apply_toFun {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [CompactSpace α] (f : C(α, β)) (a : α) :
    (liftZeroAtInfty f) a = f a

    A continuous function on a compact space is automatically a continuous function vanishing at infinity. This is not an instance to avoid type class loops.

    Algebraic structure #

    Whenever β has suitable algebraic structure and a compatible topological structure, then C₀(α, β) inherits a corresponding algebraic structure. The primary exception to this is that C₀(α, β) will not have a multiplicative identity.

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    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_zero {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] :
    0 = 0
    theorem ZeroAtInftyContinuousMap.zero_apply {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [Zero β] :
    0 x = 0
    Equations
    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_mul {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [MulZeroClass β] [ContinuousMul β] (f g : ZeroAtInftyContinuousMap α β) :
    (f * g) = f * g
    theorem ZeroAtInftyContinuousMap.mul_apply {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [MulZeroClass β] [ContinuousMul β] (f g : ZeroAtInftyContinuousMap α β) :
    (f * g) x = f x * g x
    Equations
    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_add {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β] (f g : ZeroAtInftyContinuousMap α β) :
    (f + g) = f + g
    theorem ZeroAtInftyContinuousMap.add_apply {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddZeroClass β] [ContinuousAdd β] (f g : ZeroAtInftyContinuousMap α β) :
    (f + g) x = f x + g x
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    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_smul {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {R : Type u_2} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R) (f : ZeroAtInftyContinuousMap α β) :
    (r f) = r f
    theorem ZeroAtInftyContinuousMap.smul_apply {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {R : Type u_2} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R) (f : ZeroAtInftyContinuousMap α β) (x : α) :
    (r f) x = r f x
    Equations
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    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_sub {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (f g : ZeroAtInftyContinuousMap α β) :
    (f - g) = f - g
    theorem ZeroAtInftyContinuousMap.sub_apply {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddGroup β] [TopologicalAddGroup β] (f g : ZeroAtInftyContinuousMap α β) :
    (f - g) x = f x - g x
    Equations
    theorem ZeroAtInftyContinuousMap.uniformContinuous {F : Type u_1} {β : Type v} {γ : Type w} [UniformSpace β] [UniformSpace γ] [Zero γ] [FunLike F β γ] [ZeroAtInftyContinuousMapClass F β γ] (f : F) :

    Metric structure #

    When β is a metric space, then every element of C₀(α, β) is bounded, and so there is a natural inclusion map ZeroAtInftyContinuousMap.toBCF : C₀(α, β) → (α →ᵇ β). Via this map C₀(α, β) inherits a metric as the pullback of the metric on α →ᵇ β. Moreover, this map has closed range in α →ᵇ β and consequently C₀(α, β) is a complete space whenever β is complete.

    theorem ZeroAtInftyContinuousMap.bounded {F : Type u_1} {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] [FunLike F α β] [ZeroAtInftyContinuousMapClass F α β] (f : F) :
    ∃ (C : ), ∀ (x y : α), dist (f x) (f y) C

    Construct a bounded continuous function from a continuous function vanishing at infinity.

    Equations
    • f.toBCF = { toContinuousMap := f, map_bounded' := }
    @[simp]
    theorem ZeroAtInftyContinuousMap.toBCF_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : ZeroAtInftyContinuousMap α β) (a : α) :
    f.toBCF a = f a

    The type of continuous functions vanishing at infinity, with the uniform distance induced by the inclusion ZeroAtInftyContinuousMap.toBCF, is a pseudo-metric space.

    Equations

    The type of continuous functions vanishing at infinity, with the uniform distance induced by the inclusion ZeroAtInftyContinuousMap.toBCF, is a metric space.

    Equations
    @[simp]
    theorem ZeroAtInftyContinuousMap.dist_toBCF_eq_dist {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] {f g : ZeroAtInftyContinuousMap α β} :
    dist f.toBCF g.toBCF = dist f g
    theorem ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] {ι : Type u_2} {F : ιZeroAtInftyContinuousMap α β} {f : ZeroAtInftyContinuousMap α β} {l : Filter ι} :
    Filter.Tendsto F l (nhds f) TendstoUniformly (fun (i : ι) => (F i)) (⇑f) l

    Convergence in the metric on C₀(α, β) is uniform convergence.

    Continuous functions vanishing at infinity taking values in a complete space form a complete space.

    Normed space #

    The norm structure on C₀(α, β) is the one induced by the inclusion toBCF : C₀(α, β) → (α →ᵇ b), viewed as an additive monoid homomorphism. Then C₀(α, β) is naturally a normed space over a normed field 𝕜 whenever β is as well.

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

    Star structure #

    It is possible to equip C₀(α, β) with a pointwise star operation whenever there is a continuous star : β → β for which star (0 : β) = 0. We don't have quite this weak a typeclass, but StarAddMonoid is close enough.

    The StarAddMonoid and NormedStarGroup classes on C₀(α, β) are inherited from their counterparts on α →ᵇ β. Ultimately, when β is a C⋆-ring, then so is C₀(α, β).

    Equations

    C₀ as a functor #

    For each β with sufficient structure, there is a contravariant functor C₀(-, β) from the category of topological spaces with morphisms given by CocompactMaps.

    Composition of a continuous function vanishing at infinity with a cocompact map yields another continuous function vanishing at infinity.

    Equations
    • f.comp g = { toContinuousMap := (↑f).comp g, zero_at_infty' := }
    @[simp]
    theorem ZeroAtInftyContinuousMap.coe_comp_to_continuous_fun {β : Type v} {γ : Type w} {δ : Type u_2} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : ZeroAtInftyContinuousMap γ δ) (g : CocompactMap β γ) :
    (f.comp g) = f g
    @[simp]
    theorem ZeroAtInftyContinuousMap.comp_id {γ : Type w} {δ : Type u_2} [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : ZeroAtInftyContinuousMap γ δ) :
    f.comp (CocompactMap.id γ) = f
    @[simp]
    theorem ZeroAtInftyContinuousMap.comp_assoc {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] {δ : Type u_2} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : ZeroAtInftyContinuousMap γ δ) (g : CocompactMap β γ) (h : CocompactMap α β) :
    (f.comp g).comp h = f.comp (g.comp h)
    @[simp]
    theorem ZeroAtInftyContinuousMap.zero_comp {β : Type v} {γ : Type w} {δ : Type u_2} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (g : CocompactMap β γ) :
    comp 0 g = 0

    Composition as an additive monoid homomorphism.

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    Composition as a semigroup homomorphism.

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    Composition as a linear map.

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    Composition as a non-unital algebra homomorphism.

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