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Mathlib.Topology.ContinuousMap.Bounded.Basic

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 C(α, β) :
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.

Instances For

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.

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  • One or more equations did not get rendered due to their size.
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 F α β :

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

You should also extend this typeclass when you extend BoundedContinuousFunction.

Instances
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@[deprecated BoundedContinuousFunction.coe_toContinuousMap (since := "2024-11-23")]

Alias of BoundedContinuousFunction.coe_toContinuousMap.

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 {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f 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} :
(mkOfBound f C h) = f

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

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@[simp]
theorem BoundedContinuousFunction.mkOfCompact_apply {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [CompactSpace α] (f : C(α, β)) (a : α) :
(mkOfCompact f) 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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@[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) :
(mkOfDiscrete f C h) a✝ = f a✝

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

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

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

@[deprecated BoundedContinuousFunction.isInducing_coeFn (since := "2024-10-28")]

Alias of BoundedContinuousFunction.isInducing_coeFn.


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

@[deprecated BoundedContinuousFunction.isEmbedding_coeFn (since := "2024-10-26")]

Alias of BoundedContinuousFunction.isEmbedding_coeFn.

Constant as a continuous bounded function.

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@[simp]
theorem BoundedContinuousFunction.const_apply (α : Type u) {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (b : β) :
(const α b) = fun (x : α) => b
theorem BoundedContinuousFunction.const_apply' {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (a : α) (b : β) :
(const α b) a = b

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.

Composition of a bounded continuous function and a continuous function.

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@[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)

Restrict a bounded continuous function to a set.

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@[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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theorem BoundedContinuousFunction.lipschitz_comp {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {G : βγ} {C : NNReal} (H : LipschitzWith C G) :

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.

theorem BoundedContinuousFunction.continuous_comp {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {G : βγ} {C : NNReal} (H : LipschitzWith C G) :

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) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β) (x : α) :
(extend f g h) (f x) = g x
@[simp]
theorem BoundedContinuousFunction.extend_comp {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α δ) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β) :
(extend f g h) f = g
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) :
(extend f g h) x = h x
@[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 g₂ : BoundedContinuousFunction α β) (h₁ h₂ : BoundedContinuousFunction δ β) :
dist (extend f g₁ h₁) (extend f g₂ h₂) = dist g₁ g₂ dist (h₁.restrict (Set.range f)) (h₂.restrict (Set.range f))

The indicator function of a clopen set, as a bounded continuous function.

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@[simp]
theorem BoundedContinuousFunction.indicator_apply {α : Type u} [TopologicalSpace α] (s : Set α) (hs : IsClopen s) (x : α) :
(indicator s hs) x = s.indicator 1 x
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.

@[simp]
theorem BoundedContinuousFunction.coe_one {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [One β] :
1 = 1
@[simp]
theorem BoundedContinuousFunction.coe_zero {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] :
0 = 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
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
@[simp]
@[simp]

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

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@[simp]
theorem BoundedContinuousFunction.coe_add {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f 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 g : BoundedContinuousFunction α β) {x : α} :
(f + g) x = f x + g x
@[simp]
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@[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

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

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@[simp]

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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theorem BoundedContinuousFunction.sub_apply {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Sub R] [BoundedSub R] [ContinuousSub R] (f 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 g : BoundedContinuousFunction α R) :
⇑(f - g) = f - g
@[simp]
theorem BoundedContinuousFunction.natCast_apply {α : Type u} [TopologicalSpace α] {β : Type u_2} [PseudoMetricSpace β] [NatCast β] (n : ) (x : α) :
n x = n
@[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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@[simp]
theorem BoundedContinuousFunction.coe_mul {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Mul R] [BoundedMul R] [ContinuousMul R] (f 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 g : BoundedContinuousFunction α R) (x : α) :
(f * g) x = f x * g x
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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

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.

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 y : γ) :
dist (f x) (f y) 2 * C

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) :
(ofNormedAddCommGroup f Hf C H) = f
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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@[simp]
theorem BoundedContinuousFunction.coe_ofNormedAddCommGroupDiscrete {α : Type u} {β : Type v} [TopologicalSpace α] [DiscreteTopology α] [SeminormedAddCommGroup β] (f : αβ) (C : ) (H : ∀ (x : α), f x C) :

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

Equations

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

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

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@[simp]
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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@[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

The nnnorm of a function is controlled by the supremum of the pointwise nnnorms.

IsBoundedSMul (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 IsBoundedSMul 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] [IsBoundedSMul 𝕜 β] :
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@[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] [IsBoundedSMul 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] [IsBoundedSMul 𝕜 β] (c : 𝕜) (f : BoundedContinuousFunction α β) (x : α) :
(c f) x = c f x
instance BoundedContinuousFunction.instIsScalarTower {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [IsBoundedSMul 𝕜 β] {𝕜' : Type u_3} [PseudoMetricSpace 𝕜'] [Zero 𝕜'] [SMul 𝕜' β] [IsBoundedSMul 𝕜' β] [SMul 𝕜' 𝕜] [IsScalarTower 𝕜' 𝕜 β] :
instance BoundedContinuousFunction.instSMulCommClass {α : Type u} {β : Type v} {𝕜 : Type u_2} [PseudoMetricSpace 𝕜] [TopologicalSpace α] [PseudoMetricSpace β] [Zero 𝕜] [Zero β] [SMul 𝕜 β] [IsBoundedSMul 𝕜 β] {𝕜' : Type u_3} [PseudoMetricSpace 𝕜'] [Zero 𝕜'] [SMul 𝕜' β] [IsBoundedSMul 𝕜' β] [SMulCommClass 𝕜' 𝕜 β] :
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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] [IsBoundedSMul 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] [IsBoundedSMul 𝕜 β] [BoundedAdd β] [ContinuousAdd β] (x : α) (f : BoundedContinuousFunction α β) :
(evalCLM 𝕜 x) f = f x

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.

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.
@[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.

@[simp]
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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@[simp]
theorem BoundedContinuousFunction.coe_natCast {α : Type u} [TopologicalSpace α] {R : Type u_2} [SeminormedRing R] (n : ) :
n = n
@[simp]
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.

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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@[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

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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@[simp]
theorem BoundedContinuousFunction.coe_sup {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f g : BoundedContinuousFunction α β) :
⇑(f g) = f g
@[simp]
theorem BoundedContinuousFunction.coe_inf {α : Type u} {β : Type v} [TopologicalSpace α] [NormedLatticeAddCommGroup β] (f g : BoundedContinuousFunction α β) :
⇑(f g) = f g

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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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.