mathlib documentation

topology.continuous_function.bounded

Bounded continuous functions #

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

structure bounded_continuous_function (α : Type u) (β : Type v) [topological_space α] [pseudo_metric_space 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*) [bounded_continuous_map_class F α β] (f : F).

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

Instances for bounded_continuous_function
@[class]
structure bounded_continuous_map_class (F : Type u_2) (α : Type u_3) (β : Type u_4) [topological_space α] [pseudo_metric_space β] :
Type (max u_2 u_3 u_4)

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

You should also extend this typeclass when you extend bounded_continuous_function.

Instances of this typeclass
Instances of other typeclasses for bounded_continuous_map_class
  • bounded_continuous_map_class.has_sizeof_inst
@[protected, instance]

Helper instance for when there's too many metavariables to apply fun_like.has_coe_to_fun directly.

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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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@[protected]
theorem bounded_continuous_function.bounded {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] (f : bounded_continuous_function α β) :
(C : ), (x y : α), has_dist.dist (f x) (f y) C
@[ext]
theorem bounded_continuous_function.ext {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] {f g : bounded_continuous_function α β} (h : (x : α), f x = g x) :
f = g
def bounded_continuous_function.mk_of_bound {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] (f : C(α, β)) (C : ) (h : (x y : α), has_dist.dist (f x) (f y) C) :

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

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@[simp]
theorem bounded_continuous_function.mk_of_bound_coe {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] {f : C(α, β)} {C : } {h : (x y : α), has_dist.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 bounded_continuous_function.mk_of_discrete_apply {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] [discrete_topology α] (f : α β) (C : ) (h : (x y : α), has_dist.dist (f x) (f y) C) (ᾰ : α) :
@[simp]
def bounded_continuous_function.mk_of_discrete {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] [discrete_topology α] (f : α β) (C : ) (h : (x y : α), has_dist.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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@[protected, instance]

The uniform distance between two bounded continuous functions

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

theorem bounded_continuous_function.dist_le {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] {f g : bounded_continuous_function α β} {C : } (C0 : 0 C) :
has_dist.dist f g C (x : α), has_dist.dist (f x) (g x) C

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

@[protected, instance]

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

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@[protected, instance]

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

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

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

Constant as a continuous bounded function.

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@[protected, instance]

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

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@[continuity]

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

@[continuity]

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

@[protected, instance]

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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Restrict a bounded continuous function to a set.

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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

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

theorem bounded_continuous_function.arzela_ascoli₂ {α : Type u} {β : Type v} [topological_space α] [compact_space α] [pseudo_metric_space β] (s : set β) (hs : is_compact s) (A : set (bounded_continuous_function α β)) (closed : is_closed A) (in_s : (f : bounded_continuous_function α β) (x : α), f A f x s) (H : equicontinuous coe_fn) :

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

theorem bounded_continuous_function.arzela_ascoli {α : Type u} {β : Type v} [topological_space α] [compact_space α] [pseudo_metric_space β] [t2_space β] (s : set β) (hs : is_compact s) (A : set (bounded_continuous_function α β)) (in_s : (f : bounded_continuous_function α β) (x : α), f A f x s) (H : equicontinuous coe_fn) :

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

@[simp]
@[protected, instance]

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

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@[simp]
theorem bounded_continuous_function.nsmul_apply {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] [add_monoid β] [has_lipschitz_add β] (r : ) (f : bounded_continuous_function α β) (v : α) :
(r f) v = r f v
@[protected, instance]
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@[simp]
theorem bounded_continuous_function.coe_sum {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] [add_comm_monoid β] [has_lipschitz_add β] {ι : Type u_1} (s : finset ι) (f : ι bounded_continuous_function α β) :
(s.sum (λ (i : ι), f i)) = s.sum (λ (i : ι), (f i))
theorem bounded_continuous_function.sum_apply {α : Type u} {β : Type v} [topological_space α] [pseudo_metric_space β] [add_comm_monoid β] [has_lipschitz_add β] {ι : Type u_1} (s : finset ι) (f : ι bounded_continuous_function α β) (a : α) :
(s.sum (λ (i : ι), f i)) a = s.sum (λ (i : ι), (f i) a)

The norm of a bounded continuous function is the supremum of ‖f x‖. We use Inf 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 an Inf.

theorem bounded_continuous_function.dist_le_two_norm' {β : Type v} {γ : Type w} [seminormed_add_comm_group β] {f : γ β} {C : } (hC : (x : γ), f x C) (x y : γ) :
has_dist.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 bounded_continuous_function.norm_le {α : Type u} {β : Type v} [topological_space α] [seminormed_add_comm_group β] {f : bounded_continuous_function 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

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

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

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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 seminormed_add_comm_group yields a bounded continuous function with values in ℝ.

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@[protected, instance]

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

@[protected, instance]

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

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@[protected, instance]

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

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theorem bounded_continuous_function.sub_apply {α : Type u} {β : Type v} [topological_space α] [seminormed_add_comm_group β] (f g : bounded_continuous_function α β) {x : α} :
(f - g) x = f x - g x
@[simp]
theorem bounded_continuous_function.zsmul_apply {α : Type u} {β : Type v} [topological_space α] [seminormed_add_comm_group β] (r : ) (f : bounded_continuous_function α β) (v : α) :
(r f) v = r f v
@[protected, instance]
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The nnnorm of a function is controlled by the supremum of the pointwise nnnorms

has_bounded_smul (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 has_bounded_smul structure (in particular, a has_continuous_mul 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.

@[simp]
theorem bounded_continuous_function.coe_smul {α : Type u} {β : Type v} {𝕜 : Type u_2} [pseudo_metric_space 𝕜] [topological_space α] [pseudo_metric_space β] [has_zero 𝕜] [has_zero β] [has_smul 𝕜 β] [has_bounded_smul 𝕜 β] (c : 𝕜) (f : bounded_continuous_function α β) :
(c f) = λ (x : α), c f x
theorem bounded_continuous_function.smul_apply {α : Type u} {β : Type v} {𝕜 : Type u_2} [pseudo_metric_space 𝕜] [topological_space α] [pseudo_metric_space β] [has_zero 𝕜] [has_zero β] [has_smul 𝕜 β] [has_bounded_smul 𝕜 β] (c : 𝕜) (f : bounded_continuous_function α β) (x : α) :
(c f) x = c f x
@[protected, instance]
@[protected, instance]
@[protected, instance]
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@[protected, instance]
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@[protected, instance]
noncomputable def bounded_continuous_function.module {α : Type u} {β : Type v} {𝕜 : Type u_2} [pseudo_metric_space 𝕜] [topological_space α] [pseudo_metric_space β] [semiring 𝕜] [add_comm_monoid β] [module 𝕜 β] [has_bounded_smul 𝕜 β] [has_lipschitz_add β] :
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The evaluation at a point, as a continuous linear map from α →ᵇ β to β.

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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.

@[protected]

Postcomposition of bounded continuous functions into a normed module by a continuous linear map is a continuous linear map. Upgraded version of continuous_linear_map.comp_left_continuous, similar to linear_map.comp_left.

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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.

@[protected, instance]
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@[simp]
theorem bounded_continuous_function.coe_pow {α : Type u} [topological_space α] {R : Type u_2} [semi_normed_ring R] (n : ) (f : bounded_continuous_function α R) :
(f ^ n) = f ^ n
@[simp]
theorem bounded_continuous_function.pow_apply {α : Type u} [topological_space α] {R : Type u_2} [semi_normed_ring R] (n : ) (f : bounded_continuous_function α R) (v : α) :
(f ^ n) v = f v ^ n
@[simp, norm_cast]
@[simp, norm_cast]
@[protected, instance]
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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.

@[simp]
theorem bounded_continuous_function.algebra_map_apply {α : Type u} {γ : Type w} {𝕜 : Type u_2} [normed_field 𝕜] [topological_space α] [normed_ring γ] [normed_algebra 𝕜 γ] (k : 𝕜) (a : α) :

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.

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 bounded_continuous_function.complete).

@[simp]

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

@[protected, instance]

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.