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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) [metric_space β] :
Type (max u v)

The type of bounded continuous functions from a topological space to a metric space

@[protected, instance]
def bounded_continuous_function.has_coe_to_fun {α : Type u} {β : Type v} [metric_space β] :
has_coe_to_fun →ᵇ β) (λ (_x : α →ᵇ β), α → β)
Equations
@[simp]
theorem bounded_continuous_function.coe_to_continuous_fun {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) :
def bounded_continuous_function.simps.apply {α : Type u} {β : Type v} [metric_space β] (h : α →ᵇ β) :
α → β

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

Equations
@[protected]
theorem bounded_continuous_function.bounded {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) :
∃ (C : ), ∀ (x y : α), dist (f x) (f y) C
@[protected, continuity]
theorem bounded_continuous_function.continuous {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) :
@[ext]
theorem bounded_continuous_function.ext {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} (H : ∀ (x : α), f x = g x) :
f = g
theorem bounded_continuous_function.ext_iff {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} :
f = g ∀ (x : α), f x = g x
theorem bounded_continuous_function.coe_injective {α : Type u} {β : Type v} [metric_space β] :
theorem bounded_continuous_function.bounded_range {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) :
theorem bounded_continuous_function.bounded_image {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) (s : set α) :
theorem bounded_continuous_function.eq_of_empty {α : Type u} {β : Type v} [metric_space β] [is_empty α] (f g : α →ᵇ β) :
f = g
def bounded_continuous_function.mk_of_bound {α : Type u} {β : Type v} [metric_space β] (f : C(α, β)) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) :
α →ᵇ β

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

Equations
@[simp]
theorem bounded_continuous_function.mk_of_bound_coe {α : Type u} {β : Type v} [metric_space β] {f : C(α, β)} {C : } {h : ∀ (x y : α), dist (f x) (f y) C} :
def bounded_continuous_function.mk_of_compact {α : Type u} {β : Type v} [metric_space β] (f : C(α, β)) :
α →ᵇ β

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

Equations
@[simp]
theorem bounded_continuous_function.mk_of_compact_apply {α : Type u} {β : Type v} [metric_space β] (f : C(α, β)) (a : α) :
@[simp]
theorem bounded_continuous_function.mk_of_discrete_apply {α : Type u} {β : Type v} [metric_space β] (f : α → β) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) (ᾰ : α) :
= f ᾰ
def bounded_continuous_function.mk_of_discrete {α : Type u} {β : Type v} [metric_space β] (f : α → β) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) :
α →ᵇ β

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

Equations
@[simp]
theorem bounded_continuous_function.mk_of_discrete_to_continuous_map_to_fun {α : Type u} {β : Type v} [metric_space β] (f : α → β) (C : ) (h : ∀ (x y : α), dist (f x) (f y) C) (ᾰ : α) :
= f ᾰ
@[protected, instance]
noncomputable def bounded_continuous_function.has_dist {α : Type u} {β : Type v} [metric_space β] :

The uniform distance between two bounded continuous functions

Equations
theorem bounded_continuous_function.dist_eq {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} :
dist f g = Inf {C : | 0 C ∀ (x : α), dist (f x) (g x) C}
theorem bounded_continuous_function.dist_set_exists {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} :
∃ (C : ), 0 C ∀ (x : α), dist (f x) (g x) C
theorem bounded_continuous_function.dist_coe_le_dist {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} (x : α) :
dist (f x) (g x) dist f g

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

theorem bounded_continuous_function.dist_le {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ 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 bounded_continuous_function.dist_le_iff_of_nonempty {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} {C : } [nonempty α] :
dist f g C ∀ (x : α), dist (f x) (g x) C
theorem bounded_continuous_function.dist_lt_of_nonempty_compact {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} {C : } [nonempty α] (w : ∀ (x : α), dist (f x) (g x) < C) :
dist f g < C
theorem bounded_continuous_function.dist_lt_iff_of_compact {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} {C : } (C0 : 0 < C) :
dist f g < C ∀ (x : α), dist (f x) (g x) < C
theorem bounded_continuous_function.dist_lt_iff_of_nonempty_compact {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} {C : } [nonempty α]  :
dist f g < C ∀ (x : α), dist (f x) (g x) < C
@[protected, instance]
noncomputable def bounded_continuous_function.metric_space {α : Type u} {β : Type v} [metric_space β] :

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

Equations
theorem bounded_continuous_function.dist_zero_of_empty {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} [is_empty α] :
dist f g = 0

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

theorem bounded_continuous_function.dist_eq_supr {α : Type u} {β : Type v} [metric_space β] {f g : α →ᵇ β} :
dist f g = ⨆ (x : α), dist (f x) (g x)
@[simp]
theorem bounded_continuous_function.const_apply (α : Type u) {β : Type v} [metric_space β] (b : β) :
= λ (x : α), b
@[simp]
theorem bounded_continuous_function.const_to_continuous_map (α : Type u) {β : Type v} [metric_space β] (b : β) :
def bounded_continuous_function.const (α : Type u) {β : Type v} [metric_space β] (b : β) :
α →ᵇ β

Constant as a continuous bounded function.

Equations
theorem bounded_continuous_function.const_apply' {α : Type u} {β : Type v} [metric_space β] (a : α) (b : β) :
= b
@[protected, instance]
def bounded_continuous_function.inhabited {α : Type u} {β : Type v} [metric_space β] [inhabited β] :

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

Equations
theorem bounded_continuous_function.lipschitz_evalx {α : Type u} {β : Type v} [metric_space β] (x : α) :
(λ (f : α →ᵇ β), f x)
theorem bounded_continuous_function.uniform_continuous_coe {α : Type u} {β : Type v} [metric_space β] :
theorem bounded_continuous_function.continuous_coe {α : Type u} {β : Type v} [metric_space β] :
continuous (λ (f : α →ᵇ β) (x : α), f x)
@[continuity]
theorem bounded_continuous_function.continuous_evalx {α : Type u} {β : Type v} [metric_space β] {x : α} :
continuous (λ (f : α →ᵇ β), f x)

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

@[continuity]
theorem bounded_continuous_function.continuous_eval {α : Type u} {β : Type v} [metric_space β] :
continuous (λ (p : →ᵇ β) × α), (p.fst) p.snd)

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

@[protected, instance]
def bounded_continuous_function.complete_space {α : Type u} {β : Type v} [metric_space β]  :

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

@[simp]
theorem bounded_continuous_function.comp_continuous_to_continuous_map {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α →ᵇ β) (g : C(δ, α)) :
def bounded_continuous_function.comp_continuous {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α →ᵇ β) (g : C(δ, α)) :
δ →ᵇ β

Composition of a bounded continuous function and a continuous function.

Equations
@[simp]
theorem bounded_continuous_function.comp_continuous_apply {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α →ᵇ β) (g : C(δ, α)) :
theorem bounded_continuous_function.lipschitz_comp_continuous {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (g : C(δ, α)) :
(λ (f : α →ᵇ β), f.comp_continuous g)
theorem bounded_continuous_function.continuous_comp_continuous {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (g : C(δ, α)) :
continuous (λ (f : α →ᵇ β), f.comp_continuous g)
@[simp]
theorem bounded_continuous_function.restrict_apply {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) (s : set α) :
(f.restrict s) =
def bounded_continuous_function.restrict {α : Type u} {β : Type v} [metric_space β] (f : α →ᵇ β) (s : set α) :

Restrict a bounded continuous function to a set.

Equations
def bounded_continuous_function.comp {α : Type u} {β : Type v} {γ : Type w} [metric_space β] [metric_space γ] (G : β → γ) {C : ℝ≥0} (H : G) (f : α →ᵇ β) :
α →ᵇ γ

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

Equations
theorem bounded_continuous_function.lipschitz_comp {α : Type u} {β : Type v} {γ : Type w} [metric_space β] [metric_space γ] {G : β → γ} {C : ℝ≥0} (H : G) :

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

theorem bounded_continuous_function.uniform_continuous_comp {α : Type u} {β : Type v} {γ : Type w} [metric_space β] [metric_space γ] {G : β → γ} {C : ℝ≥0} (H : G) :

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

theorem bounded_continuous_function.continuous_comp {α : Type u} {β : Type v} {γ : Type w} [metric_space β] [metric_space γ] {G : β → γ} {C : ℝ≥0} (H : G) :

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

def bounded_continuous_function.cod_restrict {α : Type u} {β : Type v} [metric_space β] (s : set β) (f : α →ᵇ β) (H : ∀ (x : α), f x s) :

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

Equations
noncomputable def bounded_continuous_function.extend {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α δ) (g : α →ᵇ β) (h : δ →ᵇ β) :
δ →ᵇ β

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

Equations
@[simp]
theorem bounded_continuous_function.extend_apply {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α δ) (g : α →ᵇ β) (h : δ →ᵇ β) (x : α) :
(f x) = g x
@[simp]
theorem bounded_continuous_function.extend_comp {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α δ) (g : α →ᵇ β) (h : δ →ᵇ β) :
= g
theorem bounded_continuous_function.extend_apply' {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} {f : α δ} {x : δ} (hx : x ) (g : α →ᵇ β) (h : δ →ᵇ β) :
x = h x
theorem bounded_continuous_function.extend_of_empty {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} [is_empty α] (f : α δ) (g : α →ᵇ β) (h : δ →ᵇ β) :
@[simp]
theorem bounded_continuous_function.dist_extend_extend {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
dist h₁) h₂) = max (dist g₁ g₂) (dist (h₁.restrict (set.range f)) (h₂.restrict (set.range f)))
theorem bounded_continuous_function.isometry_extend {α : Type u} {β : Type v} [metric_space β] {δ : Type u_1} (f : α δ) (h : δ →ᵇ β) :
isometry (λ (g : α →ᵇ β),
theorem bounded_continuous_function.arzela_ascoli₁ {α : Type u} {β : Type v} [metric_space β] (A : set →ᵇ β)) (closed : is_closed A) (H : ∀ (x : α) (ε : ), ε > 0(∃ (U : set α) (H : U 𝓝 x), ∀ (y : α), y U∀ (z : α), z U∀ (f : α →ᵇ β), f Adist (f y) (f z) < ε)) :

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

theorem bounded_continuous_function.arzela_ascoli₂ {α : Type u} {β : Type v} [metric_space β] (s : set β) (hs : is_compact s) (A : set →ᵇ β)) (closed : is_closed A) (in_s : ∀ (f : α →ᵇ β) (x : α), f Af x s) (H : ∀ (x : α) (ε : ), ε > 0(∃ (U : set α) (H : U 𝓝 x), ∀ (y : α), y U∀ (z : α), z U∀ (f : α →ᵇ β), f Adist (f y) (f z) < ε)) :

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

theorem bounded_continuous_function.arzela_ascoli {α : Type u} {β : Type v} [metric_space β] (s : set β) (hs : is_compact s) (A : set →ᵇ β)) (in_s : ∀ (f : α →ᵇ β) (x : α), f Af x s) (H : ∀ (x : α) (ε : ), ε > 0(∃ (U : set α) (H : U 𝓝 x), ∀ (y : α), y U∀ (z : α), z U∀ (f : α →ᵇ β), f Adist (f y) (f z) < ε)) :

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

theorem bounded_continuous_function.equicontinuous_of_continuity_modulus {β : Type v} [metric_space β] {α : Type u} [metric_space α] (b : ) (b_lim : (𝓝 0) (𝓝 0)) (A : set →ᵇ β)) (H : ∀ (x y : α) (f : α →ᵇ β), f Adist (f x) (f y) b (dist x y)) (x : α) (ε : ) (ε0 : 0 < ε) :
∃ (U : set α) (H : U 𝓝 x), ∀ (y : α), y U∀ (z : α), z U∀ (f : α →ᵇ β), f Adist (f y) (f z) < ε
@[protected, instance]
def bounded_continuous_function.has_zero {α : Type u} {β : Type v} [metric_space β] [add_monoid β] :
Equations
@[simp]
theorem bounded_continuous_function.coe_zero {α : Type u} {β : Type v} [metric_space β] [add_monoid β] :
0 = 0
theorem bounded_continuous_function.forall_coe_zero_iff_zero {α : Type u} {β : Type v} [metric_space β] [add_monoid β] (f : α →ᵇ β) :
(∀ (x : α), f x = 0) f = 0
@[simp]
theorem bounded_continuous_function.zero_comp_continuous {α : Type u} {β : Type v} {γ : Type w} [metric_space β] [add_monoid β] (f : C(γ, α)) :
= 0
@[protected, instance]
noncomputable def bounded_continuous_function.has_add {α : Type u} {β : Type v} [metric_space β] [add_monoid β]  :

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

Equations
@[simp]
theorem bounded_continuous_function.coe_add {α : Type u} {β : Type v} [metric_space β] [add_monoid β] (f g : α →ᵇ β) :
(f + g) = f + g
theorem bounded_continuous_function.add_apply {α : Type u} {β : Type v} [metric_space β] [add_monoid β] (f g : α →ᵇ β) {x : α} :
(f + g) x = f x + g x
theorem bounded_continuous_function.add_comp_continuous {α : Type u} {β : Type v} {γ : Type w} [metric_space β] [add_monoid β] (f g : α →ᵇ β) (h : C(γ, α)) :
(g + f).comp_continuous h = +
@[protected, instance]
noncomputable def bounded_continuous_function.add_monoid {α : Type u} {β : Type v} [metric_space β] [add_monoid β]  :
Equations
@[protected, instance]
def bounded_continuous_function.has_lipschitz_add {α : Type u} {β : Type v} [metric_space β] [add_monoid β]  :
@[simp]
theorem bounded_continuous_function.coe_fn_add_hom_apply {α : Type u} {β : Type v} [metric_space β] [add_monoid β] (x : α →ᵇ β) (ᾰ : α) :
noncomputable def bounded_continuous_function.coe_fn_add_hom {α : Type u} {β : Type v} [metric_space β] [add_monoid β]  :
→ᵇ β) →+ α → β

Coercion of a normed_group_hom is an add_monoid_hom. Similar to add_monoid_hom.coe_fn

Equations
@[simp]
theorem bounded_continuous_function.to_continuous_map_add_hom_apply (α : Type u) (β : Type v) [metric_space β] [add_monoid β] (self : α →ᵇ β) :
noncomputable def bounded_continuous_function.to_continuous_map_add_hom (α : Type u) (β : Type v) [metric_space β] [add_monoid β]  :
→ᵇ β) →+ C(α, β)

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

Equations
@[protected, instance]
noncomputable def bounded_continuous_function.add_comm_monoid {α : Type u} {β : Type v} [metric_space β]  :
Equations
@[protected, instance]
noncomputable def bounded_continuous_function.add_add_comm_monoid {α : Type u} {β : Type v} [metric_space β]  :
Equations
@[simp]
theorem bounded_continuous_function.coe_sum {α : Type u} {β : Type v} [metric_space β] {ι : Type u_1} (s : finset ι) (f : ι → α →ᵇ β) :
∑ (i : ι) in s, f i = ∑ (i : ι) in s, (f i)
theorem bounded_continuous_function.sum_apply {α : Type u} {β : Type v} [metric_space β] {ι : Type u_1} (s : finset ι) (f : ι → α →ᵇ β) (a : α) :
(∑ (i : ι) in s, f i) a = ∑ (i : ι) in s, (f i) a
@[protected, instance]
noncomputable def bounded_continuous_function.has_norm {α : Type u} {β : Type v} [normed_group β] :
Equations
theorem bounded_continuous_function.norm_def {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
theorem bounded_continuous_function.norm_eq {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
f = Inf {C : | 0 C ∀ (x : α), f x C}

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.

theorem bounded_continuous_function.norm_eq_of_nonempty {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) [h : nonempty α] :
f = Inf {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 an Inf.

@[simp]
theorem bounded_continuous_function.norm_eq_zero_of_empty {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) [h : is_empty α] :
theorem bounded_continuous_function.norm_coe_le_norm {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) (x : α) :
theorem bounded_continuous_function.dist_le_two_norm' {β : Type v} {γ : Type w} [normed_group β] {f : γ → β} {C : } (hC : ∀ (x : γ), f x C) (x y : γ) :
dist (f x) (f y) 2 * C
theorem bounded_continuous_function.dist_le_two_norm {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) (x 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 bounded_continuous_function.norm_le {α : Type u} {β : Type v} [normed_group β] {f : α →ᵇ 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 bounded_continuous_function.norm_le_of_nonempty {α : Type u} {β : Type v} [normed_group β] [nonempty α] {f : α →ᵇ β} {M : } :
f M ∀ (x : α), f x M
theorem bounded_continuous_function.norm_lt_iff_of_compact {α : Type u} {β : Type v} [normed_group β] {f : α →ᵇ β} {M : } (M0 : 0 < M) :
f < M ∀ (x : α), f x < M
theorem bounded_continuous_function.norm_lt_iff_of_nonempty_compact {α : Type u} {β : Type v} [normed_group β] [nonempty α] {f : α →ᵇ β} {M : } :
f < M ∀ (x : α), f x < M
theorem bounded_continuous_function.norm_const_le {α : Type u} {β : Type v} [normed_group β] (b : β) :

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

@[simp]
theorem bounded_continuous_function.norm_const_eq {α : Type u} {β : Type v} [normed_group β] [h : nonempty α] (b : β) :
def bounded_continuous_function.of_normed_group {α : Type u} {β : Type v} [normed_group β] (f : α → β) (Hf : continuous f) (C : ) (H : ∀ (x : α), f x C) :
α →ᵇ β

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

Equations
@[simp]
theorem bounded_continuous_function.coe_of_normed_group {α : Type u} {β : Type v} [normed_group β] (f : α → β) (Hf : continuous f) (C : ) (H : ∀ (x : α), f x C) :
= f
theorem bounded_continuous_function.norm_of_normed_group_le {α : Type u} {β : Type v} [normed_group β] {f : α → β} (hfc : continuous f) {C : } (hC : 0 C) (hfC : ∀ (x : α), f x C) :
hfC C
def bounded_continuous_function.of_normed_group_discrete {α : Type u} {β : Type v} [normed_group β] (f : α → β) (C : ) (H : ∀ (x : α), f x C) :
α →ᵇ β

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

Equations
@[simp]
theorem bounded_continuous_function.coe_of_normed_group_discrete {α : Type u} {β : Type v} [normed_group β] (f : α → β) (C : ) (H : ∀ (x : α), f x C) :
noncomputable def bounded_continuous_function.norm_comp {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :

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

Equations
• f.norm_comp = bounded_continuous_function.norm_comp._proof_1 f
@[simp]
theorem bounded_continuous_function.coe_norm_comp {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
@[simp]
theorem bounded_continuous_function.norm_norm_comp {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
theorem bounded_continuous_function.bdd_above_range_norm_comp {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
theorem bounded_continuous_function.norm_eq_supr_norm {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
f = ⨆ (x : α), f x
@[protected, instance]
noncomputable def bounded_continuous_function.has_neg {α : Type u} {β : Type v} [normed_group β] :
has_neg →ᵇ β)

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

Equations
@[protected, instance]
noncomputable def bounded_continuous_function.has_sub {α : Type u} {β : Type v} [normed_group β] :
has_sub →ᵇ β)

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

Equations
@[simp]
theorem bounded_continuous_function.coe_neg {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
theorem bounded_continuous_function.neg_apply {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) {x : α} :
(-f) x = -f x
@[protected, instance]
noncomputable def bounded_continuous_function.add_comm_group {α : Type u} {β : Type v} [normed_group β] :
Equations
@[simp]
theorem bounded_continuous_function.coe_sub {α : Type u} {β : Type v} [normed_group β] (f g : α →ᵇ β) :
(f - g) = f - g
theorem bounded_continuous_function.sub_apply {α : Type u} {β : Type v} [normed_group β] (f g : α →ᵇ β) {x : α} :
(f - g) x = f x - g x
@[protected, instance]
noncomputable def bounded_continuous_function.normed_group {α : Type u} {β : Type v} [normed_group β] :
Equations
theorem bounded_continuous_function.abs_diff_coe_le_dist {α : Type u} {β : Type v} [normed_group β] (f g : α →ᵇ β) {x : α} :
f x - g x dist f g
theorem bounded_continuous_function.coe_le_coe_add_dist {α : Type u} {x : α} {f g : α →ᵇ } :
f x g x + dist f g
theorem bounded_continuous_function.norm_comp_continuous_le {α : Type u} {β : Type v} {γ : Type w} [normed_group β] (f : α →ᵇ β) (g : C(γ, α)) :

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.

@[protected, instance]
noncomputable def bounded_continuous_function.has_scalar {α : Type u} {β : Type v} {𝕜 : Type u_1} [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] :
→ᵇ β)
Equations
@[simp]
theorem bounded_continuous_function.coe_smul {α : Type u} {β : Type v} {𝕜 : Type u_1} [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] (c : 𝕜) (f : α →ᵇ β) :
(c f) = λ (x : α), c f x
theorem bounded_continuous_function.smul_apply {α : Type u} {β : Type v} {𝕜 : Type u_1} [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] (c : 𝕜) (f : α →ᵇ β) (x : α) :
(c f) x = c f x
@[protected, instance]
def bounded_continuous_function.has_bounded_smul {α : Type u} {β : Type v} {𝕜 : Type u_1} [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] :
→ᵇ β)
@[protected, instance]
noncomputable def bounded_continuous_function.module {α : Type u} {β : Type v} {𝕜 : Type u_1} [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β]  :
→ᵇ β)
Equations
noncomputable def bounded_continuous_function.eval_clm {α : Type u} {β : Type v} (𝕜 : Type u_1) [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] (x : α) :
→ᵇ β) →L[𝕜] β

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

Equations
@[simp]
theorem bounded_continuous_function.eval_clm_apply {α : Type u} {β : Type v} (𝕜 : Type u_1) [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] (x : α) (f : α →ᵇ β) :
= f x
@[simp]
theorem bounded_continuous_function.to_continuous_map_linear_map_apply (α : Type u) (β : Type v) (𝕜 : Type u_1) [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β] (self : α →ᵇ β) :
noncomputable def bounded_continuous_function.to_continuous_map_linear_map (α : Type u) (β : Type v) (𝕜 : Type u_1) [metric_space 𝕜] [semiring 𝕜] [metric_space β] [ β] [ β]  :
→ᵇ β) →ₗ[𝕜] C(α, β)

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

Equations

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, instance]
noncomputable def bounded_continuous_function.normed_space {α : Type u} {β : Type v} {𝕜 : Type u_1} [normed_group β] [normed_field 𝕜] [ β] :
→ᵇ β)
Equations
@[protected]
noncomputable def continuous_linear_map.comp_left_continuous_bounded (α : Type u) {β : Type v} {γ : Type w} {𝕜 : Type u_1} [normed_group β] [ β] [normed_group γ] [ γ] (g : β →L[𝕜] γ) :
→ᵇ β) →L[𝕜] α →ᵇ γ

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.

Equations
@[simp]
theorem continuous_linear_map.comp_left_continuous_bounded_apply (α : Type u) {β : Type v} {γ : Type w} {𝕜 : Type u_1} [normed_group β] [ β] [normed_group γ] [ γ] (g : β →L[𝕜] γ) (f : α →ᵇ β) (x : α) :
= g (f x)

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]
noncomputable def bounded_continuous_function.ring {α : Type u} {R : Type u_1} [normed_ring R] :
ring →ᵇ R)
Equations
@[simp]
theorem bounded_continuous_function.coe_mul {α : Type u} {R : Type u_1} [normed_ring R] (f g : α →ᵇ R) :
f * g = (f) * g
theorem bounded_continuous_function.mul_apply {α : Type u} {R : Type u_1} [normed_ring R] (f g : α →ᵇ R) (x : α) :
(f * g) x = (f x) * g x
@[protected, instance]
noncomputable def bounded_continuous_function.normed_ring {α : Type u} {R : Type u_1} [normed_ring R] :
Equations

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.

@[protected, instance]
noncomputable def bounded_continuous_function.comm_ring {α : Type u} {R : Type u_1}  :
Equations
@[protected, instance]
noncomputable def bounded_continuous_function.normed_comm_ring {α : Type u} {R : Type u_1}  :
Equations

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.

noncomputable def bounded_continuous_function.C {α : Type u} {γ : Type w} {𝕜 : Type u_1} [normed_field 𝕜] [normed_ring γ] [ γ] :
𝕜 →+* α →ᵇ γ

bounded_continuous_function.const as a ring_hom.

Equations
@[protected, instance]
noncomputable def bounded_continuous_function.algebra {α : Type u} {γ : Type w} {𝕜 : Type u_1} [normed_field 𝕜] [normed_ring γ] [ γ] :
→ᵇ γ)
Equations
@[simp]
theorem bounded_continuous_function.algebra_map_apply {α : Type u} {γ : Type w} {𝕜 : Type u_1} [normed_field 𝕜] [normed_ring γ] [ γ] (k : 𝕜) (a : α) :
( →ᵇ γ)) k) a = k 1
@[protected, instance]
noncomputable def bounded_continuous_function.normed_algebra {α : Type u} {γ : Type w} {𝕜 : Type u_1} [normed_field 𝕜] [normed_ring γ] [ γ] [nonempty α] :
→ᵇ γ)
Equations

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.

@[protected, instance]
noncomputable def bounded_continuous_function.has_scalar' {α : Type u} {β : Type v} {𝕜 : Type u_1} [normed_field 𝕜] [normed_group β] [ β] :
has_scalar →ᵇ 𝕜) →ᵇ β)
Equations
@[protected, instance]
noncomputable def bounded_continuous_function.module' {α : Type u} {β : Type v} {𝕜 : Type u_1} [normed_field 𝕜] [normed_group β] [ β] :
module →ᵇ 𝕜) →ᵇ β)
Equations
theorem bounded_continuous_function.norm_smul_le {α : Type u} {β : Type v} {𝕜 : Type u_1} [normed_field 𝕜] [normed_group β] [ β] (f : α →ᵇ 𝕜) (g : α →ᵇ β) :
theorem bounded_continuous_function.nnreal.upper_bound {α : Type u_1} (f : α →ᵇ ℝ≥0) (x : α) :
f x 0

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).

@[protected, instance]
noncomputable def bounded_continuous_function.star_add_monoid {α : Type u} {β : Type v} [normed_group β]  :
Equations
@[simp]
theorem bounded_continuous_function.coe_star {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) :
(star f) =

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.

@[simp]
theorem bounded_continuous_function.star_apply {α : Type u} {β : Type v} [normed_group β] (f : α →ᵇ β) (x : α) :
(star f) x = star (f x)
@[protected, instance]
noncomputable def bounded_continuous_function.normed_star_monoid {α : Type u} {β : Type v} [normed_group β]  :
Equations
@[protected, instance]
noncomputable def bounded_continuous_function.star_module {α : Type u} {β : Type v} {𝕜 : Type u_1} [normed_field 𝕜] [star_ring 𝕜] [normed_group β] [ β] [ β] :
→ᵇ β)
Equations
@[protected, instance]
noncomputable def bounded_continuous_function.star_ring {α : Type u} {β : Type v} [normed_ring β] [star_ring β]  :
Equations
@[protected, instance]
noncomputable def bounded_continuous_function.cstar_ring {α : Type u} {β : Type v} [normed_ring β] [star_ring β] [cstar_ring β] :
Equations
@[protected, instance]
def bounded_continuous_function.partial_order {α : Type u} {β : Type v}  :
Equations
@[protected, instance]
def bounded_continuous_function.semilattice_inf {α : Type u} {β : Type v}  :

Continuous normed lattice group valued functions form a meet-semilattice

Equations
@[protected, instance]
def bounded_continuous_function.semilattice_sup {α : Type u} {β : Type v}  :
Equations
@[protected, instance]
def bounded_continuous_function.lattice {α : Type u} {β : Type v}  :
lattice →ᵇ β)
Equations
@[simp]
theorem bounded_continuous_function.coe_fn_sup {α : Type u} {β : Type v} (f g : α →ᵇ β) :
(f g) = f g
@[simp]
theorem bounded_continuous_function.coe_fn_abs {α : Type u} {β : Type v} (f : α →ᵇ β) :
@[protected, instance]
noncomputable def bounded_continuous_function.normed_lattice_add_comm_group {α : Type u} {β : Type v}  :
Equations