mathlib documentation

topology.continuous_function.compact

Continuous functions on a compact space #

Continuous functions C(α, β) from a compact space α to a metric space β are automatically bounded, and so acquire various structures inherited from α →ᵇ β.

This file transfers these structures, and restates some lemmas characterising these structures.

If you need a lemma which is proved about α →ᵇ β but not for C(α, β) when α is compact, you should restate it here. You can also use bounded_continuous_function.equiv_continuous_map_of_compact to move functions back and forth.

When α is compact, the bounded continuous maps α →ᵇ 𝕜 are additively equivalent to C(α, 𝕜).

Equations

When α is compact, and β is a metric space, the bounded continuous maps α →ᵇ β are isometric to C(α, β).

Equations
theorem continuous_map.dist_apply_le_dist {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] {f g : C(α, β)} (x : α) :

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

theorem continuous_map.dist_le {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] {f g : C(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

theorem continuous_map.dist_le_iff_of_nonempty {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] {f g : C(α, β)} {C : } [nonempty α] :
has_dist.dist f g C (x : α), has_dist.dist (f x) (g x) C
theorem continuous_map.dist_lt_iff_of_nonempty {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] {f g : C(α, β)} {C : } [nonempty α] :
has_dist.dist f g < C (x : α), has_dist.dist (f x) (g x) < C
theorem continuous_map.dist_lt_of_nonempty {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] {f g : C(α, β)} {C : } [nonempty α] (w : (x : α), has_dist.dist (f x) (g x) < C) :
theorem continuous_map.dist_lt_iff {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] {f g : C(α, β)} {C : } (C0 : 0 < C) :
has_dist.dist f g < C (x : α), has_dist.dist (f x) (g x) < C
@[protected, instance]
@[continuity]
theorem continuous_map.continuous_eval {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] :
continuous (λ (p : C(α, β) × α), (p.fst) p.snd)

See also continuous_map.continuous_eval'

@[continuity]
theorem continuous_map.continuous_eval_const {α : Type u_1} {β : Type u_2} [topological_space α] [compact_space α] [metric_space β] (x : α) :
continuous (λ (f : C(α, β)), f x)

See also continuous_map.continuous_eval_const

See also continuous_map.continuous_coe'

@[protected, instance]
noncomputable def continuous_map.has_norm {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] :
Equations
@[protected, instance]
theorem continuous_map.norm_coe_le_norm {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) (x : α) :
theorem continuous_map.dist_le_two_norm {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) (x y : α) :

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

theorem continuous_map.norm_le {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) {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 continuous_map.norm_le_of_nonempty {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) [nonempty α] {M : } :
f M (x : α), f x M
theorem continuous_map.norm_lt_iff {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) {M : } (M0 : 0 < M) :
f < M (x : α), f x < M
theorem continuous_map.nnnorm_lt_iff {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) {M : nnreal} (M0 : 0 < M) :
f‖₊ < M (x : α), f x‖₊ < M
theorem continuous_map.norm_lt_iff_of_nonempty {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) [nonempty α] {M : } :
f < M (x : α), f x < M
theorem continuous_map.nnnorm_lt_iff_of_nonempty {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) [nonempty α] {M : nnreal} :
f‖₊ < M (x : α), f x‖₊ < M
theorem continuous_map.apply_le_norm {α : Type u_1} [topological_space α] [compact_space α] (f : C(α, )) (x : α) :
theorem continuous_map.neg_norm_le_apply {α : Type u_1} [topological_space α] [compact_space α] (f : C(α, )) (x : α) :
theorem continuous_map.norm_eq_supr_norm {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (f : C(α, E)) :
f = (x : α), f x
@[protected, instance]
def continuous_map.normed_space {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] {𝕜 : Type u_4} [normed_field 𝕜] [normed_space 𝕜 E] :
Equations

When α is compact and 𝕜 is a normed field, the 𝕜-algebra of bounded continuous maps α →ᵇ β is 𝕜-linearly isometric to C(α, β).

Equations
noncomputable def continuous_map.eval_clm {α : Type u_1} {E : Type u_3} [topological_space α] [compact_space α] [normed_add_comm_group E] (𝕜 : Type u_4) [normed_field 𝕜] [normed_space 𝕜 E] (x : α) :
C(α, E) →L[𝕜] E

The evaluation at a point, as a continuous linear map from C(α, 𝕜) to 𝕜.

Equations
@[protected, instance]
def continuous_map.normed_algebra {α : Type u_1} [topological_space α] [compact_space α] {𝕜 : Type u_4} {γ : Type u_5} [normed_field 𝕜] [normed_ring γ] [normed_algebra 𝕜 γ] :
Equations

We now set up some declarations making it convenient to use uniform continuity.

theorem continuous_map.uniform_continuity {α : Type u_1} {β : Type u_2} [metric_space α] [compact_space α] [metric_space β] (f : C(α, β)) (ε : ) (h : 0 < ε) :
(δ : ) (H : δ > 0), {x y : α}, has_dist.dist x y < δ has_dist.dist (f x) (f y) < ε
noncomputable def continuous_map.modulus {α : Type u_1} {β : Type u_2} [metric_space α] [compact_space α] [metric_space β] (f : C(α, β)) (ε : ) (h : 0 < ε) :

An arbitrarily chosen modulus of uniform continuity for a given function f and ε > 0.

Equations
theorem continuous_map.modulus_pos {α : Type u_1} {β : Type u_2} [metric_space α] [compact_space α] [metric_space β] (f : C(α, β)) {ε : } {h : 0 < ε} :
0 < f.modulus ε h
theorem continuous_map.dist_lt_of_dist_lt_modulus {α : Type u_1} {β : Type u_2} [metric_space α] [compact_space α] [metric_space β] (f : C(α, β)) (ε : ) (h : 0 < ε) {a b : α} (w : has_dist.dist a b < f.modulus ε h) :
has_dist.dist (f a) (f b) < ε

We now setup variations on comp_right_* f, where f : C(X, Y) (that is, precomposition by a continuous map), as a morphism C(Y, T) → C(X, T), respecting various types of structure.

In particular:

Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.

Equations

Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.

Equations

Star structures #

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

Furthermore, if α is compact and β is a C⋆-ring, then C(α, β) is a C⋆-ring.

@[protected, instance]