# mathlibdocumentation

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.

@[simp]
theorem continuous_map.equiv_bounded_of_compact_apply (α : Type u_1) (μ : Type u_3) [metric_space μ] (f : C(α, μ)) :
@[simp]
theorem continuous_map.equiv_bounded_of_compact_symm_apply (α : Type u_1) (μ : Type u_3) [metric_space μ] (ᾰ : α →ᵇ μ) :
def continuous_map.equiv_bounded_of_compact (α : Type u_1) (μ : Type u_3) [metric_space μ] :
C(α, μ) →ᵇ μ)

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

Equations
@[simp]
theorem continuous_map.add_equiv_bounded_of_compact_symm_apply (α : Type u_1) (β : Type u_2) [normed_group β] (ᾰ : α →ᵇ β) :
def continuous_map.add_equiv_bounded_of_compact (α : Type u_1) (β : Type u_2) [normed_group β] :
C(α, β) ≃+ →ᵇ β)

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

Equations
@[simp]
theorem continuous_map.add_equiv_bounded_of_compact_apply_to_continuous_map (α : Type u_1) (β : Type u_2) [normed_group β] (ᾰ : C(α, β)) :
@[simp]
theorem continuous_map.add_equiv_bounded_of_compact_apply_apply (α : Type u_1) (β : Type u_2) [normed_group β] (f : C(α, β)) (a : α) :
= f a
@[simp]
theorem continuous_map.add_equiv_bounded_of_compact_to_equiv (α : Type u_1) (β : Type u_2) [normed_group β] :
@[instance]
def continuous_map.metric_space (α : Type u_1) (μ : Type u_3) [metric_space μ] :
Equations
theorem continuous_map.dist_le {α : Type u_1} {β : Type u_2} [normed_group β] (f g : C(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 continuous_map.dist_le_iff_of_nonempty {α : Type u_1} {β : Type u_2} [normed_group β] (f g : C(α, β)) {C : } [nonempty α] :
dist f g C ∀ (x : α), dist (f x) (g x) C
theorem continuous_map.dist_lt_of_nonempty {α : Type u_1} {β : Type u_2} [normed_group β] (f g : C(α, β)) {C : } [nonempty α] (w : ∀ (x : α), dist (f x) (g x) < C) :
dist f g < C
theorem continuous_map.dist_lt_iff {α : Type u_1} {β : Type u_2} [normed_group β] (f g : C(α, β)) {C : } (C0 : 0 < C) :
dist f g < C ∀ (x : α), dist (f x) (g x) < C
theorem continuous_map.dist_lt_iff_of_nonempty {α : Type u_1} {β : Type u_2} [normed_group β] (f g : C(α, β)) {C : } [nonempty α] :
dist f g < C ∀ (x : α), dist (f x) (g x) < C
def continuous_map.isometric_bounded_of_compact (α : Type u_1) (μ : Type u_3) [metric_space μ] :
C(α, μ) ≃ᵢ →ᵇ μ)

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

Equations
@[simp]
theorem continuous_map.isometric_bounded_of_compact_apply_to_continuous_map (α : Type u_1) (μ : Type u_3) [metric_space μ] (f : C(α, μ)) :
@[simp]
theorem continuous_map.isometric_bounded_of_compact_symm_apply (α : Type u_1) (μ : Type u_3) [metric_space μ] (ᾰ : α →ᵇ μ) :
@[simp]
theorem continuous_map.isometric_bounded_of_compact_to_equiv (α : Type u_1) (μ : Type u_3) [metric_space μ] :
@[instance]
def continuous_map.has_norm (α : Type u_1) (β : Type u_2) [normed_group β] :
Equations
@[instance]
def continuous_map.normed_group (α : Type u_1) (β : Type u_2) [normed_group β] :
Equations
theorem continuous_map.norm_coe_le_norm {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(α, β)) (x : α) :
theorem continuous_map.dist_le_two_norm {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(α, 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 continuous_map.norm_le {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(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 continuous_map.norm_le_of_nonempty {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(α, β)) [nonempty α] {M : } :
f M ∀ (x : α), f x M
theorem continuous_map.norm_lt_iff {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(α, β)) {M : } (M0 : 0 < M) :
f < M ∀ (x : α), f x < M
theorem continuous_map.norm_lt_iff_of_nonempty {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(α, β)) [nonempty α] {M : } :
f < M ∀ (x : α), f x < M
theorem continuous_map.apply_le_norm {α : Type u_1} (f : C(α, )) (x : α) :
theorem continuous_map.neg_norm_le_apply {α : Type u_1} (f : C(α, )) (x : α) :
@[simp]
theorem bounded_continuous_function.norm_forget_boundedness_eq {α : Type u_1} {β : Type u_2} [normed_group β] (f : α →ᵇ β) :
theorem continuous_map.norm_eq_supr_norm {α : Type u_1} {β : Type u_2} [normed_group β] (f : C(α, β)) :
f = ⨆ (x : α), f x
@[instance]
def continuous_map.normed_ring (α : Type u_1) {R : Type u_4} [normed_ring R] :
Equations
@[instance]
def continuous_map.normed_space (α : Type u_1) (β : Type u_2) [normed_group β] {𝕜 : Type u_4} [normed_field 𝕜] [ β] :
C(α, β)
Equations
def continuous_map.linear_isometry_bounded_of_compact (α : Type u_1) (β : Type u_2) [normed_group β] (𝕜 : Type u_4) [normed_field 𝕜] [ β] :
C(α, β) ≃ₗᵢ[𝕜] α →ᵇ β

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
@[simp]
theorem continuous_map.linear_isometry_bounded_of_compact_symm_apply (α : Type u_1) (β : Type u_2) [normed_group β] (𝕜 : Type u_4) [normed_field 𝕜] [ β] (f : α →ᵇ β) :
@[simp]
theorem continuous_map.linear_isometry_bounded_of_compact_apply_apply (α : Type u_1) (β : Type u_2) [normed_group β] (𝕜 : Type u_4) [normed_field 𝕜] [ β] (f : C(α, β)) (a : α) :
a = f a
@[simp]
theorem continuous_map.linear_isometry_bounded_of_compact_to_isometric (α : Type u_1) (β : Type u_2) [normed_group β] (𝕜 : Type u_4) [normed_field 𝕜] [ β] :
@[simp]
theorem continuous_map.linear_isometry_bounded_of_compact_to_add_equiv (α : Type u_1) (β : Type u_2) [normed_group β] (𝕜 : Type u_4) [normed_field 𝕜] [ β] :
@[simp]
theorem continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv (α : Type u_1) (β : Type u_2) [normed_group β] (𝕜 : Type u_4) [normed_field 𝕜] [ β] :
@[instance]
def continuous_map.normed_algebra (α : Type u_1) {𝕜 : Type u_4} {γ : Type u_5} [normed_field 𝕜] [normed_ring γ] [ γ] [nonempty α] :
C(α, 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] [metric_space β] (f : C(α, β)) (ε : ) (h : 0 < ε) :
∃ (δ : ) (H : δ > 0), ∀ {x y : α}, dist x y < δdist (f x) (f y) < ε
def continuous_map.modulus {α : Type u_1} {β : Type u_2} [metric_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] [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] [metric_space β] (f : C(α, β)) (ε : ) (h : 0 < ε) {a b : α} (w : dist a b < f.modulus ε h) :
dist (f a) (f b) < ε
def continuous_linear_map.comp_left_continuous_compact (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [normed_group β] [ β] [normed_group γ] [ γ] (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ)

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

Equations
@[simp]
theorem continuous_linear_map.to_linear_comp_left_continuous_compact (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [normed_group β] [ β] [normed_group γ] [ γ] (g : β →L[𝕜] γ) :
@[simp]
theorem continuous_linear_map.comp_left_continuous_compact_apply (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [normed_group β] [ β] [normed_group γ] [ γ] (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
= g (f x)

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:

• comp_right_continuous_map, the bundled continuous map (for this we need X Y compact).
• comp_right_homeomorph, when we precompose by a homeomorphism.
• comp_right_alg_hom, when T = R is a topological ring.
def continuous_map.comp_right_continuous_map {X : Type u_1} {Y : Type u_2} (T : Type u_3) [normed_group T] (f : C(X, Y)) :

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

Equations
@[simp]
theorem continuous_map.comp_right_continuous_map_apply {X : Type u_1} {Y : Type u_2} (T : Type u_3) [normed_group T] (f : C(X, Y)) (g : C(Y, T)) :
= g.comp f
def continuous_map.comp_right_homeomorph {X : Type u_1} {Y : Type u_2} (T : Type u_3) [normed_group T] (f : X ≃ₜ Y) :

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

Equations
def continuous_map.comp_right_alg_hom {X : Type u_1} {Y : Type u_2} (R : Type u_3) (f : C(X, Y)) :

Precomposition of functions into a normed ring by continuous map is an algebra homomorphism.

Equations
@[simp]
theorem continuous_map.comp_right_alg_hom_apply {X : Type u_1} {Y : Type u_2} (R : Type u_3) (f : C(X, Y)) (g : C(Y, R)) :
= g.comp f
theorem continuous_map.comp_right_alg_hom_continuous {X : Type u_1} {Y : Type u_2} (R : Type u_3) (f : C(X, Y)) :