mathlib documentation

topology.metric_space.isometry

Isometries #

We define isometries, i.e., maps between emetric spaces that preserve the edistance (on metric spaces, these are exactly the maps that preserve distances), and prove their basic properties. We also introduce isometric bijections.

Since a lot of elementary properties don't require eq_of_dist_eq_zero we start setting up the theory for pseudo_metric_space and we specialize to metric_space when needed.

def isometry {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (f : α → β) :
Prop

An isometry (also known as isometric embedding) is a map preserving the edistance between pseudoemetric spaces, or equivalently the distance between pseudometric space.

Equations
theorem isometry_emetric_iff_metric {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] {f : α → β} :
isometry f ∀ (x y : α), dist (f x) (f y) = dist x y

On pseudometric spaces, a map is an isometry if and only if it preserves distances.

theorem isometry.edist_eq {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (hf : isometry f) (x y : α) :
edist (f x) (f y) = edist x y

An isometry preserves edistances.

theorem isometry.dist_eq {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] {f : α → β} (hf : isometry f) (x y : α) :
dist (f x) (f y) = dist x y

An isometry preserves distances.

theorem isometry.lipschitz {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (h : isometry f) :
theorem isometry.antilipschitz {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (h : isometry f) :
theorem isometry.injective {β : Type v} [pseudo_emetric_space β] {α : Type u} [emetric_space α] {f : α → β} (h : isometry f) :

An isometry from an emetric space is injective

theorem isometry_subsingleton {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} [subsingleton α] :

Any map on a subsingleton is an isometry

theorem isometry_id {α : Type u} [pseudo_emetric_space α] :

The identity is an isometry

theorem isometry.comp {α : Type u} {β : Type v} {γ : Type w} [pseudo_emetric_space α] [pseudo_emetric_space β] [pseudo_emetric_space γ] {g : β → γ} {f : α → β} (hg : isometry g) (hf : isometry f) :

The composition of isometries is an isometry

theorem isometry.uniform_inducing {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (hf : isometry f) :

An isometry from a metric space is a uniform inducing map

theorem isometry.uniform_embedding {α : Type u} {β : Type v} [emetric_space α] [pseudo_emetric_space β] {f : α → β} (hf : isometry f) :

An isometry from a metric space is a uniform embedding

theorem isometry.closed_embedding {α : Type u} {β : Type v} [emetric_space α] [complete_space α] [emetric_space β] {f : α → β} (hf : isometry f) :

An isometry from a complete emetric space is a closed embedding

theorem isometry.continuous {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (hf : isometry f) :

An isometry is continuous.

theorem isometry.right_inv {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} {g : β → α} (h : isometry f) (hg : function.right_inverse g f) :

The right inverse of an isometry is an isometry.

theorem isometry.ediam_image {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (hf : isometry f) (s : set α) :

Isometries preserve the diameter in pseudoemetric spaces.

theorem isometry.ediam_range {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} (hf : isometry f) :
theorem isometry_subtype_coe {α : Type u} [pseudo_emetric_space α] {s : set α} :

The injection from a subtype is an isometry

theorem isometry.comp_continuous_on_iff {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} {γ : Type u_1} [topological_space γ] (hf : isometry f) {g : γ → α} {s : set γ} :
theorem isometry.comp_continuous_iff {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {f : α → β} {γ : Type u_1} [topological_space γ] (hf : isometry f) {g : γ → α} :
theorem isometry.diam_image {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] {f : α → β} (hf : isometry f) (s : set α) :

An isometry preserves the diameter in pseudometric spaces.

theorem isometry.diam_range {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] {f : α → β} (hf : isometry f) :
theorem add_monoid_hom.isometry_iff_norm {E : Type u_1} {F : Type u_2} [semi_normed_group E] [semi_normed_group F] (f : E →+ F) :
isometry f ∀ (x : E), f x = x
theorem add_monoid_hom.isometry_of_norm {E : Type u_1} {F : Type u_2} [semi_normed_group E] [semi_normed_group F] (f : E →+ F) (hf : ∀ (x : E), f x = x) :
@[nolint]
structure isometric (α : Type u_1) (β : Type u_2) [pseudo_emetric_space α] [pseudo_emetric_space β] :
Type (max u_1 u_2)

α and β are isometric if there is an isometric bijection between them.

@[instance]
def isometric.has_coe_to_fun {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] :
Equations
theorem isometric.coe_eq_to_equiv {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) (a : α) :
h a = (h.to_equiv) a
@[simp]
theorem isometric.coe_to_equiv {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.isometry {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.bijective {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.injective {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.surjective {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.edist_eq {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) (x y : α) :
edist (h x) (h y) = edist x y
theorem isometric.dist_eq {α : Type u_1} {β : Type u_2} [pseudo_metric_space α] [pseudo_metric_space β] (h : α ≃ᵢ β) (x y : α) :
dist (h x) (h y) = dist x y
theorem isometric.continuous {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
@[simp]
theorem isometric.ediam_image {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) (s : set α) :
theorem isometric.to_equiv_inj {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] ⦃h₁ h₂ : α ≃ᵢ β⦄ :
h₁.to_equiv = h₂.to_equivh₁ = h₂
@[ext]
theorem isometric.ext {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ (x : α), h₁ x = h₂ x) :
h₁ = h₂
def isometric.mk' {β : Type v} [pseudo_emetric_space β] {α : Type u} [emetric_space α] (f : α → β) (g : β → α) (hfg : ∀ (x : β), f (g x) = x) (hf : isometry f) :
α ≃ᵢ β

Alternative constructor for isometric bijections, taking as input an isometry, and a right inverse.

Equations
def isometric.refl (α : Type u_1) [pseudo_emetric_space α] :
α ≃ᵢ α

The identity isometry of a space.

Equations
def isometric.trans {α : Type u} {β : Type v} {γ : Type w} [pseudo_emetric_space α] [pseudo_emetric_space β] [pseudo_emetric_space γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) :
α ≃ᵢ γ

The composition of two isometric isomorphisms, as an isometric isomorphism.

Equations
@[simp]
theorem isometric.trans_apply {α : Type u} {β : Type v} {γ : Type w} [pseudo_emetric_space α] [pseudo_emetric_space β] [pseudo_emetric_space γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) :
(h₁.trans h₂) x = h₂ (h₁ x)
def isometric.symm {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
β ≃ᵢ α

The inverse of an isometric isomorphism, as an isometric isomorphism.

Equations
@[simp]
theorem isometric.symm_symm {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
h.symm.symm = h
@[simp]
theorem isometric.apply_symm_apply {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) (y : β) :
h ((h.symm) y) = y
@[simp]
theorem isometric.symm_apply_apply {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) (x : α) :
(h.symm) (h x) = x
theorem isometric.symm_apply_eq {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) {x : α} {y : β} :
(h.symm) y = x y = h x
theorem isometric.eq_symm_apply {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) {x : α} {y : β} :
x = (h.symm) y h x = y
theorem isometric.symm_comp_self {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.self_comp_symm {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
@[simp]
theorem isometric.range_eq_univ {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.image_symm {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
theorem isometric.preimage_symm {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
@[simp]
theorem isometric.symm_trans_apply {α : Type u} {β : Type v} {γ : Type w} [pseudo_emetric_space α] [pseudo_emetric_space β] [pseudo_emetric_space γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
((h₁.trans h₂).symm) x = (h₁.symm) ((h₂.symm) x)
@[simp]
theorem isometric.ediam_preimage {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) (s : set β) :
def isometric.to_homeomorph {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
α ≃ₜ β

The (bundled) homeomorphism associated to an isometric isomorphism.

Equations
@[simp]
theorem isometric.coe_to_homeomorph {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
@[simp]
theorem isometric.coe_to_homeomorph_symm {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
@[simp]
theorem isometric.to_homeomorph_to_equiv {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] (h : α ≃ᵢ β) :
@[simp]
theorem isometric.comp_continuous_on_iff {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {γ : Type u_1} [topological_space γ] (h : α ≃ᵢ β) {f : γ → α} {s : set γ} :
@[simp]
theorem isometric.comp_continuous_iff {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {γ : Type u_1} [topological_space γ] (h : α ≃ᵢ β) {f : γ → α} :
@[simp]
theorem isometric.comp_continuous_iff' {α : Type u} {β : Type v} [pseudo_emetric_space α] [pseudo_emetric_space β] {γ : Type u_1} [topological_space γ] (h : α ≃ᵢ β) {f : β → γ} :
@[instance]
def isometric.group {α : Type u} [pseudo_emetric_space α] :
group ≃ᵢ α)

The group of isometries.

Equations
@[simp]
theorem isometric.coe_one {α : Type u} [pseudo_emetric_space α] :
@[simp]
theorem isometric.coe_mul {α : Type u} [pseudo_emetric_space α] (e₁ e₂ : α ≃ᵢ α) :
e₁ * e₂ = e₁ e₂
theorem isometric.mul_apply {α : Type u} [pseudo_emetric_space α] (e₁ e₂ : α ≃ᵢ α) (x : α) :
(e₁ * e₂) x = e₁ (e₂ x)
@[simp]
theorem isometric.inv_apply_self {α : Type u} [pseudo_emetric_space α] (e : α ≃ᵢ α) (x : α) :
e⁻¹ (e x) = x
@[simp]
theorem isometric.apply_inv_self {α : Type u} [pseudo_emetric_space α] (e : α ≃ᵢ α) (x : α) :
e (e⁻¹ x) = x
def isometric.add_right {G : Type u_1} [semi_normed_group G] (x : G) :
G ≃ᵢ G

Addition y ↦ y + x as an isometry.

Equations
@[simp]
theorem isometric.coe_add_right {G : Type u_1} [semi_normed_group G] (x : G) :
(isometric.add_right x) = λ (y : G), y + x
theorem isometric.add_right_apply {G : Type u_1} [semi_normed_group G] (x y : G) :
@[simp]
def isometric.add_left {G : Type u_1} [semi_normed_group G] (x : G) :
G ≃ᵢ G

Addition y ↦ x + y as an isometry.

Equations
@[simp]
theorem isometric.coe_add_left {G : Type u_1} [semi_normed_group G] (x : G) :
@[simp]
def isometric.neg (G : Type u_1) [semi_normed_group G] :
G ≃ᵢ G

Negation x ↦ -x as an isometry.

Equations
@[simp]
@[simp]
@[simp]
@[simp]
theorem isometric.diam_image {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] (h : α ≃ᵢ β) (s : set α) :
@[simp]
theorem isometric.diam_preimage {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] (h : α ≃ᵢ β) (s : set β) :
theorem isometric.diam_univ {α : Type u} {β : Type v} [pseudo_metric_space α] [pseudo_metric_space β] (h : α ≃ᵢ β) :
def isometry.isometric_on_range {α : Type u} {β : Type v} [emetric_space α] [pseudo_emetric_space β] {f : α → β} (h : isometry f) :

An isometry induces an isometric isomorphism between the source space and the range of the isometry.

Equations
@[simp]
theorem isometry.isometric_on_range_apply {α : Type u} {β : Type v} [emetric_space α] [pseudo_emetric_space β] {f : α → β} (h : isometry f) (x : α) :
(h.isometric_on_range) x = f x, _⟩
theorem algebra_map_isometry (𝕜 : Type u_1) (𝕜' : Type u_2) [normed_field 𝕜] [semi_normed_ring 𝕜'] [semi_normed_algebra 𝕜 𝕜'] :

In a normed algebra, the inclusion of the base field in the extended field is an isometry.