Documentation

Mathlib.Analysis.NormedSpace.LinearIsometry

(Semi-)linear isometries #

In this file we define LinearIsometry σ₁₂ E E₂ (notation: E →ₛₗᵢ[σ₁₂] E₂) to be a semilinear isometric embedding of E into E₂ and LinearIsometryEquiv (notation: E ≃ₛₗᵢ[σ₁₂] E₂) to be a semilinear isometric equivalence between E and E₂. The notation for the associated purely linear concepts is E →ₗᵢ[R] E₂, E ≃ₗᵢ[R] E₂, and E →ₗᵢ⋆[R] E₂, E ≃ₗᵢ⋆[R] E₂ for the star-linear versions.

We also prove some trivial lemmas and provide convenience constructors.

Since a lot of elementary properties don't require ‖x‖ = 0 → x = 0 we start setting up the theory for SeminormedAddCommGroup and we specialize to NormedAddCommGroup when needed.

structure LinearIsometry {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] (σ₁₂ : R →+* R₂) (E : Type u_11) (E₂ : Type u_12) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] extends LinearMap :

Type (max u_11 u_12)

A σ₁₂-semilinear isometric embedding of a normed R-module into an R₂-module.

toFun : E → E₂
map_add' : ∀ (x y : E), self.toFun (x + y) = self.toFun x + self.toFun y
map_smul' : ∀ (r : R) (x : E), self.toFun (r • x) = σ₁₂ r • self.toFun x
norm_map' : ∀ (x : E), ‖self.toLinearMap x‖ = ‖x‖

Instances For

def «term_→ₛₗᵢ[_]_» :

Lean.TrailingParserDescr

A σ₁₂-semilinear isometric embedding of a normed R-module into an R₂-module.

Equations

One or more equations did not get rendered due to their size.

def «term_→ₗᵢ[_]_» :

Lean.TrailingParserDescr

A linear isometric embedding of a normed R-module into another one.

Equations

One or more equations did not get rendered due to their size.

def «term_→ₗᵢ⋆[_]_» :

Lean.TrailingParserDescr

An antilinear isometric embedding of a normed R-module into another one.

Equations

One or more equations did not get rendered due to their size.

class SemilinearIsometryClass (𝓕 : Type u_11) {R : outParam (Type u_12)} {R₂ : outParam (Type u_13)} [Semiring R] [Semiring R₂] (σ₁₂ : outParam (R →+* R₂)) (E : outParam (Type u_14)) (E₂ : outParam (Type u_15)) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] extends SemilinearMapClass :

SemilinearIsometryClass F σ E E₂ asserts F is a type of bundled σ-semilinear isometries E → E₂.

See also LinearIsometryClass F R E E₂ for the case where σ is the identity map on R.

A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

map_add : ∀ (f : 𝓕) (x y : E), f (x + y) = f x + f y
map_smulₛₗ : ∀ (f : 𝓕) (r : R) (x : E), f (r • x) = σ₁₂ r • f x
norm_map : ∀ (f : 𝓕) (x : E), ‖f x‖ = ‖x‖

Instances

@[inline, reducible]

abbrev LinearIsometryClass (𝓕 : Type u_11) (R : outParam (Type u_12)) (E : outParam (Type u_13)) (E₂ : outParam (Type u_14)) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R E₂] [FunLike 𝓕 E E₂] :

LinearIsometryClass F R E E₂ asserts F is a type of bundled R-linear isometries M → M₂.

This is an abbreviation for SemilinearIsometryClass F (RingHom.id R) E E₂.

Equations

LinearIsometryClass 𝓕 R E E₂ = SemilinearIsometryClass 𝓕 (RingHom.id R) E E₂

theorem SemilinearIsometryClass.isometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :

theorem SemilinearIsometryClass.continuous {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :

Continuous ⇑f

theorem SemilinearIsometryClass.nnnorm_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E) :

‖f x‖₊ = ‖x‖₊

theorem SemilinearIsometryClass.lipschitz {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :

LipschitzWith 1 ⇑f

theorem SemilinearIsometryClass.antilipschitz {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :

AntilipschitzWith 1 ⇑f

theorem SemilinearIsometryClass.ediam_image {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : Set E) :

EMetric.diam (⇑f '' s) = EMetric.diam s

theorem SemilinearIsometryClass.ediam_range {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :

EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ

theorem SemilinearIsometryClass.diam_image {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : Set E) :

Metric.diam (⇑f '' s) = Metric.diam s

theorem SemilinearIsometryClass.diam_range {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :

Metric.diam (Set.range ⇑f) = Metric.diam Set.univ

instance SemilinearIsometryClass.toContinuousSemilinearMapClass {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] :

ContinuousSemilinearMapClass 𝓕 σ₁₂ E E₂

Equations

⋯ = ⋯

theorem LinearIsometry.toLinearMap_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometry.toLinearMap

@[simp]

theorem LinearIsometry.toLinearMap_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E →ₛₗᵢ[σ₁₂] E₂} {g : E →ₛₗᵢ[σ₁₂] E₂} :

f.toLinearMap = g.toLinearMap ↔ f = g

instance LinearIsometry.instFunLike {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

FunLike (E →ₛₗᵢ[σ₁₂] E₂) E E₂

Equations

LinearIsometry.instFunLike = { coe := fun (f : E →ₛₗᵢ[σ₁₂] E₂) => f.toFun, coe_injective' := ⋯ }

instance LinearIsometry.instSemilinearIsometryClass {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

SemilinearIsometryClass (E →ₛₗᵢ[σ₁₂] E₂) σ₁₂ E E₂

Equations

⋯ = ⋯

@[simp]

theorem LinearIsometry.coe_toLinearMap {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

⇑f.toLinearMap = ⇑f

@[simp]

theorem LinearIsometry.coe_mk {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗ[σ₁₂] E₂) (hf : ∀ (x : E), ‖f x‖ = ‖x‖) :

⇑{ toLinearMap := f, norm_map' := hf } = ⇑f

theorem LinearIsometry.coe_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective fun (f : E →ₛₗᵢ[σ₁₂] E₂) => ⇑f

def LinearIsometry.Simps.apply {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] (σ₁₂ : R →+* R₂) (E : Type u_11) (E₂ : Type u_12) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (h : E →ₛₗᵢ[σ₁₂] E₂) :

E → E₂

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

Equations

LinearIsometry.Simps.apply σ₁₂ E E₂ h = ⇑h

theorem LinearIsometry.ext {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E →ₛₗᵢ[σ₁₂] E₂} {g : E →ₛₗᵢ[σ₁₂] E₂} (h : ∀ (x : E), f x = g x) :

f = g

theorem LinearIsometry.congr_arg {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [FunLike 𝓕 E E₂] {f : 𝓕} {x : E} {x' : E} :

x = x' → f x = f x'

theorem LinearIsometry.congr_fun {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [FunLike 𝓕 E E₂] {f : 𝓕} {g : 𝓕} (h : f = g) (x : E) :

f x = g x

theorem LinearIsometry.map_zero {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

f 0 = 0

theorem LinearIsometry.map_add {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

f (x + y) = f x + f y

theorem LinearIsometry.map_neg {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) :

f (-x) = -f x

theorem LinearIsometry.map_sub {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

f (x - y) = f x - f y

theorem LinearIsometry.map_smulₛₗ {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (c : R) (x : E) :

f (c • x) = σ₁₂ c • f x

theorem LinearIsometry.map_smul {R : Type u_1} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R E₂] (f : E →ₗᵢ[R] E₂) (c : R) (x : E) :

f (c • x) = c • f x

@[simp]

theorem LinearIsometry.norm_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) :

‖f x‖ = ‖x‖

@[simp]

theorem LinearIsometry.nnnorm_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) :

‖f x‖₊ = ‖x‖₊

theorem LinearIsometry.isometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

theorem LinearIsometry.isComplete_image_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) {s : Set E} :

IsComplete (⇑f '' s) ↔ IsComplete s

@[simp]

theorem LinearIsometry.isComplete_image_iff' {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) {s : Set E} :

IsComplete (⇑f '' s) ↔ IsComplete s

theorem LinearIsometry.isComplete_map_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) [RingHomSurjective σ₁₂] {p : Submodule R E} :

IsComplete ↑(Submodule.map f.toLinearMap p) ↔ IsComplete ↑p

theorem LinearIsometry.isComplete_map_iff' {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) [RingHomSurjective σ₁₂] {p : Submodule R E} :

IsComplete ↑(Submodule.map f p) ↔ IsComplete ↑p

instance LinearIsometry.completeSpace_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) [RingHomSurjective σ₁₂] (p : Submodule R E) [CompleteSpace ↥p] :

CompleteSpace ↥(Submodule.map f p)

Equations

⋯ = ⋯

instance LinearIsometry.completeSpace_map' {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) [RingHomSurjective σ₁₂] (p : Submodule R E) [CompleteSpace ↥p] :

CompleteSpace ↥(Submodule.map f.toLinearMap p)

Equations

⋯ = ⋯

@[simp]

theorem LinearIsometry.dist_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

dist (f x) (f y) = dist x y

@[simp]

theorem LinearIsometry.edist_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

edist (f x) (f y) = edist x y

theorem LinearIsometry.injective {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f₁ : F →ₛₗᵢ[σ₁₂] E₂) :

Function.Injective ⇑f₁

@[simp]

theorem LinearIsometry.map_eq_iff {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f₁ : F →ₛₗᵢ[σ₁₂] E₂) {x : F} {y : F} :

f₁ x = f₁ y ↔ x = y

theorem LinearIsometry.map_ne {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f₁ : F →ₛₗᵢ[σ₁₂] E₂) {x : F} {y : F} (h : x ≠ y) :

f₁ x ≠ f₁ y

theorem LinearIsometry.lipschitz {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

LipschitzWith 1 ⇑f

theorem LinearIsometry.antilipschitz {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

AntilipschitzWith 1 ⇑f

theorem LinearIsometry.continuous {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

Continuous ⇑f

@[simp]

theorem LinearIsometry.preimage_ball {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ) :

⇑f ⁻¹' Metric.ball (f x) r = Metric.ball x r

@[simp]

theorem LinearIsometry.preimage_sphere {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ) :

⇑f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r

@[simp]

theorem LinearIsometry.preimage_closedBall {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ) :

⇑f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r

theorem LinearIsometry.ediam_image {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (s : Set E) :

EMetric.diam (⇑f '' s) = EMetric.diam s

theorem LinearIsometry.ediam_range {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ

theorem LinearIsometry.diam_image {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (s : Set E) :

Metric.diam (⇑f '' s) = Metric.diam s

theorem LinearIsometry.diam_range {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

Metric.diam (Set.range ⇑f) = Metric.diam Set.univ

def LinearIsometry.toContinuousLinearMap {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

E →SL[σ₁₂] E₂

Interpret a linear isometry as a continuous linear map.

Equations

LinearIsometry.toContinuousLinearMap f = { toLinearMap := f.toLinearMap, cont := ⋯ }

theorem LinearIsometry.toContinuousLinearMap_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometry.toContinuousLinearMap

@[simp]

theorem LinearIsometry.toContinuousLinearMap_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E →ₛₗᵢ[σ₁₂] E₂} {g : E →ₛₗᵢ[σ₁₂] E₂} :

LinearIsometry.toContinuousLinearMap f = LinearIsometry.toContinuousLinearMap g ↔ f = g

@[simp]

theorem LinearIsometry.coe_toContinuousLinearMap {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometry.toContinuousLinearMap f) = ⇑f

@[simp]

theorem LinearIsometry.comp_continuous_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) {α : Type u_11} [TopologicalSpace α] {g : α → E} :

Continuous (⇑f ∘ g) ↔ Continuous g

def LinearIsometry.id {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

E →ₗᵢ[R] E

The identity linear isometry.

Equations

LinearIsometry.id = { toLinearMap := LinearMap.id, norm_map' := ⋯ }

@[simp]

theorem LinearIsometry.coe_id {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

⇑LinearIsometry.id = id

@[simp]

theorem LinearIsometry.id_apply {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (x : E) :

LinearIsometry.id x = x

@[simp]

theorem LinearIsometry.id_toLinearMap {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

LinearIsometry.id.toLinearMap = LinearMap.id

@[simp]

theorem LinearIsometry.id_toContinuousLinearMap {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

LinearIsometry.toContinuousLinearMap LinearIsometry.id = ContinuousLinearMap.id R E

instance LinearIsometry.instInhabited {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

Inhabited (E →ₗᵢ[R] E)

Equations

LinearIsometry.instInhabited = { default := LinearIsometry.id }

def LinearIsometry.comp {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₁₃ : R →+* R₃} {σ₂₃ : R₂ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (f : E →ₛₗᵢ[σ₁₂] E₂) :

E →ₛₗᵢ[σ₁₃] E₃

Composition of linear isometries.

Equations

LinearIsometry.comp g f = { toLinearMap := LinearMap.comp g.toLinearMap f.toLinearMap, norm_map' := ⋯ }

@[simp]

theorem LinearIsometry.coe_comp {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₁₃ : R →+* R₃} {σ₂₃ : R₂ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (f : E →ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometry.comp g f) = ⇑g ∘ ⇑f

@[simp]

theorem LinearIsometry.id_comp {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

LinearIsometry.comp LinearIsometry.id f = f

@[simp]

theorem LinearIsometry.comp_id {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) :

LinearIsometry.comp f LinearIsometry.id = f

theorem LinearIsometry.comp_assoc {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {R₄ : Type u_4} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} {E₄ : Type u_8} [Semiring R] [Semiring R₂] [Semiring R₃] [Semiring R₄] {σ₁₂ : R →+* R₂} {σ₁₃ : R →+* R₃} {σ₁₄ : R →+* R₄} {σ₂₃ : R₂ →+* R₃} {σ₂₄ : R₂ →+* R₄} {σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [SeminormedAddCommGroup E₄] [Module R E] [Module R₂ E₂] [Module R₃ E₃] [Module R₄ E₄] (f : E₃ →ₛₗᵢ[σ₃₄] E₄) (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (h : E →ₛₗᵢ[σ₁₂] E₂) :

LinearIsometry.comp (LinearIsometry.comp f g) h = LinearIsometry.comp f (LinearIsometry.comp g h)

instance LinearIsometry.instMonoid {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

Monoid (E →ₗᵢ[R] E)

Equations

LinearIsometry.instMonoid = Monoid.mk ⋯ ⋯ npowRec ⋯ ⋯

@[simp]

theorem LinearIsometry.coe_one {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

⇑1 = id

@[simp]

theorem LinearIsometry.coe_mul {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (f : E →ₗᵢ[R] E) (g : E →ₗᵢ[R] E) :

⇑(f * g) = ⇑f ∘ ⇑g

theorem LinearIsometry.one_def {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

1 = LinearIsometry.id

theorem LinearIsometry.mul_def {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (f : E →ₗᵢ[R] E) (g : E →ₗᵢ[R] E) :

f * g = LinearIsometry.comp f g

theorem LinearIsometry.coe_pow {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (f : E →ₗᵢ[R] E) (n : ℕ) :

⇑(f ^ n) = (⇑f)^[n]

def LinearMap.toLinearIsometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗ[σ₁₂] E₂) (hf : Isometry ⇑f) :

E →ₛₗᵢ[σ₁₂] E₂

Construct a LinearIsometry from a LinearMap satisfying Isometry.

Equations

LinearMap.toLinearIsometry f hf = { toLinearMap := f, norm_map' := ⋯ }

def Submodule.subtypeₗᵢ {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E) :

↥p →ₗᵢ[R'] E

Submodule.subtype as a LinearIsometry.

Equations

Submodule.subtypeₗᵢ p = { toLinearMap := Submodule.subtype p, norm_map' := ⋯ }

@[simp]

theorem Submodule.coe_subtypeₗᵢ {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E) :

⇑(Submodule.subtypeₗᵢ p) = ⇑(Submodule.subtype p)

@[simp]

theorem Submodule.subtypeₗᵢ_toLinearMap {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E) :

(Submodule.subtypeₗᵢ p).toLinearMap = Submodule.subtype p

@[simp]

theorem Submodule.subtypeₗᵢ_toContinuousLinearMap {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E) :

LinearIsometry.toContinuousLinearMap (Submodule.subtypeₗᵢ p) = Submodule.subtypeL p

structure LinearIsometryEquiv {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (E : Type u_11) (E₂ : Type u_12) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] extends LinearEquiv :

Type (max u_11 u_12)

A semilinear isometric equivalence between two normed vector spaces.

toFun : E → E₂
map_add' : ∀ (x y : E), self.toFun (x + y) = self.toFun x + self.toFun y
map_smul' : ∀ (r : R) (x : E), self.toFun (r • x) = σ₁₂ r • self.toFun x
invFun : E₂ → E
left_inv : Function.LeftInverse self.invFun self.toFun
right_inv : Function.RightInverse self.invFun self.toFun
norm_map' : ∀ (x : E), ‖self.toLinearEquiv x‖ = ‖x‖

Instances For

def «term_≃ₛₗᵢ[_]_» :

Lean.TrailingParserDescr

A semilinear isometric equivalence between two normed vector spaces.

Equations

One or more equations did not get rendered due to their size.

def «term_≃ₗᵢ[_]_» :

Lean.TrailingParserDescr

A linear isometric equivalence between two normed vector spaces.

Equations

One or more equations did not get rendered due to their size.

def «term_≃ₗᵢ⋆[_]_» :

Lean.TrailingParserDescr

An antilinear isometric equivalence between two normed vector spaces.

Equations

One or more equations did not get rendered due to their size.

class SemilinearIsometryEquivClass (𝓕 : Type u_11) {R : outParam (Type u_12)} {R₂ : outParam (Type u_13)} [Semiring R] [Semiring R₂] (σ₁₂ : outParam (R →+* R₂)) {σ₂₁ : outParam (R₂ →+* R)} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (E : outParam (Type u_14)) (E₂ : outParam (Type u_15)) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [EquivLike 𝓕 E E₂] extends SemilinearEquivClass :

SemilinearIsometryEquivClass F σ E E₂ asserts F is a type of bundled σ-semilinear isometric equivs E → E₂.

See also LinearIsometryEquivClass F R E E₂ for the case where σ is the identity map on R.

A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

map_add : ∀ (f : 𝓕) (a b : E), f (a + b) = f a + f b
map_smulₛₗ : ∀ (f : 𝓕) (r : R) (x : E), f (r • x) = σ₁₂ r • f x
norm_map : ∀ (f : 𝓕) (x : E), ‖f x‖ = ‖x‖

Instances

LinearIsometryEquiv.instSemilinearIsometryEquivClass

@[inline, reducible]

abbrev LinearIsometryEquivClass (𝓕 : Type u_11) (R : outParam (Type u_12)) (E : outParam (Type u_13)) (E₂ : outParam (Type u_14)) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R E₂] [EquivLike 𝓕 E E₂] :

LinearIsometryEquivClass F R E E₂ asserts F is a type of bundled R-linear isometries M → M₂.

This is an abbreviation for SemilinearIsometryEquivClass F (RingHom.id R) E E₂.

Equations

LinearIsometryEquivClass 𝓕 R E E₂ = SemilinearIsometryEquivClass 𝓕 (RingHom.id R) E E₂

instance SemilinearIsometryEquivClass.toSemilinearIsometryClass {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} (𝓕 : Type u_10) [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [EquivLike 𝓕 E E₂] [s : SemilinearIsometryEquivClass 𝓕 σ₁₂ E E₂] :

SemilinearIsometryClass 𝓕 σ₁₂ E E₂

Equations

⋯ = ⋯

theorem LinearIsometryEquiv.toLinearEquiv_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometryEquiv.toLinearEquiv

@[simp]

theorem LinearIsometryEquiv.toLinearEquiv_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {g : E ≃ₛₗᵢ[σ₁₂] E₂} :

f.toLinearEquiv = g.toLinearEquiv ↔ f = g

instance LinearIsometryEquiv.instEquivLike {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

EquivLike (E ≃ₛₗᵢ[σ₁₂] E₂) E E₂

Equations

LinearIsometryEquiv.instEquivLike = { coe := fun (e : E ≃ₛₗᵢ[σ₁₂] E₂) => e.toFun, inv := fun (e : E ≃ₛₗᵢ[σ₁₂] E₂) => e.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

instance LinearIsometryEquiv.instSemilinearIsometryEquivClass {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

SemilinearIsometryEquivClass (E ≃ₛₗᵢ[σ₁₂] E₂) σ₁₂ E E₂

Equations

⋯ = ⋯

instance LinearIsometryEquiv.instCoeFun {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

CoeFun (E ≃ₛₗᵢ[σ₁₂] E₂) fun (x : E ≃ₛₗᵢ[σ₁₂] E₂) => E → E₂

Helper instance for when there's too many metavariables to apply DFunLike.hasCoeToFun directly.

Equations

LinearIsometryEquiv.instCoeFun = { coe := DFunLike.coe }

theorem LinearIsometryEquiv.coe_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective DFunLike.coe

@[simp]

theorem LinearIsometryEquiv.coe_mk {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗ[σ₁₂] E₂) (he : ∀ (x : E), ‖e x‖ = ‖x‖) :

⇑{ toLinearEquiv := e, norm_map' := he } = ⇑e

@[simp]

theorem LinearIsometryEquiv.coe_toLinearEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑e.toLinearEquiv = ⇑e

theorem LinearIsometryEquiv.ext {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {e : E ≃ₛₗᵢ[σ₁₂] E₂} {e' : E ≃ₛₗᵢ[σ₁₂] E₂} (h : ∀ (x : E), e x = e' x) :

e = e'

theorem LinearIsometryEquiv.congr_arg {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {x : E} {x' : E} :

x = x' → f x = f x'

theorem LinearIsometryEquiv.congr_fun {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {g : E ≃ₛₗᵢ[σ₁₂] E₂} (h : f = g) (x : E) :

f x = g x

def LinearIsometryEquiv.ofBounds {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗ[σ₁₂] E₂) (h₁ : ∀ (x : E), ‖e x‖ ≤ ‖x‖) (h₂ : ∀ (y : E₂), ‖(LinearEquiv.symm e) y‖ ≤ ‖y‖) :

E ≃ₛₗᵢ[σ₁₂] E₂

Construct a LinearIsometryEquiv from a LinearEquiv and two inequalities: ∀ x, ‖e x‖ ≤ ‖x‖ and ∀ y, ‖e.symm y‖ ≤ ‖y‖.

Equations

LinearIsometryEquiv.ofBounds e h₁ h₂ = { toLinearEquiv := e, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.norm_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) :

‖e x‖ = ‖x‖

def LinearIsometryEquiv.toLinearIsometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

E →ₛₗᵢ[σ₁₂] E₂

Reinterpret a LinearIsometryEquiv as a LinearIsometry.

Equations

LinearIsometryEquiv.toLinearIsometry e = { toLinearMap := ↑e.toLinearEquiv, norm_map' := ⋯ }

theorem LinearIsometryEquiv.toLinearIsometry_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometryEquiv.toLinearIsometry

@[simp]

theorem LinearIsometryEquiv.toLinearIsometry_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {g : E ≃ₛₗᵢ[σ₁₂] E₂} :

LinearIsometryEquiv.toLinearIsometry f = LinearIsometryEquiv.toLinearIsometry g ↔ f = g

@[simp]

theorem LinearIsometryEquiv.coe_toLinearIsometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometryEquiv.toLinearIsometry e) = ⇑e

theorem LinearIsometryEquiv.isometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

def LinearIsometryEquiv.toIsometryEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

E ≃ᵢ E₂

Reinterpret a LinearIsometryEquiv as an IsometryEquiv.

Equations

LinearIsometryEquiv.toIsometryEquiv e = { toEquiv := LinearEquiv.toEquiv e.toLinearEquiv, isometry_toFun := ⋯ }

theorem LinearIsometryEquiv.toIsometryEquiv_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometryEquiv.toIsometryEquiv

@[simp]

theorem LinearIsometryEquiv.toIsometryEquiv_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {g : E ≃ₛₗᵢ[σ₁₂] E₂} :

LinearIsometryEquiv.toIsometryEquiv f = LinearIsometryEquiv.toIsometryEquiv g ↔ f = g

@[simp]

theorem LinearIsometryEquiv.coe_toIsometryEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometryEquiv.toIsometryEquiv e) = ⇑e

theorem LinearIsometryEquiv.range_eq_univ {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

Set.range ⇑e = Set.univ

def LinearIsometryEquiv.toHomeomorph {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

E ≃ₜ E₂

Reinterpret a LinearIsometryEquiv as a Homeomorph.

Equations

LinearIsometryEquiv.toHomeomorph e = IsometryEquiv.toHomeomorph (LinearIsometryEquiv.toIsometryEquiv e)

theorem LinearIsometryEquiv.toHomeomorph_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometryEquiv.toHomeomorph

@[simp]

theorem LinearIsometryEquiv.toHomeomorph_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {g : E ≃ₛₗᵢ[σ₁₂] E₂} :

LinearIsometryEquiv.toHomeomorph f = LinearIsometryEquiv.toHomeomorph g ↔ f = g

@[simp]

theorem LinearIsometryEquiv.coe_toHomeomorph {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometryEquiv.toHomeomorph e) = ⇑e

theorem LinearIsometryEquiv.continuous {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

Continuous ⇑e

theorem LinearIsometryEquiv.continuousAt {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {x : E} :

ContinuousAt (⇑e) x

theorem LinearIsometryEquiv.continuousOn {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {s : Set E} :

ContinuousOn (⇑e) s

theorem LinearIsometryEquiv.continuousWithinAt {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {s : Set E} {x : E} :

ContinuousWithinAt (⇑e) s x

def LinearIsometryEquiv.toContinuousLinearEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

E ≃SL[σ₁₂] E₂

Interpret a LinearIsometryEquiv as a ContinuousLinearEquiv.

Equations

One or more equations did not get rendered due to their size.

theorem LinearIsometryEquiv.toContinuousLinearEquiv_injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

Function.Injective LinearIsometryEquiv.toContinuousLinearEquiv

@[simp]

theorem LinearIsometryEquiv.toContinuousLinearEquiv_inj {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f : E ≃ₛₗᵢ[σ₁₂] E₂} {g : E ≃ₛₗᵢ[σ₁₂] E₂} :

LinearIsometryEquiv.toContinuousLinearEquiv f = LinearIsometryEquiv.toContinuousLinearEquiv g ↔ f = g

@[simp]

theorem LinearIsometryEquiv.coe_toContinuousLinearEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometryEquiv.toContinuousLinearEquiv e) = ⇑e

def LinearIsometryEquiv.refl (R : Type u_1) (E : Type u_5) [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

E ≃ₗᵢ[R] E

Identity map as a LinearIsometryEquiv.

Equations

LinearIsometryEquiv.refl R E = { toLinearEquiv := LinearEquiv.refl R E, norm_map' := ⋯ }

def LinearIsometryEquiv.ulift (R : Type u_1) (E : Type u_5) [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

ULift.{u_11, u_5} E ≃ₗᵢ[R] E

Linear isometry equiv between a space and its lift to another universe.

Equations

LinearIsometryEquiv.ulift R E = let __src := ContinuousLinearEquiv.ulift; { toLinearEquiv := __src.toLinearEquiv, norm_map' := ⋯ }

instance LinearIsometryEquiv.instInhabited {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

Inhabited (E ≃ₗᵢ[R] E)

Equations

LinearIsometryEquiv.instInhabited = { default := LinearIsometryEquiv.refl R E }

@[simp]

theorem LinearIsometryEquiv.coe_refl {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

⇑(LinearIsometryEquiv.refl R E) = id

def LinearIsometryEquiv.symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

E₂ ≃ₛₗᵢ[σ₂₁] E

The inverse LinearIsometryEquiv.

Equations

LinearIsometryEquiv.symm e = { toLinearEquiv := LinearEquiv.symm e.toLinearEquiv, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.apply_symm_apply {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) :

e ((LinearIsometryEquiv.symm e) x) = x

@[simp]

theorem LinearIsometryEquiv.symm_apply_apply {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) :

(LinearIsometryEquiv.symm e) (e x) = x

theorem LinearIsometryEquiv.map_eq_zero_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {x : E} :

e x = 0 ↔ x = 0

@[simp]

theorem LinearIsometryEquiv.symm_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.symm (LinearIsometryEquiv.symm e) = e

@[simp]

theorem LinearIsometryEquiv.toLinearEquiv_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearEquiv.symm e.toLinearEquiv = (LinearIsometryEquiv.symm e).toLinearEquiv

@[simp]

theorem LinearIsometryEquiv.toIsometryEquiv_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

IsometryEquiv.symm (LinearIsometryEquiv.toIsometryEquiv e) = LinearIsometryEquiv.toIsometryEquiv (LinearIsometryEquiv.symm e)

@[simp]

theorem LinearIsometryEquiv.toHomeomorph_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

Homeomorph.symm (LinearIsometryEquiv.toHomeomorph e) = LinearIsometryEquiv.toHomeomorph (LinearIsometryEquiv.symm e)

def LinearIsometryEquiv.Simps.apply {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (E : Type u_11) (E₂ : Type u_12) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (h : E ≃ₛₗᵢ[σ₁₂] E₂) :

E → E₂

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

Equations

LinearIsometryEquiv.Simps.apply σ₁₂ E E₂ h = ⇑h

def LinearIsometryEquiv.Simps.symm_apply {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (E : Type u_11) (E₂ : Type u_12) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (h : E ≃ₛₗᵢ[σ₁₂] E₂) :

E₂ → E

See Note [custom simps projection]

Equations

LinearIsometryEquiv.Simps.symm_apply σ₁₂ E E₂ h = ⇑(LinearIsometryEquiv.symm h)

def LinearIsometryEquiv.trans {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (e' : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :

E ≃ₛₗᵢ[σ₁₃] E₃

Composition of LinearIsometryEquivs as a LinearIsometryEquiv.

Equations

LinearIsometryEquiv.trans e e' = { toLinearEquiv := LinearEquiv.trans e.toLinearEquiv e'.toLinearEquiv, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.coe_trans {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :

⇑(LinearIsometryEquiv.trans e₁ e₂) = ⇑e₂ ∘ ⇑e₁

@[simp]

theorem LinearIsometryEquiv.trans_apply {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) (c : E) :

(LinearIsometryEquiv.trans e₁ e₂) c = e₂ (e₁ c)

@[simp]

theorem LinearIsometryEquiv.toLinearEquiv_trans {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (e' : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :

(LinearIsometryEquiv.trans e e').toLinearEquiv = LinearEquiv.trans e.toLinearEquiv e'.toLinearEquiv

@[simp]

theorem LinearIsometryEquiv.trans_refl {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.trans e (LinearIsometryEquiv.refl R₂ E₂) = e

@[simp]

theorem LinearIsometryEquiv.refl_trans {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.trans (LinearIsometryEquiv.refl R E) e = e

@[simp]

theorem LinearIsometryEquiv.self_trans_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.trans e (LinearIsometryEquiv.symm e) = LinearIsometryEquiv.refl R E

@[simp]

theorem LinearIsometryEquiv.symm_trans_self {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.trans (LinearIsometryEquiv.symm e) e = LinearIsometryEquiv.refl R₂ E₂

@[simp]

theorem LinearIsometryEquiv.symm_comp_self {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑(LinearIsometryEquiv.symm e) ∘ ⇑e = id

@[simp]

theorem LinearIsometryEquiv.self_comp_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑e ∘ ⇑(LinearIsometryEquiv.symm e) = id

@[simp]

theorem LinearIsometryEquiv.symm_trans {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :

LinearIsometryEquiv.symm (LinearIsometryEquiv.trans e₁ e₂) = LinearIsometryEquiv.trans (LinearIsometryEquiv.symm e₂) (LinearIsometryEquiv.symm e₁)

theorem LinearIsometryEquiv.coe_symm_trans {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :

⇑(LinearIsometryEquiv.symm (LinearIsometryEquiv.trans e₁ e₂)) = ⇑(LinearIsometryEquiv.symm e₁) ∘ ⇑(LinearIsometryEquiv.symm e₂)

theorem LinearIsometryEquiv.trans_assoc {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {R₄ : Type u_4} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} {E₄ : Type u_8} [Semiring R] [Semiring R₂] [Semiring R₃] [Semiring R₄] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R} {σ₁₄ : R →+* R₄} {σ₄₁ : R₄ →+* R} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} {σ₂₄ : R₂ →+* R₄} {σ₄₂ : R₄ →+* R₂} {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomInvPair σ₁₄ σ₄₁] [RingHomInvPair σ₄₁ σ₁₄] [RingHomInvPair σ₂₄ σ₄₂] [RingHomInvPair σ₄₂ σ₂₄] [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [RingHomCompTriple σ₄₂ σ₂₁ σ₄₁] [RingHomCompTriple σ₄₃ σ₃₂ σ₄₂] [RingHomCompTriple σ₄₃ σ₃₁ σ₄₁] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [SeminormedAddCommGroup E₄] [Module R E] [Module R₂ E₂] [Module R₃ E₃] [Module R₄ E₄] (eEE₂ : E ≃ₛₗᵢ[σ₁₂] E₂) (eE₂E₃ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) (eE₃E₄ : E₃ ≃ₛₗᵢ[σ₃₄] E₄) :

LinearIsometryEquiv.trans eEE₂ (LinearIsometryEquiv.trans eE₂E₃ eE₃E₄) = LinearIsometryEquiv.trans (LinearIsometryEquiv.trans eEE₂ eE₂E₃) eE₃E₄

instance LinearIsometryEquiv.instGroup {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

Group (E ≃ₗᵢ[R] E)

Equations

LinearIsometryEquiv.instGroup = Group.mk ⋯

@[simp]

theorem LinearIsometryEquiv.coe_one {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

⇑1 = id

@[simp]

theorem LinearIsometryEquiv.coe_mul {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E) (e' : E ≃ₗᵢ[R] E) :

⇑(e * e') = ⇑e ∘ ⇑e'

@[simp]

theorem LinearIsometryEquiv.coe_inv {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E) :

⇑e⁻¹ = ⇑(LinearIsometryEquiv.symm e)

theorem LinearIsometryEquiv.one_def {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

1 = LinearIsometryEquiv.refl R E

theorem LinearIsometryEquiv.mul_def {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E) (e' : E ≃ₗᵢ[R] E) :

e * e' = LinearIsometryEquiv.trans e' e

theorem LinearIsometryEquiv.inv_def {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E) :

e⁻¹ = LinearIsometryEquiv.symm e

Lemmas about mixing the group structure with definitions. Because we have multiple ways to express LinearIsometryEquiv.refl, LinearIsometryEquiv.symm, and LinearIsometryEquiv.trans, we want simp lemmas for every combination. The assumption made here is that if you're using the group structure, you want to preserve it after simp.

This copies the approach used by the lemmas near Equiv.Perm.trans_one.

@[simp]

theorem LinearIsometryEquiv.trans_one {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.trans e 1 = e

@[simp]

theorem LinearIsometryEquiv.one_trans {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LinearIsometryEquiv.trans 1 e = e

@[simp]

theorem LinearIsometryEquiv.refl_mul {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E) :

LinearIsometryEquiv.refl R E * e = e

@[simp]

theorem LinearIsometryEquiv.mul_refl {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E) :

e * LinearIsometryEquiv.refl R E = e

instance LinearIsometryEquiv.instCoeTCContinuousLinearEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

CoeTC (E ≃ₛₗᵢ[σ₁₂] E₂) (E ≃SL[σ₁₂] E₂)

Reinterpret a LinearIsometryEquiv as a ContinuousLinearEquiv.

Equations

LinearIsometryEquiv.instCoeTCContinuousLinearEquiv = { coe := fun (e : E ≃ₛₗᵢ[σ₁₂] E₂) => { toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } }

instance LinearIsometryEquiv.instCoeTCContinuousLinearMap {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] :

CoeTC (E ≃ₛₗᵢ[σ₁₂] E₂) (E →SL[σ₁₂] E₂)

Equations

LinearIsometryEquiv.instCoeTCContinuousLinearMap = { coe := fun (e : E ≃ₛₗᵢ[σ₁₂] E₂) => ↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } }

@[simp]

theorem LinearIsometryEquiv.coe_coe {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } = ⇑e

@[simp]

theorem LinearIsometryEquiv.coe_coe'' {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

⇑↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } = ⇑e

theorem LinearIsometryEquiv.map_zero {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

e 0 = 0

theorem LinearIsometryEquiv.map_add {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

e (x + y) = e x + e y

theorem LinearIsometryEquiv.map_sub {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

e (x - y) = e x - e y

theorem LinearIsometryEquiv.map_smulₛₗ {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (c : R) (x : E) :

e (c • x) = σ₁₂ c • e x

theorem LinearIsometryEquiv.map_smul {R : Type u_1} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R E₂] {e : E ≃ₗᵢ[R] E₂} (c : R) (x : E) :

e (c • x) = c • e x

@[simp]

theorem LinearIsometryEquiv.nnnorm_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) :

‖e x‖₊ = ‖x‖₊

@[simp]

theorem LinearIsometryEquiv.dist_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

dist (e x) (e y) = dist x y

@[simp]

theorem LinearIsometryEquiv.edist_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (y : E) :

edist (e x) (e y) = edist x y

theorem LinearIsometryEquiv.bijective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

Function.Bijective ⇑e

theorem LinearIsometryEquiv.injective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

Function.Injective ⇑e

theorem LinearIsometryEquiv.surjective {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

Function.Surjective ⇑e

theorem LinearIsometryEquiv.map_eq_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {x : E} {y : E} :

e x = e y ↔ x = y

theorem LinearIsometryEquiv.map_ne {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {x : E} {y : E} (h : x ≠ y) :

e x ≠ e y

theorem LinearIsometryEquiv.lipschitz {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

LipschitzWith 1 ⇑e

theorem LinearIsometryEquiv.antilipschitz {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) :

AntilipschitzWith 1 ⇑e

theorem LinearIsometryEquiv.image_eq_preimage {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (s : Set E) :

⇑e '' s = ⇑(LinearIsometryEquiv.symm e) ⁻¹' s

@[simp]

theorem LinearIsometryEquiv.ediam_image {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (s : Set E) :

EMetric.diam (⇑e '' s) = EMetric.diam s

@[simp]

theorem LinearIsometryEquiv.diam_image {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (s : Set E) :

Metric.diam (⇑e '' s) = Metric.diam s

@[simp]

theorem LinearIsometryEquiv.preimage_ball {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) (r : ℝ) :

⇑e ⁻¹' Metric.ball x r = Metric.ball ((LinearIsometryEquiv.symm e) x) r

@[simp]

theorem LinearIsometryEquiv.preimage_sphere {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) (r : ℝ) :

⇑e ⁻¹' Metric.sphere x r = Metric.sphere ((LinearIsometryEquiv.symm e) x) r

@[simp]

theorem LinearIsometryEquiv.preimage_closedBall {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) (r : ℝ) :

⇑e ⁻¹' Metric.closedBall x r = Metric.closedBall ((LinearIsometryEquiv.symm e) x) r

@[simp]

theorem LinearIsometryEquiv.image_ball {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ) :

⇑e '' Metric.ball x r = Metric.ball (e x) r

@[simp]

theorem LinearIsometryEquiv.image_sphere {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ) :

⇑e '' Metric.sphere x r = Metric.sphere (e x) r

@[simp]

theorem LinearIsometryEquiv.image_closedBall {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ) :

⇑e '' Metric.closedBall x r = Metric.closedBall (e x) r

@[simp]

theorem LinearIsometryEquiv.comp_continuousOn_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {α : Type u_11} [TopologicalSpace α] {f : α → E} {s : Set α} :

ContinuousOn (⇑e ∘ f) s ↔ ContinuousOn f s

@[simp]

theorem LinearIsometryEquiv.comp_continuous_iff {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {α : Type u_11} [TopologicalSpace α] {f : α → E} :

Continuous (⇑e ∘ f) ↔ Continuous f

instance LinearIsometryEquiv.completeSpace_map {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (p : Submodule R E) [CompleteSpace ↥p] :

CompleteSpace ↥(Submodule.map (↑e.toLinearEquiv) p)

Equations

⋯ = ⋯

noncomputable def LinearIsometryEquiv.ofSurjective {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f : F →ₛₗᵢ[σ₁₂] E₂) (hfr : Function.Surjective ⇑f) :

F ≃ₛₗᵢ[σ₁₂] E₂

Construct a linear isometry equiv from a surjective linear isometry.

Equations

LinearIsometryEquiv.ofSurjective f hfr = let __src := LinearEquiv.ofBijective f.toLinearMap ⋯; { toLinearEquiv := __src, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.coe_ofSurjective {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f : F →ₛₗᵢ[σ₁₂] E₂) (hfr : Function.Surjective ⇑f) :

⇑(LinearIsometryEquiv.ofSurjective f hfr) = ⇑f

def LinearIsometryEquiv.ofLinearIsometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E) (h₁ : LinearMap.comp f.toLinearMap g = LinearMap.id) (h₂ : LinearMap.comp g f.toLinearMap = LinearMap.id) :

E ≃ₛₗᵢ[σ₁₂] E₂

If a linear isometry has an inverse, it is a linear isometric equivalence.

Equations

LinearIsometryEquiv.ofLinearIsometry f g h₁ h₂ = let __src := LinearEquiv.ofLinear f.toLinearMap g h₁ h₂; { toLinearEquiv := __src, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.coe_ofLinearIsometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E) (h₁ : LinearMap.comp f.toLinearMap g = LinearMap.id) (h₂ : LinearMap.comp g f.toLinearMap = LinearMap.id) :

⇑(LinearIsometryEquiv.ofLinearIsometry f g h₁ h₂) = ⇑f

@[simp]

theorem LinearIsometryEquiv.coe_ofLinearIsometry_symm {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E) (h₁ : LinearMap.comp f.toLinearMap g = LinearMap.id) (h₂ : LinearMap.comp g f.toLinearMap = LinearMap.id) :

⇑(LinearIsometryEquiv.symm (LinearIsometryEquiv.ofLinearIsometry f g h₁ h₂)) = ⇑g

def LinearIsometryEquiv.neg (R : Type u_1) {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

E ≃ₗᵢ[R] E

The negation operation on a normed space E, considered as a linear isometry equivalence.

Equations

LinearIsometryEquiv.neg R = let __src := LinearEquiv.neg R; { toLinearEquiv := __src, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.coe_neg {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

⇑(LinearIsometryEquiv.neg R) = fun (x : E) => -x

@[simp]

theorem LinearIsometryEquiv.symm_neg {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] :

LinearIsometryEquiv.symm (LinearIsometryEquiv.neg R) = LinearIsometryEquiv.neg R

def LinearIsometryEquiv.prodAssoc (R : Type u_1) (E : Type u_5) (E₂ : Type u_6) (E₃ : Type u_7) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R E₂] [Module R E₃] :

(E × E₂) × E₃ ≃ₗᵢ[R] E × E₂ × E₃

The natural equivalence (E × E₂) × E₃ ≃ E × (E₂ × E₃) is a linear isometry.

Equations

One or more equations did not get rendered due to their size.

@[simp]

theorem LinearIsometryEquiv.coe_prodAssoc (R : Type u_1) (E : Type u_5) (E₂ : Type u_6) (E₃ : Type u_7) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R E₂] [Module R E₃] :

⇑(LinearIsometryEquiv.prodAssoc R E E₂ E₃) = ⇑(Equiv.prodAssoc E E₂ E₃)

@[simp]

theorem LinearIsometryEquiv.coe_prodAssoc_symm (R : Type u_1) (E : Type u_5) (E₂ : Type u_6) (E₃ : Type u_7) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R E₂] [Module R E₃] :

⇑(LinearIsometryEquiv.symm (LinearIsometryEquiv.prodAssoc R E E₂ E₃)) = ⇑(Equiv.prodAssoc E E₂ E₃).symm

@[simp]

theorem LinearIsometryEquiv.ofTop_toLinearEquiv (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) :

(LinearIsometryEquiv.ofTop E p hp).toLinearEquiv = LinearEquiv.ofTop p hp

@[simp]

theorem LinearIsometryEquiv.ofTop_apply (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) (self : ↥p) :

(LinearIsometryEquiv.ofTop E p hp) self = ↑self

@[simp]

theorem LinearIsometryEquiv.ofTop_symm_apply_coe (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) (x : E) :

↑((LinearIsometryEquiv.symm (LinearIsometryEquiv.ofTop E p hp)) x) = x

def LinearIsometryEquiv.ofTop (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) :

↥p ≃ₗᵢ[R] E

If p is a submodule that is equal to ⊤, then LinearIsometryEquiv.ofTop p hp is the "identity" equivalence between p and E.

Equations

LinearIsometryEquiv.ofTop E p hp = let __src := Submodule.subtypeₗᵢ p; { toLinearEquiv := LinearEquiv.ofTop p hp, norm_map' := ⋯ }

def LinearIsometryEquiv.ofEq {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] (p : Submodule R' E) (q : Submodule R' E) (hpq : p = q) :

↥p ≃ₗᵢ[R'] ↥q

LinearEquiv.ofEq as a LinearIsometryEquiv.

Equations

LinearIsometryEquiv.ofEq p q hpq = let __src := LinearEquiv.ofEq p q hpq; { toLinearEquiv := __src, norm_map' := ⋯ }

@[simp]

theorem LinearIsometryEquiv.coe_ofEq_apply {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] {p : Submodule R' E} {q : Submodule R' E} (h : p = q) (x : ↥p) :

↑((LinearIsometryEquiv.ofEq p q h) x) = ↑x

@[simp]

theorem LinearIsometryEquiv.ofEq_symm {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] {p : Submodule R' E} {q : Submodule R' E} (h : p = q) :

LinearIsometryEquiv.symm (LinearIsometryEquiv.ofEq p q h) = LinearIsometryEquiv.ofEq q p ⋯

@[simp]

theorem LinearIsometryEquiv.ofEq_rfl {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] {p : Submodule R' E} :

LinearIsometryEquiv.ofEq p p ⋯ = LinearIsometryEquiv.refl R' ↥p

theorem Basis.ext_linearIsometry {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {ι : Type u_11} (b : Basis ι R E) {f₁ : E →ₛₗᵢ[σ₁₂] E₂} {f₂ : E →ₛₗᵢ[σ₁₂] E₂} (h : ∀ (i : ι), f₁ (b i) = f₂ (b i)) :

f₁ = f₂

Two linear isometries are equal if they are equal on basis vectors.

theorem Basis.ext_linearIsometryEquiv {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {ι : Type u_11} (b : Basis ι R E) {f₁ : E ≃ₛₗᵢ[σ₁₂] E₂} {f₂ : E ≃ₛₗᵢ[σ₁₂] E₂} (h : ∀ (i : ι), f₁ (b i) = f₂ (b i)) :

f₁ = f₂

Two linear isometric equivalences are equal if they are equal on basis vectors.

@[simp]

theorem LinearIsometry.equivRange_apply_coe {E : Type u_5} {F : Type u_9} [SeminormedAddCommGroup E] [NormedAddCommGroup F] {R : Type u_11} {S : Type u_12} [Semiring R] [Ring S] [Module S E] [Module R F] {σ₁₂ : R →+* S} {σ₂₁ : S →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : F →ₛₗᵢ[σ₁₂] E) (a : F) :

↑((LinearIsometry.equivRange f) a) = f a

noncomputable def LinearIsometry.equivRange {E : Type u_5} {F : Type u_9} [SeminormedAddCommGroup E] [NormedAddCommGroup F] {R : Type u_11} {S : Type u_12} [Semiring R] [Ring S] [Module S E] [Module R F] {σ₁₂ : R →+* S} {σ₂₁ : S →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : F →ₛₗᵢ[σ₁₂] E) :

F ≃ₛₗᵢ[σ₁₂] ↥(LinearMap.range f.toLinearMap)

Reinterpret a LinearIsometry as a LinearIsometryEquiv to the range.

Equations

LinearIsometry.equivRange f = { toLinearEquiv := LinearEquiv.ofInjective f.toLinearMap ⋯, norm_map' := ⋯ }