(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)
- toFun : E → E₂
- map_add' : ∀ (x y : E), AddHom.toFun s.toAddHom (x + y) = AddHom.toFun s.toAddHom x + AddHom.toFun s.toAddHom y
- map_smul' : ∀ (r : R) (x : E), AddHom.toFun s.toAddHom (r • x) = ↑σ₁₂ r • AddHom.toFun s.toAddHom x
A σ₁₂-semilinear isometric embedding of a normed R-module into an R₂-module.
Instances For
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]
extends
SemilinearMapClass
:
Type (max (max u_11 u_14) u_15)
- coe : 𝓕 → E → E₂
- coe_injective' : Function.Injective FunLike.coe
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.
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]
:
Type (max (max u_11 u_13) u_14)
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.
Instances For
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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
(f : 𝓕)
:
Isometry ↑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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
(f : 𝓕)
:
Continuous ↑f
@[simp]
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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
(f : 𝓕)
(x : E)
:
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]
[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]
[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]
[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]
[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]
[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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
(f : 𝓕)
:
Metric.diam (Set.range ↑f) = Metric.diam Set.univ
instance
SemilinearIsometryClass.instContinuousSemilinearMapClassToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroup
{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]
[s : SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
:
ContinuousSemilinearMapClass 𝓕 σ₁₂ E E₂
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
instance
LinearIsometry.instSemilinearIsometryClassLinearIsometry
{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
@[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 => ↑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.
Instances For
theorem
LinearIsometry.congr_arg
{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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
{f : 𝓕}
{x : E}
{x' : E}
:
theorem
LinearIsometry.congr_fun
{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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
{f : 𝓕}
{g : 𝓕}
(h : f = g)
(x : E)
:
↑f x = ↑g 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)
:
@[simp]
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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
(f : 𝓕)
{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]
[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]
[SemilinearIsometryClass 𝓕 σ₁₂ E E₂]
(f : 𝓕)
[RingHomSurjective σ₁₂]
(p : Submodule R E)
[CompleteSpace { x // x ∈ p }]
:
CompleteSpace { x // x ∈ Submodule.map f p }
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 { x // x ∈ p }]
:
CompleteSpace { x // x ∈ Submodule.map f.toLinearMap p }
@[simp]
@[simp]
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)
:
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)
:
Interpret a linear isometry as a continuous linear map.
Instances For
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.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)
:
@[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]
:
The identity linear isometry.
Instances For
@[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.instInhabitedLinearIsometryIdToNonAssocSemiring
{R : Type u_1}
{E : Type u_5}
[Semiring R]
[SeminormedAddCommGroup E]
[Module R E]
:
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)
:
Composition of linear isometries.
Instances For
@[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.instMonoidLinearIsometryIdToNonAssocSemiring
{R : Type u_1}
{E : Type u_5}
[Semiring R]
[SeminormedAddCommGroup E]
[Module R E]
:
@[simp]
theorem
LinearIsometry.coe_one
{R : Type u_1}
{E : Type u_5}
[Semiring R]
[SeminormedAddCommGroup E]
[Module R E]
:
↑1 = id
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
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)
:
Construct a LinearIsometry from a LinearMap satisfying Isometry.
Instances For
def
Submodule.subtypeₗᵢ
{E : Type u_5}
[SeminormedAddCommGroup E]
{R' : Type u_11}
[Ring R']
[Module R' E]
(p : Submodule R' E)
:
Submodule.subtype as a LinearIsometry.
Instances For
@[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)
:
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)
- toFun : E → E₂
- map_add' : ∀ (x y : E), AddHom.toFun s.toAddHom (x + y) = AddHom.toFun s.toAddHom x + AddHom.toFun s.toAddHom y
- map_smul' : ∀ (r : R) (x : E), AddHom.toFun s.toAddHom (r • x) = ↑σ₁₂ r • AddHom.toFun s.toAddHom x
- invFun : E₂ → E
- left_inv : Function.LeftInverse s.invFun s.toFun
- right_inv : Function.RightInverse s.invFun s.toFun
A semilinear isometric equivalence between two normed vector spaces.
Instances For
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]
extends
SemilinearEquivClass
:
Type (max (max u_11 u_14) u_15)
- coe : 𝓕 → E → E₂
- inv : 𝓕 → E₂ → E
- left_inv : ∀ (e : 𝓕), Function.LeftInverse (EquivLike.inv e) (EquivLike.coe e)
- right_inv : ∀ (e : 𝓕), Function.RightInverse (EquivLike.inv e) (EquivLike.coe e)
- coe_injective' : ∀ (e g : 𝓕), EquivLike.coe e = EquivLike.coe g → EquivLike.inv e = EquivLike.inv g → e = g
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.
Instances
@[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]
:
Type (max (max u_11 u_13) u_14)
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.
Instances For
instance
SemilinearIsometryEquivClass.instSemilinearIsometryClass
{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]
[s : SemilinearIsometryEquivClass 𝓕 σ₁₂ E E₂]
:
SemilinearIsometryClass 𝓕 σ₁₂ E E₂
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}
:
instance
LinearIsometryEquiv.instSemilinearIsometryEquivClassLinearIsometryEquiv
{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
instance
LinearIsometryEquiv.instCoeFunLinearIsometryEquivForAll
{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]
:
Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun
directly.
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 FunLike.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}
:
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‖)
:
Construct a LinearIsometryEquiv from a LinearEquiv and two inequalities:
∀ x, ‖e x‖ ≤ ‖x‖ and ∀ y, ‖e.symm y‖ ≤ ‖y‖.
Instances For
@[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)
:
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)
:
Reinterpret a LinearIsometryEquiv as a LinearIsometry.
Instances For
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}
:
@[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)
:
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)
:
Isometry ↑e
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.
Instances For
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}
:
@[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)
:
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.
Instances For
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}
:
@[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)
:
Interpret a LinearIsometryEquiv as a ContinuousLinearEquiv.
Instances For
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}
:
@[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)
:
def
LinearIsometryEquiv.refl
(R : Type u_1)
(E : Type u_5)
[Semiring R]
[SeminormedAddCommGroup E]
[Module R E]
:
Identity map as a LinearIsometryEquiv.
Instances For
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.
Instances For
instance
LinearIsometryEquiv.instInhabitedLinearIsometryEquivIdToNonAssocSemiringIds
{R : Type u_1}
{E : Type u_5}
[Semiring R]
[SeminormedAddCommGroup E]
[Module 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)
:
The inverse LinearIsometryEquiv.
Instances For
@[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}
:
@[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)
:
@[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)
:
@[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)
:
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.
Instances For
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]
Instances For
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)
:
Composition of LinearIsometryEquivs as a LinearIsometryEquiv.
Instances For
@[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)
:
@[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)
:
@[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)
:
@[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)
:
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.instGroupLinearIsometryEquivIdToNonAssocSemiringIds
{R : Type u_1}
{E : Type u_5}
[Semiring R]
[SeminormedAddCommGroup E]
[Module R E]
:
@[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_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)
:
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.instCoeTCLinearIsometryEquivContinuousLinearEquivToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroup
{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]
:
Reinterpret a LinearIsometryEquiv as a ContinuousLinearEquiv.
instance
LinearIsometryEquiv.instCoeTCLinearIsometryEquivContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroup
{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]
:
@[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)
:
↑(ContinuousLinearEquiv.mk e.toLinearEquiv) = ↑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)
:
↑↑(ContinuousLinearEquiv.mk e.toLinearEquiv) = ↑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)
:
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)
:
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)
:
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)
:
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)
:
@[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)
:
@[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)
:
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)
:
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)
:
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)
:
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}
:
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 { x // x ∈ p }]
:
CompleteSpace { x // x ∈ Submodule.map (↑e.toLinearEquiv) p }
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)
:
Construct a linear isometry equiv from a surjective linear isometry.
Instances For
@[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)
:
If a linear isometry has an inverse, it is a linear isometric equivalence.
Instances For
@[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]
:
The negation operation on a normed space E, considered as a linear isometry equivalence.
Instances For
@[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 => -x
@[simp]
theorem
LinearIsometryEquiv.symm_neg
{R : Type u_1}
{E : Type u_5}
[Semiring R]
[SeminormedAddCommGroup E]
[Module R E]
:
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]
:
The natural equivalence (E × E₂) × E₃ ≃ E × (E₂ × E₃) is a linear isometry.
Instances For
@[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_apply
(E : Type u_5)
[SeminormedAddCommGroup E]
{R : Type u_12}
[Ring R]
[Module R E]
(p : Submodule R E)
(hp : p = ⊤)
(self : { x // x ∈ p })
:
↑(LinearIsometryEquiv.ofTop E p hp) self = ↑self
@[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_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 = ⊤)
:
If p is a submodule that is equal to ⊤, then LinearIsometryEquiv.ofTop p hp is the
"identity" equivalence between p and E.
Instances For
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)
:
LinearEquiv.ofEq as a LinearIsometryEquiv.
Instances For
@[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 : { x // 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 (_ : 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 (_ : p = p) = LinearIsometryEquiv.refl R' { x // x ∈ 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] { x // x ∈ LinearMap.range f.toLinearMap }
Reinterpret a LinearIsometry as a LinearIsometryEquiv to the range.