Continuous linear equivalences

Continuous semilinear / linear / star-linear equivalences between topological modules are denoted by M ≃SL[σ] M₂, M ≃L[R] M₂ and M ≃L⋆[R] M₂.

structure ContinuousLinearEquiv {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_3) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_4) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends M ≃ₛₗ[σ] M₂ : Type (max u_3 u_4)

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

• toFun : M → M₂
• map_add' (x y : M) : (↑self.toLinearEquiv).toFun (x + y) = (↑self.toLinearEquiv).toFun x + (↑self.toLinearEquiv).toFun y
• map_smul' (m : R) (x : M) : (↑self.toLinearEquiv).toFun (m • x) = σ m • (↑self.toLinearEquiv).toFun x
• invFun : M₂ → M
• left_inv : Function.LeftInverse self.invFun (↑self.toLinearEquiv).toFun
• right_inv : Function.RightInverse self.invFun (↑self.toLinearEquiv).toFun
• continuous_toFun : Continuous (↑self.toLinearEquiv).toFun

  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

• continuous_invFun : Continuous self.invFun

  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

def «term_≃SL[_]_» : Lean.TrailingParserDescr

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

Equations

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

def «term_≃L[_]_» : Lean.TrailingParserDescr

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

Equations

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

class ContinuousSemilinearEquivClass (F : Type u_1) {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam (Type u_4)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_5)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] extends SemilinearEquivClass F σ M M₂ : Prop

ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ 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 : M) : f (a + b) = f a + f b
• map_smulₛₗ (f : F) (r : R) (x : M) : f (r • x) = σ r • f x
• map_continuous (f : F) : Continuous ⇑f

  ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ 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.

• inv_continuous (f : F) : Continuous (EquivLike.inv f)

  ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ 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.

@[reducible, inline]

abbrev ContinuousLinearEquivClass (F : Type u_1) (R : outParam (Type u_2)) [Semiring R] (M : outParam (Type u_3)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_4)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] [EquivLike F M M₂] : Prop

ContinuousLinearEquivClass F σ M M₂ asserts F is a type of bundled continuous R-linear equivs M → M₂. This is an abbreviation for ContinuousSemilinearEquivClass F (RingHom.id R) M M₂.

Equations

• ContinuousLinearEquivClass F R M M₂ = ContinuousSemilinearEquivClass F (RingHom.id R) M M₂

@[instance 100]

instance ContinuousSemilinearEquivClass.continuousSemilinearMapClass (F : Type u_1) {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_4) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] [s : ContinuousSemilinearEquivClass F σ M M₂] : ContinuousSemilinearMapClass F σ M M₂

@[instance 100]

instance ContinuousSemilinearEquivClass.instHomeomorphClass (F : Type u_1) {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_4) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] [s : ContinuousSemilinearEquivClass F σ M M₂] : HomeomorphClass F M M₂

def ContinuousLinearMap.iInfKerProjEquiv (R : Type u_1) [Semiring R] {ι : Type u_4} (φ : ι → Type u_5) [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] {I J : Set ι} [DecidablePred fun (i : ι) => i ∈ I] (hd : Disjoint I J) (hu : Set.univ ⊆ I ∪ J) : ↥(⨅ i ∈ J, LinearMap.ker (proj i)) ≃L[R] (i : ↑I) → φ ↑i

If I and J are complementary index sets, the product of the kernels of the Jth projections of φ is linearly equivalent to the product over I.

Equations

• ContinuousLinearMap.iInfKerProjEquiv R φ hd hu = { toLinearEquiv := LinearMap.iInfKerProjEquiv R φ hd hu, continuous_toFun := ⋯, continuous_invFun := ⋯ }

def ContinuousLinearEquiv.toContinuousLinearMap {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : M₁ →SL[σ₁₂] M₂

A continuous linear equivalence induces a continuous linear map.

Equations

• ↑e = { toLinearMap := ↑e.toLinearEquiv, cont := ⋯ }

instance ContinuousLinearEquiv.ContinuousLinearMap.coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Coe (M₁ ≃SL[σ₁₂] M₂) (M₁ →SL[σ₁₂] M₂)

Coerce continuous linear equivs to continuous linear maps.

Equations

• ContinuousLinearEquiv.ContinuousLinearMap.coe = { coe := ContinuousLinearEquiv.toContinuousLinearMap }

instance ContinuousLinearEquiv.equivLike {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : EquivLike (M₁ ≃SL[σ₁₂] M₂) M₁ M₂

Equations

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

instance ContinuousLinearEquiv.continuousSemilinearEquivClass {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : ContinuousSemilinearEquivClass (M₁ ≃SL[σ₁₂] M₂) σ₁₂ M₁ M₂

theorem ContinuousLinearEquiv.coe_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : ↑e b = e b

@[simp]

theorem ContinuousLinearEquiv.coe_toLinearEquiv {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ ≃SL[σ₁₂] M₂) : ⇑f.toLinearEquiv = ⇑f

@[simp]

theorem ContinuousLinearEquiv.coe_coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : ⇑↑e = ⇑e

theorem ContinuousLinearEquiv.toLinearEquiv_injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Function.Injective toLinearEquiv

theorem ContinuousLinearEquiv.ext {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f g : M₁ ≃SL[σ₁₂] M₂} (h : ⇑f = ⇑g) : f = g

theorem ContinuousLinearEquiv.ext_iff {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f g : M₁ ≃SL[σ₁₂] M₂} : f = g ↔ ⇑f = ⇑g

theorem ContinuousLinearEquiv.coe_injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Function.Injective toContinuousLinearMap

@[simp]

theorem ContinuousLinearEquiv.coe_inj {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {e e' : M₁ ≃SL[σ₁₂] M₂} : ↑e = ↑e' ↔ e = e'

def ContinuousLinearEquiv.toHomeomorph {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : M₁ ≃ₜ M₂

A continuous linear equivalence induces a homeomorphism.

Equations

• e.toHomeomorph = { toEquiv := e.toEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.coe_toHomeomorph {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.toHomeomorph = ⇑e

theorem ContinuousLinearEquiv.isOpenMap {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : IsOpenMap ⇑e

theorem ContinuousLinearEquiv.image_closure {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : ⇑e '' closure s = closure (⇑e '' s)

theorem ContinuousLinearEquiv.preimage_closure {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : ⇑e ⁻¹' closure s = closure (⇑e ⁻¹' s)

@[simp]

theorem ContinuousLinearEquiv.isClosed_image {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} : IsClosed (⇑e '' s) ↔ IsClosed s

theorem ContinuousLinearEquiv.map_nhds_eq {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : Filter.map (⇑e) (nhds x) = nhds (e x)

theorem ContinuousLinearEquiv.map_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : e 0 = 0

theorem ContinuousLinearEquiv.map_add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (x y : M₁) : e (x + y) = e x + e y

@[simp]

theorem ContinuousLinearEquiv.map_smulₛₗ {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (x : M₁) : e (c • x) = σ₁₂ c • e x

theorem ContinuousLinearEquiv.map_smul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (e : M₁ ≃L[R₁] M₂) (c : R₁) (x : M₁) : e (c • x) = c • e x

theorem ContinuousLinearEquiv.map_eq_zero_iff {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : e x = 0 ↔ x = 0

theorem ContinuousLinearEquiv.continuous {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : Continuous ⇑e

theorem ContinuousLinearEquiv.continuousOn {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} : ContinuousOn (⇑e) s

theorem ContinuousLinearEquiv.continuousAt {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : ContinuousAt (⇑e) x

theorem ContinuousLinearEquiv.continuousWithinAt {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} {x : M₁} : ContinuousWithinAt (⇑e) s x

theorem ContinuousLinearEquiv.comp_continuousOn_iff {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {α : Type u_8} [TopologicalSpace α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} {s : Set α} : ContinuousOn (⇑e ∘ f) s ↔ ContinuousOn f s

theorem ContinuousLinearEquiv.comp_continuous_iff {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {α : Type u_8} [TopologicalSpace α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} : Continuous (⇑e ∘ f) ↔ Continuous f

theorem ContinuousLinearEquiv.ext₁ {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] {f g : R₁ ≃L[R₁] M₁} (h : f 1 = g 1) : f = g

An extensionality lemma for R ≃L[R] M.

def ContinuousLinearEquiv.refl (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : M₁ ≃L[R₁] M₁

The identity map as a continuous linear equivalence.

Equations

• ContinuousLinearEquiv.refl R₁ M₁ = { toLinearEquiv := LinearEquiv.refl R₁ M₁, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.refl_apply (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (x : M₁) : (ContinuousLinearEquiv.refl R₁ M₁) x = x

@[simp]

theorem ContinuousLinearEquiv.coe_refl {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : ↑(ContinuousLinearEquiv.refl R₁ M₁) = ContinuousLinearMap.id R₁ M₁

@[simp]

theorem ContinuousLinearEquiv.coe_refl' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : ⇑(ContinuousLinearEquiv.refl R₁ M₁) = id

def ContinuousLinearEquiv.symm {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁

The inverse of a continuous linear equivalence as a continuous linear equivalence

Equations

• e.symm = { toLinearEquiv := e.symm, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.symm_toLinearEquiv {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.toLinearEquiv = e.symm

@[simp]

theorem ContinuousLinearEquiv.symm_toHomeomorph {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : e.toHomeomorph.symm = e.symm.toHomeomorph

def ContinuousLinearEquiv.Simps.apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (h : M₁ ≃SL[σ₁₂] M₂) : M₁ → M₂

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

Equations

• ContinuousLinearEquiv.Simps.apply h = ⇑h

def ContinuousLinearEquiv.Simps.symm_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (h : M₁ ≃SL[σ₁₂] M₂) : M₂ → M₁

See Note [custom simps projection]

Equations

• ContinuousLinearEquiv.Simps.symm_apply h = ⇑h.symm

theorem ContinuousLinearEquiv.symm_map_nhds_eq {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : Filter.map (⇑e.symm) (nhds (e x)) = nhds x

def ContinuousLinearEquiv.trans {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] {σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : M₁ ≃SL[σ₁₃] M₃

The composition of two continuous linear equivalences as a continuous linear equivalence.

Equations

• e₁.trans e₂ = { toLinearEquiv := e₁.trans e₂.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.trans_toLinearEquiv {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] {σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : (e₁.trans e₂).toLinearEquiv = e₁.trans e₂.toLinearEquiv

def ContinuousLinearEquiv.prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) : (M₁ × M₃) ≃L[R₁] M₂ × M₄

Product of two continuous linear equivalences. The map comes from Equiv.prodCongr.

Equations

• e.prod e' = { toLinearEquiv := e.prodCongr e'.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.prod_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) (x : M₁ × M₃) : (e.prod e') x = (e x.1, e' x.2)

@[simp]

theorem ContinuousLinearEquiv.coe_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) : ↑(e.prod e') = (↑e).prodMap ↑e'

theorem ContinuousLinearEquiv.prod_symm {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) : (e.prod e').symm = e.symm.prod e'.symm

def ContinuousLinearEquiv.prodComm (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : (M₁ × M₂) ≃L[R₁] M₂ × M₁

Product of modules is commutative up to continuous linear isomorphism.

Equations

• ContinuousLinearEquiv.prodComm R₁ M₁ M₂ = { toLinearEquiv := LinearEquiv.prodComm R₁ M₁ M₂, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.prodComm_toLinearEquiv (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : (prodComm R₁ M₁ M₂).toLinearEquiv = LinearEquiv.prodComm R₁ M₁ M₂

@[simp]

theorem ContinuousLinearEquiv.prodComm_apply (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (a✝ : M₁ × M₂) : (prodComm R₁ M₁ M₂) a✝ = a✝.swap

@[simp]

theorem ContinuousLinearEquiv.prodComm_symm (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : (prodComm R₁ M₁ M₂).symm = prodComm R₁ M₂ M₁

theorem ContinuousLinearEquiv.bijective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : Function.Bijective ⇑e

theorem ContinuousLinearEquiv.injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : Function.Injective ⇑e

theorem ContinuousLinearEquiv.surjective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : Function.Surjective ⇑e

@[simp]

theorem ContinuousLinearEquiv.trans_apply {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] {σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) (c : M₁) : (e₁.trans e₂) c = e₂ (e₁ c)

@[simp]

theorem ContinuousLinearEquiv.apply_symm_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (c : M₂) : e (e.symm c) = c

@[simp]

theorem ContinuousLinearEquiv.symm_apply_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : e.symm (e b) = b

@[simp]

theorem ContinuousLinearEquiv.symm_trans_apply {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] {σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] (e₁ : M₂ ≃SL[σ₂₁] M₁) (e₂ : M₃ ≃SL[σ₃₂] M₂) (c : M₁) : (e₂.trans e₁).symm c = e₂.symm (e₁.symm c)

@[simp]

theorem ContinuousLinearEquiv.symm_image_image {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : ⇑e.symm '' (⇑e '' s) = s

@[simp]

theorem ContinuousLinearEquiv.image_symm_image {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : ⇑e '' (⇑e.symm '' s) = s

@[simp]

theorem ContinuousLinearEquiv.comp_coe {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] {σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] (f : M₁ ≃SL[σ₁₂] M₂) (f' : M₂ ≃SL[σ₂₃] M₃) : (↑f').comp ↑f = ↑(f.trans f')

@[simp]

theorem ContinuousLinearEquiv.coe_comp_coe_symm {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : (↑e).comp ↑e.symm = ContinuousLinearMap.id R₂ M₂

@[simp]

theorem ContinuousLinearEquiv.coe_symm_comp_coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : (↑e.symm).comp ↑e = ContinuousLinearMap.id R₁ M₁

@[simp]

theorem ContinuousLinearEquiv.symm_comp_self {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.symm ∘ ⇑e = id

@[simp]

theorem ContinuousLinearEquiv.self_comp_symm {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e ∘ ⇑e.symm = id

@[simp]

theorem ContinuousLinearEquiv.symm_symm {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.symm = e

theorem ContinuousLinearEquiv.symm_bijective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Function.Bijective ContinuousLinearEquiv.symm

@[simp]

theorem ContinuousLinearEquiv.refl_symm {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : (ContinuousLinearEquiv.refl R₁ M₁).symm = ContinuousLinearEquiv.refl R₁ M₁

theorem ContinuousLinearEquiv.symm_symm_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : e.symm.symm x = e x

theorem ContinuousLinearEquiv.symm_apply_eq {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {x : M₂} {y : M₁} : e.symm x = y ↔ x = e y

theorem ContinuousLinearEquiv.eq_symm_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {x : M₂} {y : M₁} : y = e.symm x ↔ e y = x

theorem ContinuousLinearEquiv.image_eq_preimage {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : ⇑e '' s = ⇑e.symm ⁻¹' s

theorem ContinuousLinearEquiv.image_symm_eq_preimage {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : ⇑e.symm '' s = ⇑e ⁻¹' s

@[simp]

theorem ContinuousLinearEquiv.symm_preimage_preimage {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : ⇑e.symm ⁻¹' (⇑e ⁻¹' s) = s

@[simp]

theorem ContinuousLinearEquiv.preimage_symm_preimage {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : ⇑e ⁻¹' (⇑e.symm ⁻¹' s) = s

theorem ContinuousLinearEquiv.isUniformEmbedding {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {E₁ : Type u_8} {E₂ : Type u_9} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [IsUniformAddGroup E₁] [IsUniformAddGroup E₂] (e : E₁ ≃SL[σ₁₂] E₂) : IsUniformEmbedding ⇑e

theorem LinearEquiv.isUniformEmbedding {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {E₁ : Type u_8} {E₂ : Type u_9} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [IsUniformAddGroup E₁] [IsUniformAddGroup E₂] (e : E₁ ≃ₛₗ[σ₁₂] E₂) (h₁ : Continuous ⇑e) (h₂ : Continuous ⇑e.symm) : IsUniformEmbedding ⇑e

def ContinuousLinearEquiv.equivOfInverse {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse ⇑f₂ ⇑f₁) (h₂ : Function.RightInverse ⇑f₂ ⇑f₁) : M₁ ≃SL[σ₁₂] M₂

Create a ContinuousLinearEquiv from two ContinuousLinearMaps that are inverse of each other. See also equivOfInverse'.

Equations

• ContinuousLinearEquiv.equivOfInverse f₁ f₂ h₁ h₂ = { toLinearMap := ↑f₁, invFun := ⇑f₂, left_inv := h₁, right_inv := h₂, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.equivOfInverse_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse ⇑f₂ ⇑f₁) (h₂ : Function.RightInverse ⇑f₂ ⇑f₁) (x : M₁) : (equivOfInverse f₁ f₂ h₁ h₂) x = f₁ x

@[simp]

theorem ContinuousLinearEquiv.symm_equivOfInverse {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse ⇑f₂ ⇑f₁) (h₂ : Function.RightInverse ⇑f₂ ⇑f₁) : (equivOfInverse f₁ f₂ h₁ h₂).symm = equivOfInverse f₂ f₁ h₂ h₁

def ContinuousLinearEquiv.equivOfInverse' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : f₁.comp f₂ = ContinuousLinearMap.id R₂ M₂) (h₂ : f₂.comp f₁ = ContinuousLinearMap.id R₁ M₁) : M₁ ≃SL[σ₁₂] M₂

Create a ContinuousLinearEquiv from two ContinuousLinearMaps that are inverse of each other, in the ContinuousLinearMap.comp sense. See also equivOfInverse.

Equations

• ContinuousLinearEquiv.equivOfInverse' f₁ f₂ h₁ h₂ = ContinuousLinearEquiv.equivOfInverse f₁ f₂ ⋯ ⋯

@[simp]

theorem ContinuousLinearEquiv.equivOfInverse'_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : f₁.comp f₂ = ContinuousLinearMap.id R₂ M₂) (h₂ : f₂.comp f₁ = ContinuousLinearMap.id R₁ M₁) (x : M₁) : (equivOfInverse' f₁ f₂ h₁ h₂) x = f₁ x

@[simp]

theorem ContinuousLinearEquiv.symm_equivOfInverse' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : f₁.comp f₂ = ContinuousLinearMap.id R₂ M₂) (h₂ : f₂.comp f₁ = ContinuousLinearMap.id R₁ M₁) : (equivOfInverse' f₁ f₂ h₁ h₂).symm = equivOfInverse' f₂ f₁ h₂ h₁

The inverse of equivOfInverse' is obtained by swapping the order of its parameters.

instance ContinuousLinearEquiv.automorphismGroup {R₁ : Type u_1} [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : Group (M₁ ≃L[R₁] M₁)

The continuous linear equivalences from M to itself form a group under composition.

Equations

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

def ContinuousLinearEquiv.ulift {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : ULift.{u_9, u_4} M₁ ≃L[R₁] M₁

The continuous linear equivalence between ULift M₁ and M₁.

This is a continuous version of ULift.moduleEquiv.

Equations

• ContinuousLinearEquiv.ulift = { toLinearEquiv := ULift.moduleEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }

def ContinuousLinearEquiv.arrowCongrEquiv {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] {R₄ : Type u_8} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) : (M₁ →SL[σ₁₄] M₄) ≃ (M₂ →SL[σ₂₃] M₃)

A pair of continuous (semi)linear equivalences generates an equivalence between the spaces of continuous linear maps. See also ContinuousLinearEquiv.arrowCongr.

Equations

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

@[simp]

theorem ContinuousLinearEquiv.arrowCongrEquiv_symm_apply {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] {R₄ : Type u_8} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) (f : M₂ →SL[σ₂₃] M₃) : (e₁₂.arrowCongrEquiv e₄₃).symm f = (↑e₄₃.symm).comp (f.comp ↑e₁₂)

@[simp]

theorem ContinuousLinearEquiv.arrowCongrEquiv_apply {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_7} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] {R₄ : Type u_8} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) (f : M₁ →SL[σ₁₄] M₄) : (e₁₂.arrowCongrEquiv e₄₃) f = (↑e₄₃).comp (f.comp ↑e₁₂.symm)

def ContinuousLinearEquiv.piCongrLeft (R : Type u_9) [Semiring R] {ι : Type u_10} {ι' : Type u_11} (φ : ι → Type u_12) [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] [(i : ι) → TopologicalSpace (φ i)] (e : ι' ≃ ι) : ((i' : ι') → φ (e i')) ≃L[R] (i : ι) → φ i

Combine a family of linear equivalences into a linear equivalence of pi-types. This is Equiv.piCongrLeft as a ContinuousLinearEquiv.

Equations

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

def ContinuousLinearEquiv.sumPiEquivProdPi (R : Type u_9) [Semiring R] (S : Type u_10) (T : Type u_11) (A : S ⊕ T → Type u_12) [(st : S ⊕ T) → AddCommMonoid (A st)] [(st : S ⊕ T) → Module R (A st)] [(st : S ⊕ T) → TopologicalSpace (A st)] : ((st : S ⊕ T) → A st) ≃L[R] ((s : S) → A (Sum.inl s)) × ((t : T) → A (Sum.inr t))

The product over S ⊕ T of a family of topological modules is isomorphic (topologically and alegbraically) to the product of (the product over S) and (the product over T).

This is Equiv.sumPiEquivProdPi as a ContinuousLinearEquiv.

Equations

• ContinuousLinearEquiv.sumPiEquivProdPi R S T A = { toLinearEquiv := LinearEquiv.sumPiEquivProdPi R S T A, continuous_toFun := ⋯, continuous_invFun := ⋯ }

def ContinuousLinearEquiv.piUnique {α : Type u_9} [Unique α] (R : Type u_10) [Semiring R] (f : α → Type u_11) [(x : α) → AddCommMonoid (f x)] [(x : α) → Module R (f x)] [(x : α) → TopologicalSpace (f x)] : ((t : α) → f t) ≃L[R] f default

The product Π t : α, f t of a family of topological modules is isomorphic (both topologically and algebraically) to the space f ⬝ when α only contains ⬝.

This is Equiv.piUnique as a ContinuousLinearEquiv.

Equations

• ContinuousLinearEquiv.piUnique R f = { toLinearEquiv := LinearEquiv.piUnique R f, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.piUnique_apply {α : Type u_9} [Unique α] (R : Type u_10) [Semiring R] (f : α → Type u_11) [(x : α) → AddCommMonoid (f x)] [(x : α) → Module R (f x)] [(x : α) → TopologicalSpace (f x)] : ⇑(piUnique R f) = fun (f : (i : α) → f i) => f default

@[simp]

theorem ContinuousLinearEquiv.piUnique_symm_apply {α : Type u_9} [Unique α] (R : Type u_10) [Semiring R] (f : α → Type u_11) [(x : α) → AddCommMonoid (f x)] [(x : α) → Module R (f x)] [(x : α) → TopologicalSpace (f x)] : ⇑(piUnique R f).symm = uniqueElim

def ContinuousLinearEquiv.piCongrRight {R₁ : Type u_1} [Semiring R₁] {ι : Type u_9} {M : ι → Type u_10} [(i : ι) → TopologicalSpace (M i)] [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R₁ (M i)] {N : ι → Type u_11} [(i : ι) → TopologicalSpace (N i)] [(i : ι) → AddCommMonoid (N i)] [(i : ι) → Module R₁ (N i)] (f : (i : ι) → M i ≃L[R₁] N i) : ((i : ι) → M i) ≃L[R₁] (i : ι) → N i

Combine a family of continuous linear equivalences into a continuous linear equivalence of pi-types.

Equations

• ContinuousLinearEquiv.piCongrRight f = { toLinearEquiv := LinearEquiv.piCongrRight fun (i : ι) => ↑(f i), continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.piCongrRight_apply {R₁ : Type u_1} [Semiring R₁] {ι : Type u_9} {M : ι → Type u_10} [(i : ι) → TopologicalSpace (M i)] [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R₁ (M i)] {N : ι → Type u_11} [(i : ι) → TopologicalSpace (N i)] [(i : ι) → AddCommMonoid (N i)] [(i : ι) → Module R₁ (N i)] (f : (i : ι) → M i ≃L[R₁] N i) (m : (i : ι) → M i) (i : ι) : (piCongrRight f) m i = (f i) (m i)

@[simp]

theorem ContinuousLinearEquiv.piCongrRight_symm_apply {R₁ : Type u_1} [Semiring R₁] {ι : Type u_9} {M : ι → Type u_10} [(i : ι) → TopologicalSpace (M i)] [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R₁ (M i)] {N : ι → Type u_11} [(i : ι) → TopologicalSpace (N i)] [(i : ι) → AddCommMonoid (N i)] [(i : ι) → Module R₁ (N i)] (f : (i : ι) → M i ≃L[R₁] N i) (n : (i : ι) → N i) (i : ι) : (piCongrRight f).symm n i = (f i).symm (n i)

def ContinuousLinearEquiv.smulLeft {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {G : Type u_9} [Group G] [DistribMulAction G M₁] [ContinuousConstSMul G M₁] [SMulCommClass G R₁ M₁] : G →* M₁ ≃L[R₁] M₁

Scalar multiplication by a group element as a continuous linear equivalence.

Equations

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

@[simp]

theorem ContinuousLinearEquiv.smulLeft_apply_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {G : Type u_9} [Group G] [DistribMulAction G M₁] [ContinuousConstSMul G M₁] [SMulCommClass G R₁ M₁] (g : G) (a✝ : M₁) : (smulLeft g) a✝ = g • a✝

@[simp]

theorem ContinuousLinearEquiv.smulLeft_apply_toLinearEquiv {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {G : Type u_9} [Group G] [DistribMulAction G M₁] [ContinuousConstSMul G M₁] [SMulCommClass G R₁ M₁] (g : G) : (smulLeft g).toLinearEquiv = DistribMulAction.toLinearEquiv R₁ M₁ g

Automorphisms as continuous linear equivalences and as units of the ring of endomorphisms

The next theorems cover the identification between M ≃L[R] Mand the group of units of the ring M →L[R] M.

def ContinuousLinearEquiv.ofUnit {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] (f : (M →L[R] M)ˣ) : M ≃L[R] M

An invertible continuous linear map f determines a continuous equivalence from M to itself.

Equations

• ContinuousLinearEquiv.ofUnit f = { toFun := ⇑↑f, map_add' := ⋯, map_smul' := ⋯, invFun := ⇑f.inv, left_inv := ⋯, right_inv := ⋯, continuous_toFun := ⋯, continuous_invFun := ⋯ }

def ContinuousLinearEquiv.toUnit {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] (f : M ≃L[R] M) : (M →L[R] M)ˣ

A continuous equivalence from M to itself determines an invertible continuous linear map.

Equations

• f.toUnit = { val := ↑f, inv := ↑f.symm, val_inv := ⋯, inv_val := ⋯ }

def ContinuousLinearEquiv.unitsEquiv (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : (M →L[R] M)ˣ ≃* M ≃L[R] M

The units of the algebra of continuous R-linear endomorphisms of M is multiplicatively equivalent to the type of continuous linear equivalences between M and itself.

Equations

• ContinuousLinearEquiv.unitsEquiv R M = { toFun := ContinuousLinearEquiv.ofUnit, invFun := ContinuousLinearEquiv.toUnit, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.unitsEquiv_apply (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] (f : (M →L[R] M)ˣ) (x : M) : ((unitsEquiv R M) f) x = ↑f x

Units of a ring as linear automorphisms

def ContinuousLinearEquiv.unitsEquivAut (R : Type u_1) [Semiring R] [TopologicalSpace R] [ContinuousMul R] : Rˣ ≃ R ≃L[R] R

Continuous linear equivalences R ≃L[R] R are enumerated by Rˣ.

Equations

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

@[simp]

theorem ContinuousLinearEquiv.unitsEquivAut_apply {R : Type u_1} [Semiring R] [TopologicalSpace R] [ContinuousMul R] (u : Rˣ) (x : R) : ((unitsEquivAut R) u) x = x * ↑u

@[simp]

theorem ContinuousLinearEquiv.unitsEquivAut_apply_symm {R : Type u_1} [Semiring R] [TopologicalSpace R] [ContinuousMul R] (u : Rˣ) (x : R) : ((unitsEquivAut R) u).symm x = x * ↑u⁻¹

@[simp]

theorem ContinuousLinearEquiv.unitsEquivAut_symm_apply {R : Type u_1} [Semiring R] [TopologicalSpace R] [ContinuousMul R] (e : R ≃L[R] R) : ↑((unitsEquivAut R).symm e) = e 1

def ContinuousLinearEquiv.funUnique (ι : Type u_1) (R : Type u_2) (M : Type u_3) [Unique ι] [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : (ι → M) ≃L[R] M

If ι has a unique element, then ι → M is continuously linear equivalent to M.

Equations

• ContinuousLinearEquiv.funUnique ι R M = { toLinearEquiv := LinearEquiv.funUnique ι R M, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.coe_funUnique {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Unique ι] [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : ⇑(funUnique ι R M) = Function.eval default

@[simp]

theorem ContinuousLinearEquiv.coe_funUnique_symm {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Unique ι] [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : ⇑(funUnique ι R M).symm = Function.const ι

def ContinuousLinearEquiv.piFinTwo (R : Type u_2) [Semiring R] (M : Fin 2 → Type u_4) [(i : Fin 2) → AddCommMonoid (M i)] [(i : Fin 2) → Module R (M i)] [(i : Fin 2) → TopologicalSpace (M i)] : ((i : Fin 2) → M i) ≃L[R] M 0 × M 1

Continuous linear equivalence between dependent functions (i : Fin 2) → M i and M 0 × M 1.

Equations

• ContinuousLinearEquiv.piFinTwo R M = { toLinearEquiv := LinearEquiv.piFinTwo R M, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.piFinTwo_apply (R : Type u_2) [Semiring R] (M : Fin 2 → Type u_4) [(i : Fin 2) → AddCommMonoid (M i)] [(i : Fin 2) → Module R (M i)] [(i : Fin 2) → TopologicalSpace (M i)] : ⇑(piFinTwo R M) = fun (f : (i : Fin 2) → M i) => (f 0, f 1)

@[simp]

theorem ContinuousLinearEquiv.piFinTwo_symm_apply (R : Type u_2) [Semiring R] (M : Fin 2 → Type u_4) [(i : Fin 2) → AddCommMonoid (M i)] [(i : Fin 2) → Module R (M i)] [(i : Fin 2) → TopologicalSpace (M i)] : ⇑(piFinTwo R M).symm = fun (p : M 0 × M 1) => Fin.cons p.1 (Fin.cons p.2 finZeroElim)

def ContinuousLinearEquiv.finTwoArrow (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : (Fin 2 → M) ≃L[R] M × M

Continuous linear equivalence between vectors in M² = Fin 2 → M and M × M.

Equations

• ContinuousLinearEquiv.finTwoArrow R M = { toLinearEquiv := LinearEquiv.finTwoArrow R M, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.finTwoArrow_symm_apply (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : ⇑(finTwoArrow R M).symm = fun (x : M × M) => ![x.1, x.2]

@[simp]

theorem ContinuousLinearEquiv.finTwoArrow_apply (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] : ⇑(finTwoArrow R M) = fun (f : Fin 2 → M) => (f 0, f 1)

def Fin.consEquivL {n : ℕ} (R : Type u_4) (M : Fin n.succ → Type u_5) [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [(i : Fin n.succ) → Module R (M i)] [(i : Fin n.succ) → TopologicalSpace (M i)] : (M 0 × ((i : Fin n) → M i.succ)) ≃L[R] (i : Fin n.succ) → M i

Fin.consEquiv as a continuous linear equivalence.

Equations

• Fin.consEquivL R M = { toLinearEquiv := Fin.consLinearEquiv R M, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem Fin.consEquivL_symm_apply {n : ℕ} (R : Type u_4) (M : Fin n.succ → Type u_5) [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [(i : Fin n.succ) → Module R (M i)] [(i : Fin n.succ) → TopologicalSpace (M i)] (a✝ : (i : Fin (n + 1)) → M i) : (consEquivL R M).symm a✝ = (a✝ 0, tail a✝)

@[simp]

theorem Fin.consEquivL_apply {n : ℕ} (R : Type u_4) (M : Fin n.succ → Type u_5) [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [(i : Fin n.succ) → Module R (M i)] [(i : Fin n.succ) → TopologicalSpace (M i)] (a✝ : M 0 × ((i : Fin n) → M i.succ)) (i : Fin (n + 1)) : (consEquivL R M) a✝ i = cons a✝.1 a✝.2 i

@[reducible, inline]

abbrev ContinuousLinearMap.finCons {n : ℕ} {R : Type u_4} {M : Fin n.succ → Type u_5} {N : Type u_6} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [(i : Fin n.succ) → Module R (M i)] [(i : Fin n.succ) → TopologicalSpace (M i)] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : N →L[R] M 0) (fs : N →L[R] (i : Fin n) → M i.succ) : N →L[R] (i : Fin n.succ) → M i

Fin.cons in the codomain of continuous linear maps.

Equations

• f.finCons fs = (↑(Fin.consEquivL R M)).comp (f.prod fs)

def ContinuousLinearEquiv.skewProd {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_4} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type u_5} [TopologicalSpace M₄] [AddCommGroup M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [IsTopologicalAddGroup M₄] (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) : (M × M₃) ≃L[R] M₂ × M₄

Equivalence given by a block lower diagonal matrix. e and e' are diagonal square blocks, and f is a rectangular block below the diagonal.

Equations

• e.skewProd e' f = { toLinearEquiv := e.skewProd e'.toLinearEquiv ↑f, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.skewProd_apply {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_4} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type u_5} [TopologicalSpace M₄] [AddCommGroup M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [IsTopologicalAddGroup M₄] (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x : M × M₃) : (e.skewProd e' f) x = (e x.1, e' x.2 + f x.1)

@[simp]

theorem ContinuousLinearEquiv.skewProd_symm_apply {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_4} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type u_5} [TopologicalSpace M₄] [AddCommGroup M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [IsTopologicalAddGroup M₄] (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x : M₂ × M₄) : (e.skewProd e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1)))

def ContinuousLinearEquiv.neg (R : Type u_1) [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [ContinuousNeg M] : M ≃L[R] M

The negation map as a continuous linear equivalence.

Equations

• ContinuousLinearEquiv.neg R = { toLinearEquiv := LinearEquiv.neg R, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem ContinuousLinearEquiv.coe_neg {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [ContinuousNeg M] : ⇑(neg R) = -id

@[simp]

theorem ContinuousLinearEquiv.neg_apply {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [ContinuousNeg M] (x : M) : (neg R) x = -x

@[simp]

theorem ContinuousLinearEquiv.symm_neg {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [ContinuousNeg M] : (neg R).symm = neg R

theorem ContinuousLinearEquiv.map_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (e : M ≃SL[σ₁₂] M₂) (x y : M) : e (x - y) = e x - e y

theorem ContinuousLinearEquiv.map_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (e : M ≃SL[σ₁₂] M₂) (x : M) : e (-x) = -e x

def ContinuousLinearEquiv.equivOfRightInverse {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [IsTopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) : M ≃L[R] M₂ × ↥(LinearMap.ker f₁)

A pair of continuous linear maps such that f₁ ∘ f₂ = id generates a continuous linear equivalence e between M and M₂ × f₁.ker such that (e x).2 = x for x ∈ f₁.ker, (e x).1 = f₁ x, and (e (f₂ y)).2 = 0. The map is given by e x = (f₁ x, x - f₂ (f₁ x)).

Equations

• ContinuousLinearEquiv.equivOfRightInverse f₁ f₂ h = ContinuousLinearEquiv.equivOfInverse (f₁.prod (f₁.projKerOfRightInverse f₂ h)) (f₂.coprod (LinearMap.ker f₁).subtypeL) ⋯ ⋯

@[simp]

theorem ContinuousLinearEquiv.fst_equivOfRightInverse {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [IsTopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : M) : ((equivOfRightInverse f₁ f₂ h) x).1 = f₁ x

@[simp]

theorem ContinuousLinearEquiv.snd_equivOfRightInverse {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [IsTopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : M) : ↑((equivOfRightInverse f₁ f₂ h) x).2 = x - f₂ (f₁ x)

@[simp]

theorem ContinuousLinearEquiv.equivOfRightInverse_symm_apply {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [IsTopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (y : M₂ × ↥(LinearMap.ker f₁)) : (equivOfRightInverse f₁ f₂ h).symm y = f₂ y.1 + ↑y.2

def ContinuousLinearMap.IsInvertible {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] (f : M →L[R] M₂) : Prop

A continuous linear map is invertible if it is the forward direction of a continuous linear equivalence.

Equations

• f.IsInvertible = ∃ (A : M ≃L[R] M₂), ↑A = f

noncomputable def ContinuousLinearMap.inverse {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] : (M →L[R] M₂) → M₂ →L[R] M

Introduce a function inverse from M →L[R] M₂ to M₂ →L[R] M, which sends f to f.symm if f is a continuous linear equivalence and to 0 otherwise. This definition is somewhat ad hoc, but one needs a fully (rather than partially) defined inverse function for some purposes, including for calculus.

Equations

• f.inverse = if h : f.IsInvertible then ↑(Classical.choose h).symm else 0

@[simp]

theorem ContinuousLinearMap.isInvertible_equiv {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {f : M ≃L[R] M₂} : (↑f).IsInvertible

@[simp]

theorem ContinuousLinearMap.inverse_equiv {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] (e : M ≃L[R] M₂) : (↑e).inverse = ↑e.symm

By definition, if f is invertible then inverse f = f.symm.

@[simp]

theorem ContinuousLinearMap.inverse_of_not_isInvertible {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {f : M →L[R] M₂} (hf : ¬f.IsInvertible) : f.inverse = 0

By definition, if f is not invertible then inverse f = 0.

@[deprecated ContinuousLinearMap.inverse_of_not_isInvertible (since := "2024-10-29")]

theorem ContinuousLinearMap.inverse_non_equiv {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {f : M →L[R] M₂} (hf : ¬f.IsInvertible) : f.inverse = 0

Alias of ContinuousLinearMap.inverse_of_not_isInvertible.

---

By definition, if f is not invertible then inverse f = 0.

@[simp]

theorem ContinuousLinearMap.isInvertible_zero_iff {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] : IsInvertible 0 ↔ Subsingleton M ∧ Subsingleton M₂

@[simp]

theorem ContinuousLinearMap.inverse_zero {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] : inverse 0 = 0

theorem ContinuousLinearMap.IsInvertible.comp {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {g : M₂ →L[R] M₃} {f : M →L[R] M₂} (hg : g.IsInvertible) (hf : f.IsInvertible) : (g.comp f).IsInvertible

theorem ContinuousLinearMap.IsInvertible.of_inverse {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {f : M →L[R] M₂} {g : M₂ →L[R] M} (hf : f.comp g = id R M₂) (hg : g.comp f = id R M) : f.IsInvertible

theorem ContinuousLinearMap.inverse_eq {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {f : M →L[R] M₂} {g : M₂ →L[R] M} (hf : f.comp g = id R M₂) (hg : g.comp f = id R M) : f.inverse = g

theorem ContinuousLinearMap.IsInvertible.inverse_apply_eq {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {f : M →L[R] M₂} {x : M} {y : M₂} (hf : f.IsInvertible) : f.inverse y = x ↔ y = f x

@[simp]

theorem ContinuousLinearMap.isInvertible_equiv_comp {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {e : M₂ ≃L[R] M₃} {f : M →L[R] M₂} : ((↑e).comp f).IsInvertible ↔ f.IsInvertible

@[simp]

theorem ContinuousLinearMap.isInvertible_comp_equiv {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {e : M₃ ≃L[R] M} {f : M →L[R] M₂} : (f.comp ↑e).IsInvertible ↔ f.IsInvertible

@[simp]

theorem ContinuousLinearMap.inverse_equiv_comp {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {e : M₂ ≃L[R] M₃} {f : M →L[R] M₂} : ((↑e).comp f).inverse = f.inverse.comp ↑e.symm

@[simp]

theorem ContinuousLinearMap.inverse_comp_equiv {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {e : M₃ ≃L[R] M} {f : M →L[R] M₂} : (f.comp ↑e).inverse = (↑e.symm).comp f.inverse

theorem ContinuousLinearMap.IsInvertible.inverse_comp_of_left {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {g : M₂ →L[R] M₃} {f : M →L[R] M₂} (hg : g.IsInvertible) : (g.comp f).inverse = f.inverse.comp g.inverse

theorem ContinuousLinearMap.IsInvertible.inverse_comp_apply_of_left {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {g : M₂ →L[R] M₃} {f : M →L[R] M₂} {v : M₃} (hg : g.IsInvertible) : (g.comp f).inverse v = f.inverse (g.inverse v)

theorem ContinuousLinearMap.IsInvertible.inverse_comp_of_right {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {g : M₂ →L[R] M₃} {f : M →L[R] M₂} (hf : f.IsInvertible) : (g.comp f).inverse = f.inverse.comp g.inverse

theorem ContinuousLinearMap.IsInvertible.inverse_comp_apply_of_right {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [TopologicalSpace M] [TopologicalSpace M₂] [TopologicalSpace M₃] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {g : M₂ →L[R] M₃} {f : M →L[R] M₂} {v : M₃} (hf : f.IsInvertible) : (g.comp f).inverse v = f.inverse (g.inverse v)

@[simp]

theorem ContinuousLinearMap.ringInverse_equiv {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] (e : M ≃L[R] M) : Ring.inverse ↑e = (↑e).inverse

@[deprecated ContinuousLinearMap.ringInverse_equiv (since := "2025-04-22")]

theorem ContinuousLinearMap.ring_inverse_equiv {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] (e : M ≃L[R] M) : Ring.inverse ↑e = (↑e).inverse

Alias of ContinuousLinearMap.ringInverse_equiv.

theorem ContinuousLinearMap.inverse_eq_ringInverse {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] (e : M ≃L[R] M₂) (f : M →L[R] M₂) : f.inverse = (Ring.inverse ((↑e.symm).comp f)).comp ↑e.symm

The function ContinuousLinearEquiv.inverse can be written in terms of Ring.inverse for the ring of self-maps of the domain.

@[deprecated ContinuousLinearMap.inverse_eq_ringInverse (since := "2025-04-22")]

theorem ContinuousLinearMap.to_ring_inverse {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] (e : M ≃L[R] M₂) (f : M →L[R] M₂) : f.inverse = (Ring.inverse ((↑e.symm).comp f)).comp ↑e.symm

Alias of ContinuousLinearMap.inverse_eq_ringInverse.

---

The function ContinuousLinearEquiv.inverse can be written in terms of Ring.inverse for the ring of self-maps of the domain.

theorem ContinuousLinearMap.ringInverse_eq_inverse {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] : Ring.inverse = inverse

@[deprecated ContinuousLinearMap.ringInverse_eq_inverse (since := "2025-04-22")]

theorem ContinuousLinearMap.ring_inverse_eq_map_inverse {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] : Ring.inverse = inverse

Alias of ContinuousLinearMap.ringInverse_eq_inverse.

@[simp]

theorem ContinuousLinearMap.inverse_id {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] : (id R M).inverse = id R M

theorem Submodule.ClosedComplemented.exists_submodule_equiv_prod {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [IsTopologicalAddGroup M] {p : Submodule R M} (hp : p.ClosedComplemented) : ∃ (q : Submodule R M) (e : M ≃L[R] ↥p × ↥q), (∀ (x : ↥p), e ↑x = (x, 0)) ∧ (∀ (y : ↥q), e ↑y = (0, y)) ∧ ∀ (x : ↥p × ↥q), e.symm x = ↑x.1 + ↑x.2

If p is a closed complemented submodule, then there exists a submodule q and a continuous linear equivalence M ≃L[R] (p × q) such that e (x : p) = (x, 0), e (y : q) = (0, y), and e.symm x = x.1 + x.2.

In fact, the properties of e imply the properties of e.symm and vice versa, but we provide both for convenience.

def MulOpposite.opContinuousLinearEquiv (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] [TopologicalSpace M] : M ≃L[R] Mᵐᵒᵖ

The function op is a continuous linear equivalence.

Equations

• MulOpposite.opContinuousLinearEquiv R = { toLinearEquiv := MulOpposite.opLinearEquiv R, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem MulOpposite.opContinuousLinearEquiv_apply (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] [TopologicalSpace M] (a✝ : M) : (opContinuousLinearEquiv R) a✝ = op a✝

@[simp]

theorem MulOpposite.opContinuousLinearEquiv_symm_apply (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] [TopologicalSpace M] (a✝ : Mᵐᵒᵖ) : (opContinuousLinearEquiv R).symm a✝ = unop a✝