fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
IsInvertible.inverse_comp_of_left {g : M₂ →L[R] M₃} {f : M →L[R] M₂}
(hg : g.IsInvertible) : (g ∘L f).inverse = f.inverse ∘L g.inverse := by
rcases hg with ⟨N, rfl⟩
simp | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | IsInvertible.inverse_comp_of_left | null |
IsInvertible.inverse_comp_apply_of_left {g : M₂ →L[R] M₃} {f : M →L[R] M₂} {v : M₃}
(hg : g.IsInvertible) : (g ∘L f).inverse v = f.inverse (g.inverse v) := by
simp only [hg.inverse_comp_of_left, coe_comp', Function.comp_apply] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | IsInvertible.inverse_comp_apply_of_left | null |
IsInvertible.inverse_comp_of_right {g : M₂ →L[R] M₃} {f : M →L[R] M₂}
(hf : f.IsInvertible) : (g ∘L f).inverse = f.inverse ∘L g.inverse := by
rcases hf with ⟨M, rfl⟩
simp | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | IsInvertible.inverse_comp_of_right | null |
IsInvertible.inverse_comp_apply_of_right {g : M₂ →L[R] M₃} {f : M →L[R] M₂} {v : M₃}
(hf : f.IsInvertible) : (g ∘L f).inverse v = f.inverse (g.inverse v) := by
simp only [hf.inverse_comp_of_right, coe_comp', Function.comp_apply]
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | IsInvertible.inverse_comp_apply_of_right | null |
ringInverse_equiv (e : M ≃L[R] M) : Ring.inverse ↑e = inverse (e : M →L[R] M) := by
suffices Ring.inverse ((ContinuousLinearEquiv.unitsEquiv _ _).symm e : M →L[R] M) = inverse ↑e by
convert this
simp
rfl
@[deprecated (since := "2025-04-22")] alias ring_inverse_equiv := ringInverse_equiv | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ringInverse_equiv | null |
inverse_eq_ringInverse (e : M ≃L[R] M₂) (f : M →L[R] M₂) :
inverse f = Ring.inverse ((e.symm : M₂ →L[R] M).comp f) ∘L e.symm := by
by_cases h₁ : f.IsInvertible
· obtain ⟨e', he'⟩ := h₁
rw [← he']
change _ = Ring.inverse (e'.trans e.symm : M →L[R] M) ∘L (e.symm : M₂ →L[R] M)
ext
simp
· suffices ¬IsUnit ((e.symm : M₂ →L[R] M).comp f) by simp [this, h₁]
contrapose! h₁
rcases h₁ with ⟨F, hF⟩
use (ContinuousLinearEquiv.unitsEquiv _ _ F).trans e
ext
dsimp
rw [hF]
simp
@[deprecated (since := "2025-04-22")] alias to_ring_inverse := inverse_eq_ringInverse | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | inverse_eq_ringInverse | The function `ContinuousLinearEquiv.inverse` can be written in terms of `Ring.inverse` for the
ring of self-maps of the domain. |
ringInverse_eq_inverse : Ring.inverse = inverse (R := R) (M := M) := by
ext
simp [inverse_eq_ringInverse (ContinuousLinearEquiv.refl R M)]
@[deprecated (since := "2025-04-22")]
alias ring_inverse_eq_map_inverse := ringInverse_eq_inverse
@[simp] theorem inverse_id : (id R M).inverse = id R M := by
rw [← ringInverse_eq_inverse]
exact Ring.inverse_one _ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ringInverse_eq_inverse | null |
coprod_comp_prodComm [ContinuousAdd M] (f : M₂ →L[R] M) (g : M₃ →L[R] M) :
f.coprod g ∘L ContinuousLinearEquiv.prodComm R M₃ M₂ = g.coprod f := by
ext <;> simp | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coprod_comp_prodComm | Composition of a map on a product with the exchange of the product factors |
ClosedComplemented.exists_submodule_equiv_prod [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, e.symm x = x.1 + x.2) :=
let ⟨f, hf⟩ := hp
⟨LinearMap.ker f, .equivOfRightInverse _ p.subtypeL hf,
fun _ ↦ by ext <;> simp [hf], fun _ ↦ by ext <;> simp, fun _ ↦ rfl⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ClosedComplemented.exists_submodule_equiv_prod | 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. |
@[simps!]
opContinuousLinearEquiv : M ≃L[R] Mᵐᵒᵖ where
__ := MulOpposite.opLinearEquiv R | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | opContinuousLinearEquiv | The function `op` is a continuous linear equivalence. |
unique_topology_of_t2 {t : TopologicalSpace 𝕜} (h₁ : @IsTopologicalAddGroup 𝕜 t _)
(h₂ : @ContinuousSMul 𝕜 𝕜 _ hnorm.toUniformSpace.toTopologicalSpace t) (h₃ : @T2Space 𝕜 t) :
t = hnorm.toUniformSpace.toTopologicalSpace := by
refine IsTopologicalAddGroup.ext h₁ inferInstance (le_antisymm ?_ ?_)
· -- To show `𝓣 ≤ 𝓣₀`, we have to show that closed balls are `𝓣`-neighborhoods of 0.
rw [Metric.nhds_basis_closedBall.ge_iff]
intro ε hε
rcases NormedField.exists_norm_lt 𝕜 hε with ⟨ξ₀, hξ₀, hξ₀ε⟩
have : {ξ₀}ᶜ ∈ @nhds 𝕜 t 0 := IsOpen.mem_nhds isOpen_compl_singleton <|
mem_compl_singleton_iff.mpr <| Ne.symm <| norm_ne_zero_iff.mp hξ₀.ne.symm
have : balancedCore 𝕜 {ξ₀}ᶜ ∈ @nhds 𝕜 t 0 := balancedCore_mem_nhds_zero this
refine mem_of_superset this fun ξ hξ => ?_
by_cases hξ0 : ξ = 0
· rw [hξ0]
exact Metric.mem_closedBall_self hε.le
· rw [mem_closedBall_zero_iff]
by_contra! h
suffices (ξ₀ * ξ⁻¹) • ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ by
rw [smul_eq_mul, mul_assoc, inv_mul_cancel₀ hξ0, mul_one] at this
exact notMem_compl_iff.mpr (mem_singleton ξ₀) ((balancedCore_subset _) this)
refine (balancedCore_balanced _).smul_mem ?_ hξ
rw [norm_mul, norm_inv, mul_inv_le_iff₀ (norm_pos_iff.mpr hξ0), one_mul]
exact (hξ₀ε.trans h).le
· -- Finally, to show `𝓣₀ ≤ 𝓣`, we simply argue that `id = (fun x ↦ x • 1)` is continuous from
calc
@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0 =
map id (@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0) :=
map_id.symm
_ = map (fun x => id x • (1 : 𝕜)) (@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0) := by
conv_rhs =>
congr
ext
rw [smul_eq_mul, mul_one]
_ ≤ @nhds 𝕜 t ((0 : 𝕜) • (1 : 𝕜)) :=
(@Tendsto.smul_const _ _ _ hnorm.toUniformSpace.toTopologicalSpace t _ _ _ _ _
tendsto_id (1 : 𝕜))
_ = @nhds 𝕜 t 0 := by rw [zero_smul] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | unique_topology_of_t2 | The space of continuous linear maps between finite-dimensional spaces is finite-dimensional. -/
instance [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] : FiniteDimensional 𝕜 (E →L[𝕜] F) :=
FiniteDimensional.of_injective (ContinuousLinearMap.coeLM 𝕜 : (E →L[𝕜] F) →ₗ[𝕜] E →ₗ[𝕜] F)
ContinuousLinearMap.coe_injective
end Field
section NormedField
variable {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] {F : Type w} [AddCommGroup F]
[Module 𝕜 F] [TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F] {F' : Type x}
[AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [IsTopologicalAddGroup F']
[ContinuousSMul 𝕜 F']
/-- If `𝕜` is a nontrivially normed field, any T2 topology on `𝕜` which makes it a topological
vector space over itself (with the norm topology) is *equal* to the norm topology. |
LinearMap.continuous_of_isClosed_ker (l : E →ₗ[𝕜] 𝕜)
(hl : IsClosed (LinearMap.ker l : Set E)) :
Continuous l := by
by_cases H : finrank 𝕜 (LinearMap.range l) = 0
· rw [Submodule.finrank_eq_zero, LinearMap.range_eq_bot] at H
rw [H]
exact continuous_zero
· -- In the case where `l` is surjective, we factor it as `φ : (E ⧸ l.ker) ≃ₗ[𝕜] 𝕜`. Note that
have : finrank 𝕜 (LinearMap.range l) = 1 :=
le_antisymm (finrank_self 𝕜 ▸ (LinearMap.range l).finrank_le) (zero_lt_iff.mpr H)
have hi : Function.Injective ((LinearMap.ker l).liftQ l (le_refl _)) := by
rw [← LinearMap.ker_eq_bot]
exact Submodule.ker_liftQ_eq_bot _ _ _ (le_refl _)
have hs : Function.Surjective ((LinearMap.ker l).liftQ l (le_refl _)) := by
rw [← LinearMap.range_eq_top, Submodule.range_liftQ]
exact Submodule.eq_top_of_finrank_eq ((finrank_self 𝕜).symm ▸ this)
let φ : (E ⧸ LinearMap.ker l) ≃ₗ[𝕜] 𝕜 :=
LinearEquiv.ofBijective ((LinearMap.ker l).liftQ l (le_refl _)) ⟨hi, hs⟩
have hlφ : (l : E → 𝕜) = φ ∘ (LinearMap.ker l).mkQ := by ext; rfl
suffices Continuous φ.toEquiv by
rw [hlφ]
exact this.comp continuous_quot_mk
have : induced φ.toEquiv.symm inferInstance = hnorm.toUniformSpace.toTopologicalSpace := by
refine unique_topology_of_t2 (topologicalAddGroup_induced φ.symm.toLinearMap)
(continuousSMul_induced φ.symm.toMulActionHom) ?_
rw [t2Space_iff]
exact fun x y hxy =>
@separated_by_continuous _ _ (induced _ _) _ _ _ continuous_induced_dom _ _
(φ.toEquiv.symm.injective.ne hxy)
simp_rw [this.symm, Equiv.induced_symm]
exact continuous_coinduced_rng | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearMap.continuous_of_isClosed_ker | Any linear form on a topological vector space over a nontrivially normed field is continuous if
its kernel is closed. |
LinearMap.continuous_iff_isClosed_ker (l : E →ₗ[𝕜] 𝕜) :
Continuous l ↔ IsClosed (LinearMap.ker l : Set E) :=
⟨fun h => isClosed_singleton.preimage h, l.continuous_of_isClosed_ker⟩ | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearMap.continuous_iff_isClosed_ker | Any linear form on a topological vector space over a nontrivially normed field is continuous if
and only if its kernel is closed. |
LinearMap.continuous_of_nonzero_on_open (l : E →ₗ[𝕜] 𝕜) (s : Set E) (hs₁ : IsOpen s)
(hs₂ : s.Nonempty) (hs₃ : ∀ x ∈ s, l x ≠ 0) : Continuous l := by
refine l.continuous_of_isClosed_ker (l.isClosed_or_dense_ker.resolve_right fun hl => ?_)
rcases hs₂ with ⟨x, hx⟩
have : x ∈ interior (LinearMap.ker l : Set E)ᶜ := by
rw [mem_interior_iff_mem_nhds]
exact mem_of_superset (hs₁.mem_nhds hx) hs₃
rwa [hl.interior_compl] at this
variable [CompleteSpace 𝕜] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearMap.continuous_of_nonzero_on_open | Over a nontrivially normed field, any linear form which is nonzero on a nonempty open set is
automatically continuous. |
private continuous_equivFun_basis_aux [T2Space E] {ι : Type v} [Fintype ι]
(ξ : Basis ι 𝕜 E) : Continuous ξ.equivFun := by
letI : UniformSpace E := IsTopologicalAddGroup.toUniformSpace E
letI : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
suffices ∀ n, Fintype.card ι = n → Continuous ξ.equivFun by exact this _ rfl
intro n hn
induction n generalizing ι E with
| zero =>
rw [Fintype.card_eq_zero_iff] at hn
exact continuous_of_const fun x y => funext hn.elim
| succ n IH =>
haveI : FiniteDimensional 𝕜 E := .of_fintype_basis ξ
have H₁ : ∀ s : Submodule 𝕜 E, finrank 𝕜 s = n → IsClosed (s : Set E) := by
intro s s_dim
letI : IsUniformAddGroup s := s.toAddSubgroup.isUniformAddGroup
let b := Basis.ofVectorSpace 𝕜 s
have U : IsUniformEmbedding b.equivFun.symm.toEquiv := by
have : Fintype.card (Basis.ofVectorSpaceIndex 𝕜 s) = n := by
rw [← s_dim]
exact (finrank_eq_card_basis b).symm
have : Continuous b.equivFun := IH b this
exact
b.equivFun.symm.isUniformEmbedding b.equivFun.symm.toLinearMap.continuous_on_pi this
have : IsComplete (s : Set E) :=
completeSpace_coe_iff_isComplete.1 ((completeSpace_congr U).1 inferInstance)
exact this.isClosed
have H₂ : ∀ f : E →ₗ[𝕜] 𝕜, Continuous f := by
intro f
by_cases H : finrank 𝕜 (LinearMap.range f) = 0
· rw [Submodule.finrank_eq_zero, LinearMap.range_eq_bot] at H
rw [H]
exact continuous_zero
· have : finrank 𝕜 (LinearMap.ker f) = n := by
have Z := f.finrank_range_add_finrank_ker
rw [finrank_eq_card_basis ξ, hn] at Z
have : finrank 𝕜 (LinearMap.range f) = 1 :=
le_antisymm (finrank_self 𝕜 ▸ (LinearMap.range f).finrank_le) (zero_lt_iff.mpr H)
rw [this, add_comm, Nat.add_one] at Z
exact Nat.succ.inj Z
have : IsClosed (LinearMap.ker f : Set E) := H₁ _ this
exact LinearMap.continuous_of_isClosed_ker f this
rw [continuous_pi_iff]
intro i
change Continuous (ξ.coord i)
exact H₂ (ξ.coord i) | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | continuous_equivFun_basis_aux | This version imposes `ι` and `E` to live in the same universe, so you should instead use
`continuous_equivFun_basis` which gives the same result without universe restrictions. |
@[local instance]
isModuleTopologyOfFiniteDimensional [T2Space E] [FiniteDimensional 𝕜 E] :
IsModuleTopology 𝕜 E :=
let b := Basis.ofVectorSpace 𝕜 E
have continuousEquiv : E ≃L[𝕜] (Basis.ofVectorSpaceIndex 𝕜 E) → 𝕜 :=
{ __ := b.equivFun
continuous_toFun := continuous_equivFun_basis_aux b
continuous_invFun := IsModuleTopology.continuous_of_linearMap (R := 𝕜)
(A := (Basis.ofVectorSpaceIndex 𝕜 E) → 𝕜) (B := E) b.equivFun.symm }
IsModuleTopology.iso continuousEquiv.symm | lemma | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | isModuleTopologyOfFiniteDimensional | A finite-dimensional t2 vector space over a complete field must carry the module topology.
Not declared as a global instance only for performance reasons. |
LinearMap.continuous_of_finiteDimensional [T2Space E] [FiniteDimensional 𝕜 E]
(f : E →ₗ[𝕜] F') : Continuous f :=
IsModuleTopology.continuous_of_linearMap f | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearMap.continuous_of_finiteDimensional | Any linear map on a finite-dimensional space over a complete field is continuous. |
LinearMap.continuousLinearMapClassOfFiniteDimensional [T2Space E] [FiniteDimensional 𝕜 E] :
ContinuousLinearMapClass (E →ₗ[𝕜] F') 𝕜 E F' :=
{ LinearMap.semilinearMapClass with map_continuous := fun f => f.continuous_of_finiteDimensional } | instance | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearMap.continuousLinearMapClassOfFiniteDimensional | null |
continuous_equivFun_basis [T2Space E] {ι : Type*} [Finite ι] (ξ : Basis ι 𝕜 E) :
Continuous ξ.equivFun :=
haveI : FiniteDimensional 𝕜 E := .of_fintype_basis ξ
ξ.equivFun.toLinearMap.continuous_of_finiteDimensional | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | continuous_equivFun_basis | In finite dimensions over a non-discrete complete normed field, the canonical identification
(in terms of a basis) with `𝕜^n` (endowed with the product topology) is continuous.
This is the key fact which makes all linear maps from a T2 finite-dimensional TVS over such a field
continuous (see `LinearMap.continuous_of_finiteDimensional`), which in turn implies that all
norms are equivalent in finite dimensions. |
toContinuousLinearMap : (E →ₗ[𝕜] F') ≃ₗ[𝕜] E →L[𝕜] F' where
toFun f := ⟨f, f.continuous_of_finiteDimensional⟩
invFun := (↑)
map_add' _ _ := rfl
map_smul' _ _ := rfl
right_inv _ := ContinuousLinearMap.coe_injective rfl | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toContinuousLinearMap | The continuous linear map induced by a linear map on a finite-dimensional space |
_root_.Module.End.toContinuousLinearMap (E : Type v) [NormedAddCommGroup E]
[NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] : (E →ₗ[𝕜] E) ≃ₐ[𝕜] (E →L[𝕜] E) :=
{ LinearMap.toContinuousLinearMap with
map_mul' := fun _ _ ↦ rfl
commutes' := fun _ ↦ rfl }
@[simp] | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | _root_.Module.End.toContinuousLinearMap | Algebra equivalence between the linear maps and continuous linear maps on a finite-dimensional
space. |
coe_toContinuousLinearMap' (f : E →ₗ[𝕜] F') : ⇑(LinearMap.toContinuousLinearMap f) = f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearMap' | null |
coe_toContinuousLinearMap (f : E →ₗ[𝕜] F') :
((LinearMap.toContinuousLinearMap f) : E →ₗ[𝕜] F') = f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearMap | null |
coe_toContinuousLinearMap_symm :
⇑(toContinuousLinearMap : (E →ₗ[𝕜] F') ≃ₗ[𝕜] E →L[𝕜] F').symm =
((↑) : (E →L[𝕜] F') → E →ₗ[𝕜] F') :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearMap_symm | null |
det_toContinuousLinearMap (f : E →ₗ[𝕜] E) :
(LinearMap.toContinuousLinearMap f).det = LinearMap.det f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | det_toContinuousLinearMap | null |
ker_toContinuousLinearMap (f : E →ₗ[𝕜] F') :
ker (LinearMap.toContinuousLinearMap f) = ker f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | ker_toContinuousLinearMap | null |
range_toContinuousLinearMap (f : E →ₗ[𝕜] F') :
range (LinearMap.toContinuousLinearMap f) = range f :=
rfl | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | range_toContinuousLinearMap | null |
isOpenMap_of_finiteDimensional (f : F →ₗ[𝕜] E) (hf : Function.Surjective f) :
IsOpenMap f :=
IsModuleTopology.isOpenMap_of_surjective hf | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | isOpenMap_of_finiteDimensional | A surjective linear map `f` with finite-dimensional codomain is an open map. |
canLiftContinuousLinearMap : CanLift (E →ₗ[𝕜] F) (E →L[𝕜] F) (↑) fun _ => True :=
⟨fun f _ => ⟨LinearMap.toContinuousLinearMap f, rfl⟩⟩ | instance | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | canLiftContinuousLinearMap | null |
toContinuousLinearMap_eq_iff_eq_toLinearMap (f : E →ₗ[𝕜] E) (g : E →L[𝕜] E) :
f.toContinuousLinearMap = g ↔ f = g.toLinearMap := by
simp [ContinuousLinearMap.ext_iff, LinearMap.ext_iff] | lemma | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toContinuousLinearMap_eq_iff_eq_toLinearMap | null |
_root_.ContinuousLinearMap.toLinearMap_eq_iff_eq_toContinuousLinearMap (g : E →L[𝕜] E)
(f : E →ₗ[𝕜] E) : g.toLinearMap = f ↔ g = f.toContinuousLinearMap := by
simp [ContinuousLinearMap.ext_iff, LinearMap.ext_iff] | lemma | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | _root_.ContinuousLinearMap.toLinearMap_eq_iff_eq_toContinuousLinearMap | null |
toContinuousLinearEquiv (e : E ≃ₗ[𝕜] F) : E ≃L[𝕜] F :=
{ e with
continuous_toFun := e.toLinearMap.continuous_of_finiteDimensional
continuous_invFun :=
haveI : FiniteDimensional 𝕜 F := e.finiteDimensional
e.symm.toLinearMap.continuous_of_finiteDimensional }
@[simp] | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toContinuousLinearEquiv | The continuous linear equivalence induced by a linear equivalence on a finite-dimensional
space. |
coe_toContinuousLinearEquiv (e : E ≃ₗ[𝕜] F) : (e.toContinuousLinearEquiv : E →ₗ[𝕜] F) = e :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearEquiv | null |
coe_toContinuousLinearEquiv' (e : E ≃ₗ[𝕜] F) : (e.toContinuousLinearEquiv : E → F) = e :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearEquiv' | null |
coe_toContinuousLinearEquiv_symm (e : E ≃ₗ[𝕜] F) :
(e.toContinuousLinearEquiv.symm : F →ₗ[𝕜] E) = e.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearEquiv_symm | null |
coe_toContinuousLinearEquiv_symm' (e : E ≃ₗ[𝕜] F) :
(e.toContinuousLinearEquiv.symm : F → E) = e.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearEquiv_symm' | null |
toLinearEquiv_toContinuousLinearEquiv (e : E ≃ₗ[𝕜] F) :
e.toContinuousLinearEquiv.toLinearEquiv = e := by
ext x
rfl | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toLinearEquiv_toContinuousLinearEquiv | null |
toLinearEquiv_toContinuousLinearEquiv_symm (e : E ≃ₗ[𝕜] F) :
e.toContinuousLinearEquiv.symm.toLinearEquiv = e.symm := by
ext x
rfl | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toLinearEquiv_toContinuousLinearEquiv_symm | null |
canLiftContinuousLinearEquiv :
CanLift (E ≃ₗ[𝕜] F) (E ≃L[𝕜] F) ContinuousLinearEquiv.toLinearEquiv fun _ => True :=
⟨fun f _ => ⟨_, f.toLinearEquiv_toContinuousLinearEquiv⟩⟩ | instance | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | canLiftContinuousLinearEquiv | null |
FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq
(cond : finrank 𝕜 E = finrank 𝕜 F) : Nonempty (E ≃L[𝕜] F) :=
(nonempty_linearEquiv_of_finrank_eq cond).map LinearEquiv.toContinuousLinearEquiv | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq | Two finite-dimensional topological vector spaces over a complete normed field are continuously
linearly equivalent if they have the same (finite) dimension. |
FiniteDimensional.nonempty_continuousLinearEquiv_iff_finrank_eq :
Nonempty (E ≃L[𝕜] F) ↔ finrank 𝕜 E = finrank 𝕜 F :=
⟨fun ⟨h⟩ => h.toLinearEquiv.finrank_eq, fun h =>
FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq h⟩ | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | FiniteDimensional.nonempty_continuousLinearEquiv_iff_finrank_eq | Two finite-dimensional topological vector spaces over a complete normed field are continuously
linearly equivalent if and only if they have the same (finite) dimension. |
ContinuousLinearEquiv.ofFinrankEq (cond : finrank 𝕜 E = finrank 𝕜 F) : E ≃L[𝕜] F :=
(LinearEquiv.ofFinrankEq E F cond).toContinuousLinearEquiv | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | ContinuousLinearEquiv.ofFinrankEq | A continuous linear equivalence between two finite-dimensional topological vector spaces over a
complete normed field of the same (finite) dimension. |
constrL (v : Basis ι 𝕜 E) (f : ι → F) : E →L[𝕜] F :=
haveI : FiniteDimensional 𝕜 E := FiniteDimensional.of_fintype_basis v
LinearMap.toContinuousLinearMap (v.constr 𝕜 f)
@[simp] | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | constrL | Construct a continuous linear map given the value at a finite basis. |
coe_constrL (v : Basis ι 𝕜 E) (f : ι → F) : (v.constrL f : E →ₗ[𝕜] F) = v.constr 𝕜 f :=
rfl | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_constrL | null |
@[simps! apply]
equivFunL (v : Basis ι 𝕜 E) : E ≃L[𝕜] ι → 𝕜 :=
{ v.equivFun with
continuous_toFun :=
haveI : FiniteDimensional 𝕜 E := FiniteDimensional.of_fintype_basis v
v.equivFun.toLinearMap.continuous_of_finiteDimensional
continuous_invFun := by
change Continuous v.equivFun.symm.toFun
exact v.equivFun.symm.toLinearMap.continuous_of_finiteDimensional }
@[simp] | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | equivFunL | The continuous linear equivalence between a vector space over `𝕜` with a finite basis and
functions from its basis indexing type to `𝕜`. |
equivFunL_symm_apply_repr (v : Basis ι 𝕜 E) (x : E) :
v.equivFunL.symm (v.repr x) = x :=
v.equivFunL.symm_apply_apply x
@[simp] | lemma | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | equivFunL_symm_apply_repr | null |
constrL_apply {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) (f : ι → F) (e : E) :
v.constrL f e = ∑ i, v.equivFun e i • f i :=
v.constr_apply_fintype 𝕜 _ _
@[simp 1100] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | constrL_apply | null |
constrL_basis (v : Basis ι 𝕜 E) (f : ι → F) (i : ι) : v.constrL f (v i) = f i :=
v.constr_basis 𝕜 _ _ | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | constrL_basis | null |
toContinuousLinearEquivOfDetNeZero (f : E →L[𝕜] E) (hf : f.det ≠ 0) : E ≃L[𝕜] E :=
((f : E →ₗ[𝕜] E).equivOfDetNeZero hf).toContinuousLinearEquiv
@[simp] | def | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toContinuousLinearEquivOfDetNeZero | Builds a continuous linear equivalence from a continuous linear map on a finite-dimensional
vector space whose determinant is nonzero. |
coe_toContinuousLinearEquivOfDetNeZero (f : E →L[𝕜] E) (hf : f.det ≠ 0) :
(f.toContinuousLinearEquivOfDetNeZero hf : E →L[𝕜] E) = f := by
ext x
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | coe_toContinuousLinearEquivOfDetNeZero | null |
toContinuousLinearEquivOfDetNeZero_apply (f : E →L[𝕜] E) (hf : f.det ≠ 0) (x : E) :
f.toContinuousLinearEquivOfDetNeZero hf x = f x :=
rfl | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | toContinuousLinearEquivOfDetNeZero_apply | null |
_root_.Matrix.toLin_finTwoProd_toContinuousLinearMap (a b c d : 𝕜) :
LinearMap.toContinuousLinearMap
(Matrix.toLin (Basis.finTwoProd 𝕜) (Basis.finTwoProd 𝕜) !![a, b; c, d]) =
(a • ContinuousLinearMap.fst 𝕜 𝕜 𝕜 + b • ContinuousLinearMap.snd 𝕜 𝕜 𝕜).prod
(c • ContinuousLinearMap.fst 𝕜 𝕜 𝕜 + d • ContinuousLinearMap.snd 𝕜 𝕜 𝕜) :=
ContinuousLinearMap.ext <| Matrix.toLin_finTwoProd_apply _ _ _ _ | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | _root_.Matrix.toLin_finTwoProd_toContinuousLinearMap | null |
FiniteDimensional.complete [FiniteDimensional 𝕜 E] : CompleteSpace E := by
set e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm
have : IsUniformEmbedding e.toEquiv.symm := e.symm.isUniformEmbedding
exact (completeSpace_congr this).1 inferInstance
variable {𝕜 E} | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | FiniteDimensional.complete | null |
Submodule.complete_of_finiteDimensional (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 s] :
IsComplete (s : Set E) :=
haveI : IsUniformAddGroup s := s.toAddSubgroup.isUniformAddGroup
completeSpace_coe_iff_isComplete.1 (FiniteDimensional.complete 𝕜 s) | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | Submodule.complete_of_finiteDimensional | A finite-dimensional subspace is complete. |
Submodule.closed_of_finiteDimensional
[T2Space E] (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 s] :
IsClosed (s : Set E) :=
letI := IsTopologicalAddGroup.toUniformSpace E
haveI : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
s.complete_of_finiteDimensional.isClosed | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | Submodule.closed_of_finiteDimensional | A finite-dimensional subspace is closed. |
LinearMap.isClosedEmbedding_of_injective [T2Space E] [FiniteDimensional 𝕜 E] {f : E →ₗ[𝕜] F}
(hf : LinearMap.ker f = ⊥) : IsClosedEmbedding f :=
let g := LinearEquiv.ofInjective f (LinearMap.ker_eq_bot.mp hf)
{ IsEmbedding.subtypeVal.comp g.toContinuousLinearEquiv.toHomeomorph.isEmbedding with
isClosed_range := by
haveI := f.finiteDimensional_range
simpa [LinearMap.coe_range f] using (LinearMap.range f).closed_of_finiteDimensional } | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearMap.isClosedEmbedding_of_injective | An injective linear map with finite-dimensional domain is a closed embedding. |
isClosedEmbedding_smul_left [T2Space E] {c : E} (hc : c ≠ 0) :
IsClosedEmbedding fun x : 𝕜 => x • c :=
LinearMap.isClosedEmbedding_of_injective (LinearMap.ker_toSpanSingleton 𝕜 E hc) | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | isClosedEmbedding_smul_left | null |
isClosedMap_smul_left [T2Space E] (c : E) : IsClosedMap fun x : 𝕜 => x • c := by
by_cases hc : c = 0
· simp_rw [hc, smul_zero]
exact isClosedMap_const
· exact (isClosedEmbedding_smul_left hc).isClosedMap | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | isClosedMap_smul_left | null |
ContinuousLinearMap.exists_right_inverse_of_surjective [FiniteDimensional 𝕜 F]
(f : E →L[𝕜] F) (hf : LinearMap.range f = ⊤) :
∃ g : F →L[𝕜] E, f.comp g = ContinuousLinearMap.id 𝕜 F :=
let ⟨g, hg⟩ := (f : E →ₗ[𝕜] F).exists_rightInverse_of_surjective hf
⟨LinearMap.toContinuousLinearMap g, ContinuousLinearMap.coe_inj.1 hg⟩ | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | ContinuousLinearMap.exists_right_inverse_of_surjective | null |
LocallyCompactSpace.of_finiteDimensional_of_complete (K V : Type*)
[NontriviallyNormedField K] [CompleteSpace K] [LocallyCompactSpace K]
[AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V]
[Module K V] [ContinuousSMul K V] [FiniteDimensional K V] :
LocallyCompactSpace V :=
suffices LocallyCompactSpace (SeparationQuotient V) from
SeparationQuotient.isInducing_mk.locallyCompactSpace <|
SeparationQuotient.range_mk (X := V) ▸ isClosed_univ.isLocallyClosed
let ⟨_, ⟨b⟩⟩ := Basis.exists_basis K (SeparationQuotient V)
have := FiniteDimensional.fintypeBasisIndex b
b.equivFun.toContinuousLinearEquiv.toHomeomorph.isOpenEmbedding.locallyCompactSpace | theorem | Topology | [
"Mathlib.Analysis.LocallyConvex.BalancedCoreHull",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.RingTheory.LocalRing.Basic",
"Mathlib.Topology.Algebra.Module.Determinant",
"Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LocallyCompactSpace.of_finiteDimensional_of_complete | If `K` is a complete field and `V` is a finite-dimensional vector space over `K` (equipped with
any topology so that `V` is a topological `K`-module, meaning `[IsTopologicalAddGroup V]`
and `[ContinuousSMul K V]`), and `K` is locally compact, then `V` is locally compact.
This is not an instance because `K` cannot be inferred. |
ContinuousLinearMap {R : Type*} {S : Type*} [Semiring R] [Semiring S] (σ : R →+* S)
(M : Type*) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type*) [TopologicalSpace M₂]
[AddCommMonoid M₂] [Module R M] [Module S M₂] extends M →ₛₗ[σ] M₂ where
cont : Continuous toFun := by continuity
attribute [inherit_doc ContinuousLinearMap] ContinuousLinearMap.cont
@[inherit_doc]
notation:25 M " →SL[" σ "] " M₂ => ContinuousLinearMap σ M M₂
@[inherit_doc]
notation:25 M " →L[" R "] " M₂ => ContinuousLinearMap (RingHom.id R) M M₂ | structure | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ContinuousLinearMap | Continuous linear maps 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
ring `R`. |
ContinuousSemilinearMapClass (F : Type*) {R S : outParam Type*} [Semiring R] [Semiring S]
(σ : outParam <| R →+* S) (M : outParam Type*) [TopologicalSpace M] [AddCommMonoid M]
(M₂ : outParam Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
[Module S M₂] [FunLike F M M₂] : Prop
extends SemilinearMapClass F σ M M₂, ContinuousMapClass F M M₂ | class | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ContinuousSemilinearMapClass | `ContinuousSemilinearMapClass F σ M M₂` asserts `F` is a type of bundled continuous
`σ`-semilinear maps `M → M₂`. See also `ContinuousLinearMapClass 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`. |
ContinuousLinearMapClass (F : Type*) (R : outParam Type*) [Semiring R]
(M : outParam Type*) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam Type*)
[TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] [FunLike F M M₂] :=
ContinuousSemilinearMapClass F (RingHom.id R) M M₂ | abbrev | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ContinuousLinearMapClass | `ContinuousLinearMapClass F R M M₂` asserts `F` is a type of bundled continuous
`R`-linear maps `M → M₂`. This is an abbreviation for
`ContinuousSemilinearMapClass F (RingHom.id R) M M₂`. |
StrongDual (R : Type*) [Semiring R] [TopologicalSpace R]
(M : Type*) [TopologicalSpace M] [AddCommMonoid M] [Module R M] : Type _ := M →L[R] R | abbrev | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | StrongDual | The *strong dual* of a topological vector space `M` over a ring `R`. This is the space of
continuous linear functionals and is equipped with the topology of uniform convergence
on bounded subsets. `StrongDual R M` is an abbreviation for `M →L[R] R`. |
LinearMap.coe : Coe (M₁ →SL[σ₁₂] M₂) (M₁ →ₛₗ[σ₁₂] M₂) := ⟨toLinearMap⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | LinearMap.coe | Coerce continuous linear maps to linear maps. |
coe_injective : Function.Injective ((↑) : (M₁ →SL[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₂] M₂) := by
intro f g H
cases f
cases g
congr | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_injective | null |
funLike : FunLike (M₁ →SL[σ₁₂] M₂) M₁ M₂ where
coe f := f.toLinearMap
coe_injective' _ _ h := coe_injective (DFunLike.coe_injective h) | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | funLike | null |
continuousSemilinearMapClass :
ContinuousSemilinearMapClass (M₁ →SL[σ₁₂] M₂) σ₁₂ M₁ M₂ where
map_add f := map_add f.toLinearMap
map_continuous f := f.2
map_smulₛₗ f := f.toLinearMap.map_smul' | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | continuousSemilinearMapClass | null |
coe_mk (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ →ₛₗ[σ₁₂] M₂) = f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_mk | null |
coe_mk' (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ → M₂) = f :=
rfl
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_mk' | null |
protected continuous (f : M₁ →SL[σ₁₂] M₂) : Continuous f :=
f.2 | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | continuous | null |
protected uniformContinuous {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂]
[AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [IsUniformAddGroup E₁]
[IsUniformAddGroup E₂] (f : E₁ →SL[σ₁₂] E₂) : UniformContinuous f :=
uniformContinuous_addMonoidHom_of_continuous f.continuous
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | uniformContinuous | null |
coe_inj {f g : M₁ →SL[σ₁₂] M₂} : (f : M₁ →ₛₗ[σ₁₂] M₂) = g ↔ f = g :=
coe_injective.eq_iff | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_inj | null |
coeFn_injective : @Function.Injective (M₁ →SL[σ₁₂] M₂) (M₁ → M₂) (↑) :=
DFunLike.coe_injective | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coeFn_injective | null |
toContinuousAddMonoidHom_injective :
Function.Injective ((↑) : (M₁ →SL[σ₁₂] M₂) → ContinuousAddMonoidHom M₁ M₂) :=
(DFunLike.coe_injective.of_comp_iff _).1 DFunLike.coe_injective
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_injective | null |
toContinuousAddMonoidHom_inj {f g : M₁ →SL[σ₁₂] M₂} :
(f : ContinuousAddMonoidHom M₁ M₂) = g ↔ f = g :=
toContinuousAddMonoidHom_injective.eq_iff | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_inj | null |
Simps.apply (h : M₁ →SL[σ₁₂] M₂) : M₁ → M₂ :=
h | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | Simps.apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
Simps.coe (h : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂ :=
h
initialize_simps_projections ContinuousLinearMap (toFun → apply, toLinearMap → coe, as_prefix coe)
@[ext] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | Simps.coe | See Note [custom simps projection]. |
ext {f g : M₁ →SL[σ₁₂] M₂} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext f g h | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ext | null |
protected copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : M₁ →SL[σ₁₂] M₂ where
toLinearMap := f.toLinearMap.copy f' h
cont := show Continuous f' from h.symm ▸ f.continuous
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | copy | Copy of a `ContinuousLinearMap` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. |
coe_copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : ⇑(f.copy f' h) = f' :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_copy | null |
copy_eq (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : f.copy f' h = f :=
DFunLike.ext' h | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | copy_eq | null |
range_coeFn_eq :
Set.range ((⇑) : (M₁ →SL[σ₁₂] M₂) → (M₁ → M₂)) =
{f | Continuous f} ∩ Set.range ((⇑) : (M₁ →ₛₗ[σ₁₂] M₂) → (M₁ → M₂)) := by
ext f
constructor
· rintro ⟨f, rfl⟩
exact ⟨f.continuous, f, rfl⟩
· rintro ⟨hfc, f, rfl⟩
exact ⟨⟨f, hfc⟩, rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | range_coeFn_eq | null |
protected map_zero (f : M₁ →SL[σ₁₂] M₂) : f (0 : M₁) = 0 :=
map_zero f | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_zero | null |
protected map_add (f : M₁ →SL[σ₁₂] M₂) (x y : M₁) : f (x + y) = f x + f y :=
map_add f x y
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_add | null |
protected map_smulₛₗ (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (x : M₁) : f (c • x) = σ₁₂ c • f x :=
(toLinearMap _).map_smulₛₗ _ _ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_smulₛₗ | null |
protected map_smul [Module R₁ M₂] (f : M₁ →L[R₁] M₂) (c : R₁) (x : M₁) :
f (c • x) = c • f x := by simp only [RingHom.id_apply, ContinuousLinearMap.map_smulₛₗ]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_smul | null |
map_smul_of_tower {R S : Type*} [Semiring S] [SMul R M₁] [Module S M₁] [SMul R M₂]
[Module S M₂] [LinearMap.CompatibleSMul M₁ M₂ R S] (f : M₁ →L[S] M₂) (c : R) (x : M₁) :
f (c • x) = c • f x :=
LinearMap.CompatibleSMul.map_smul (f : M₁ →ₗ[S] M₂) c x
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_smul_of_tower | null |
coe_coe (f : M₁ →SL[σ₁₂] M₂) : ⇑(f : M₁ →ₛₗ[σ₁₂] M₂) = f :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_coe | null |
ext_ring [TopologicalSpace R₁] {f g : R₁ →L[R₁] M₁} (h : f 1 = g 1) : f = g :=
coe_inj.1 <| LinearMap.ext_ring h | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ext_ring | null |
eqOn_closure_span [T2Space M₂] {s : Set M₁} {f g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn f g s) :
Set.EqOn f g (closure (Submodule.span R₁ s : Set M₁)) :=
(LinearMap.eqOn_span' h).closure f.continuous g.continuous | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | eqOn_closure_span | If two continuous linear maps are equal on a set `s`, then they are equal on the closure
of the `Submodule.span` of this set. |
ext_on [T2Space M₂] {s : Set M₁} (hs : Dense (Submodule.span R₁ s : Set M₁))
{f g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn f g s) : f = g :=
ext fun x => eqOn_closure_span h (hs x) | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ext_on | If the submodule generated by a set `s` is dense in the ambient module, then two continuous
linear maps equal on `s` are equal. |
_root_.Submodule.topologicalClosure_map [RingHomSurjective σ₁₂] [TopologicalSpace R₁]
[TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂]
[ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (s : Submodule R₁ M₁) :
s.topologicalClosure.map (f : M₁ →ₛₗ[σ₁₂] M₂) ≤
(s.map (f : M₁ →ₛₗ[σ₁₂] M₂)).topologicalClosure :=
image_closure_subset_closure_image f.continuous | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.topologicalClosure_map | Under a continuous linear map, the image of the `TopologicalClosure` of a submodule is
contained in the `TopologicalClosure` of its image. |
_root_.DenseRange.topologicalClosure_map_submodule [RingHomSurjective σ₁₂]
[TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁]
[ContinuousSMul R₂ M₂] [ContinuousAdd M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : DenseRange f)
{s : Submodule R₁ M₁} (hs : s.topologicalClosure = ⊤) :
(s.map (f : M₁ →ₛₗ[σ₁₂] M₂)).topologicalClosure = ⊤ := by
rw [SetLike.ext'_iff] at hs ⊢
simp only [Submodule.topologicalClosure_coe, Submodule.top_coe, ← dense_iff_closure_eq] at hs ⊢
exact hf'.dense_image f.continuous hs | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.DenseRange.topologicalClosure_map_submodule | Under a dense continuous linear map, a submodule whose `TopologicalClosure` is `⊤` is sent to
another such submodule. That is, the image of a dense set under a map with dense range is dense. |
instSMul : SMul S₂ (M₁ →SL[σ₁₂] M₂) where
smul c f := ⟨c • (f : M₁ →ₛₗ[σ₁₂] M₂), (f.2.const_smul _ : Continuous fun x => c • f x)⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | instSMul | null |
mulAction : MulAction S₂ (M₁ →SL[σ₁₂] M₂) where
one_smul _f := ext fun _x => one_smul _ _
mul_smul _a _b _f := ext fun _x => mul_smul _ _ _ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | mulAction | null |
smul_apply (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (c • f) x = c • f x :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smul_apply | null |
coe_smul (c : S₂) (f : M₁ →SL[σ₁₂] M₂) :
↑(c • f) = c • (f : M₁ →ₛₗ[σ₁₂] M₂) :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_smul | null |
coe_smul' (c : S₂) (f : M₁ →SL[σ₁₂] M₂) :
↑(c • f) = c • (f : M₁ → M₂) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_smul' | null |
isScalarTower [SMul S₂ T₂] [IsScalarTower S₂ T₂ M₂] :
IsScalarTower S₂ T₂ (M₁ →SL[σ₁₂] M₂) :=
⟨fun a b f => ext fun x => smul_assoc a b (f x)⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | isScalarTower | null |
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