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
⌀ |
|---|---|---|---|---|---|---|
constOfIsEmpty_toAlternatingMap [IsEmpty ι] (m : N) :
(constOfIsEmpty R M ι m).toAlternatingMap = AlternatingMap.constOfIsEmpty R M ι m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
constOfIsEmpty_toAlternatingMap
| null |
compContinuousLinearMap (g : M [⋀^ι]→L[R] N) (f : M' →L[R] M) : M' [⋀^ι]→L[R] N :=
{ g.toAlternatingMap.compLinearMap (f : M' →ₗ[R] M) with
toContinuousMultilinearMap := g.1.compContinuousLinearMap fun _ => f }
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
compContinuousLinearMap
|
If `g` is continuous alternating and `f` is a continuous linear map, then `g (f m₁, ..., f mₙ)`
is again a continuous alternating map, that we call `g.compContinuousLinearMap f`.
|
compContinuousLinearMap_apply (g : M [⋀^ι]→L[R] N) (f : M' →L[R] M) (m : ι → M') :
g.compContinuousLinearMap f m = g (f ∘ m) :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
compContinuousLinearMap_apply
| null |
_root_.ContinuousLinearMap.compContinuousAlternatingMap (g : N →L[R] N') (f : M [⋀^ι]→L[R] N) :
M [⋀^ι]→L[R] N' :=
{ (g : N →ₗ[R] N').compAlternatingMap f.toAlternatingMap with
toContinuousMultilinearMap := g.compContinuousMultilinearMap f.1 }
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearMap.compContinuousAlternatingMap
|
Composing a continuous alternating map with a continuous linear map gives again a
continuous alternating map.
|
_root_.ContinuousLinearMap.compContinuousAlternatingMap_coe (g : N →L[R] N')
(f : M [⋀^ι]→L[R] N) : ⇑(g.compContinuousAlternatingMap f) = g ∘ f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearMap.compContinuousAlternatingMap_coe
| null |
@[simps -fullyApplied apply]
_root_.ContinuousLinearEquiv.continuousAlternatingMapCongrLeftEquiv (e : M ≃L[R] M') :
M [⋀^ι]→L[R] N ≃ M' [⋀^ι]→L[R] N where
toFun f := f.compContinuousLinearMap ↑e.symm
invFun f := f.compContinuousLinearMap ↑e
left_inv f := by ext; simp [Function.comp_def]
right_inv f := by ext; simp [Function.comp_def]
@[deprecated (since := "2025-04-16")]
alias _root_.ContinuousLinearEquiv.continuousAlternatingMapComp :=
ContinuousLinearEquiv.continuousAlternatingMapCongrLeftEquiv
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearEquiv.continuousAlternatingMapCongrLeftEquiv
|
A continuous linear equivalence of domains
defines an equivalence between continuous alternating maps.
This is available as a continuous linear isomorphism at
`ContinuousLinearEquiv.continuousAlternatingMapCongrLeft`.
This is `ContinuousAlternatingMap.compContinuousLinearMap` as an equivalence.
|
@[simps -fullyApplied apply]
_root_.ContinuousLinearEquiv.continuousAlternatingMapCongrRightEquiv (e : N ≃L[R] N') :
M [⋀^ι]→L[R] N ≃ M [⋀^ι]→L[R] N' where
toFun := (e : N →L[R] N').compContinuousAlternatingMap
invFun := (e.symm : N' →L[R] N).compContinuousAlternatingMap
left_inv f := by ext; simp [(· ∘ ·)]
right_inv f := by ext; simp [(· ∘ ·)]
@[deprecated (since := "2025-04-16")]
alias _root_.ContinuousLinearEquiv.compContinuousAlternatingMap :=
ContinuousLinearEquiv.continuousAlternatingMapCongrRightEquiv
set_option linter.deprecated false in
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearEquiv.continuousAlternatingMapCongrRightEquiv
|
A continuous linear equivalence of codomains
defines an equivalence between continuous alternating maps.
|
_root_.ContinuousLinearEquiv.compContinuousAlternatingMap_coe
(e : N ≃L[R] N') (f : M [⋀^ι]→L[R] N) : ⇑(e.compContinuousAlternatingMap f) = e ∘ f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearEquiv.compContinuousAlternatingMap_coe
| null |
_root_.ContinuousLinearEquiv.continuousAlternatingMapCongrEquiv
(e : M ≃L[R] M') (e' : N ≃L[R] N') : M [⋀^ι]→L[R] N ≃ M' [⋀^ι]→L[R] N' :=
e.continuousAlternatingMapCongrLeftEquiv.trans e'.continuousAlternatingMapCongrRightEquiv
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearEquiv.continuousAlternatingMapCongrEquiv
|
Continuous linear equivalences between domains and codomains
define an equivalence between the spaces of continuous alternating maps.
|
@[simps]
piEquiv {ι' : Type*} {N : ι' → Type*} [∀ i, AddCommMonoid (N i)] [∀ i, TopologicalSpace (N i)]
[∀ i, Module R (N i)] : (∀ i, M [⋀^ι]→L[R] N i) ≃ M [⋀^ι]→L[R] ∀ i, N i where
toFun := pi
invFun f i := (ContinuousLinearMap.proj i : _ →L[R] N i).compContinuousAlternatingMap f
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
piEquiv
|
`ContinuousAlternatingMap.pi` as an `Equiv`.
|
cons_add (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (x y : M) :
f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m) :=
f.toMultilinearMap.cons_add m x y
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
cons_add
|
In the specific case of continuous alternating maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `Π(i : Fin (n+1)), M i` using `cons`, one can express directly the
additivity of an alternating map along the first variable.
|
vecCons_add (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (x y : M) :
f (vecCons (x + y) m) = f (vecCons x m) + f (vecCons y m) :=
f.toMultilinearMap.cons_add m x y
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
vecCons_add
|
In the specific case of continuous alternating maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `Π(i : Fin (n+1)), M i` using `cons`, one can express directly the
additivity of an alternating map along the first variable.
|
cons_smul (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (c : R)
(x : M) : f (Fin.cons (c • x) m) = c • f (Fin.cons x m) :=
f.toMultilinearMap.cons_smul m c x
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
cons_smul
|
In the specific case of continuous alternating maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `Π(i : Fin (n+1)), M i` using `cons`, one can express directly the
multiplicativity of an alternating map along the first variable.
|
vecCons_smul (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (c : R)
(x : M) : f (vecCons (c • x) m) = c • f (vecCons x m) :=
f.toMultilinearMap.cons_smul m c x
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
vecCons_smul
|
In the specific case of continuous alternating maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `Π(i : Fin (n+1)), M i` using `cons`, one can express directly the
multiplicativity of an alternating map along the first variable.
|
map_piecewise_add [DecidableEq ι] (m m' : ι → M) (t : Finset ι) :
f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m') :=
f.toMultilinearMap.map_piecewise_add _ _ _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_piecewise_add
| null |
map_add_univ [DecidableEq ι] [Fintype ι] (m m' : ι → M) :
f (m + m') = ∑ s : Finset ι, f (s.piecewise m m') :=
f.toMultilinearMap.map_add_univ _ _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_add_univ
|
Additivity of a continuous alternating map along all coordinates at the same time,
writing `f (m + m')` as the sum of `f (s.piecewise m m')` over all sets `s`.
|
map_sum_finset :
(f fun i => ∑ j ∈ A i, g' i j) = ∑ r ∈ piFinset A, f fun i => g' i (r i) :=
f.toMultilinearMap.map_sum_finset _ _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_sum_finset
|
If `f` is continuous alternating, then `f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ)` is the
sum of `f (g₁ (r 1), ..., gₙ (r n))` where `r` ranges over all functions with `r 1 ∈ A₁`, ...,
`r n ∈ Aₙ`. This follows from multilinearity by expanding successively with respect to each
coordinate.
|
map_sum [∀ i, Fintype (α i)] :
(f fun i => ∑ j, g' i j) = ∑ r : ∀ i, α i, f fun i => g' i (r i) :=
f.toMultilinearMap.map_sum _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_sum
|
If `f` is continuous alternating, then `f (Σ_{j₁} g₁ j₁, ..., Σ_{jₙ} gₙ jₙ)` is the sum of
`f (g₁ (r 1), ..., gₙ (r n))` where `r` ranges over all functions `r`. This follows from
multilinearity by expanding successively with respect to each coordinate.
|
restrictScalars (f : M [⋀^ι]→L[A] N) : M [⋀^ι]→L[R] N :=
{ f with toContinuousMultilinearMap := f.1.restrictScalars R }
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
restrictScalars
|
Reinterpret a continuous `A`-alternating map as a continuous `R`-alternating map, if `A` is an
algebra over `R` and their actions on all involved modules agree with the action of `R` on `A`.
|
coe_restrictScalars (f : M [⋀^ι]→L[A] N) : ⇑(f.restrictScalars R) = f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
coe_restrictScalars
| null |
@[simp]
map_update_sub [DecidableEq ι] (m : ι → M) (i : ι) (x y : M) :
f (update m i (x - y)) = f (update m i x) - f (update m i y) :=
f.toMultilinearMap.map_update_sub _ _ _ _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_update_sub
| null |
@[simp]
coe_neg : ⇑(-f) = -f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
coe_neg
| null |
neg_apply (m : ι → M) : (-f) m = -f m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
neg_apply
| null |
@[simp] coe_sub : ⇑(f - g) = ⇑f - ⇑g := rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
coe_sub
| null |
sub_apply (m : ι → M) : (f - g) m = f m - g m := rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
sub_apply
| null |
map_piecewise_smul [DecidableEq ι] (c : ι → R) (m : ι → M) (s : Finset ι) :
f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m :=
f.toMultilinearMap.map_piecewise_smul _ _ _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_piecewise_smul
| null |
map_smul_univ [Fintype ι] (c : ι → R) (m : ι → M) :
(f fun i => c i • m i) = (∏ i, c i) • f m :=
f.toMultilinearMap.map_smul_univ _ _
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
map_smul_univ
|
Multiplicativity of a continuous alternating map along all coordinates at the same time,
writing `f (fun i ↦ c i • m i)` as `(∏ i, c i) • f m`.
|
@[ext]
ext_ring [Finite ι] [TopologicalSpace R] ⦃f g : R [⋀^ι]→L[R] M⦄
(h : f (fun _ ↦ 1) = g (fun _ ↦ 1)) : f = g :=
toAlternatingMap_injective <| AlternatingMap.ext_ring h
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
ext_ring
|
If two continuous `R`-alternating maps from `R` are equal on 1, then they are equal.
This is the alternating version of `ContinuousLinearMap.ext_ring`.
|
uniqueOfCommRing [Finite ι] [Nontrivial ι] [TopologicalSpace R] :
Unique (R [⋀^ι]→L[R] N) where
uniq _ := toAlternatingMap_injective <| Subsingleton.elim _ _
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
uniqueOfCommRing
|
The only continuous `R`-alternating map from two or more copies of `R` is the zero map.
|
@[simps]
toContinuousMultilinearMapLinear :
M [⋀^ι]→L[A] N →ₗ[R] ContinuousMultilinearMap A (fun _ : ι => M) N where
toFun := toContinuousMultilinearMap
map_add' _ _ := rfl
map_smul' _ _ := rfl
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
toContinuousMultilinearMapLinear
|
The space of continuous alternating maps over an algebra over `R` is a module over `R`, for the
pointwise addition and scalar multiplication. -/
instance : Module R (M [⋀^ι]→L[A] N) :=
Function.Injective.module _ toMultilinearAddHom toContinuousMultilinearMap_injective fun _ _ =>
rfl
/-- Linear map version of the map `toMultilinearMap` associating to a continuous alternating map
the corresponding multilinear map.
|
@[simps -fullyApplied apply]
toAlternatingMapLinear : (M [⋀^ι]→L[A] N) →ₗ[R] (M [⋀^ι]→ₗ[A] N) where
toFun := toAlternatingMap
map_add' := by simp
map_smul' := by simp
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
toAlternatingMapLinear
|
Linear map version of the map `toAlternatingMap`
associating to a continuous alternating map the corresponding alternating map.
|
@[simps +simpRhs]
piLinearEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
[∀ i, TopologicalSpace (M' i)] [∀ i, ContinuousAdd (M' i)] [∀ i, Module R (M' i)]
[∀ i, Module A (M' i)] [∀ i, SMulCommClass A R (M' i)] [∀ i, ContinuousConstSMul R (M' i)] :
(∀ i, M [⋀^ι]→L[A] M' i) ≃ₗ[R] M [⋀^ι]→L[A] ∀ i, M' i :=
{ piEquiv with
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl }
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
piLinearEquiv
|
`ContinuousAlternatingMap.pi` as a `LinearEquiv`.
|
@[simps! toContinuousMultilinearMap apply]
smulRight : M [⋀^ι]→L[R] N :=
{ f.toAlternatingMap.smulRight z with toContinuousMultilinearMap := f.1.smulRight z }
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
smulRight
|
Given a continuous `R`-alternating map `f` taking values in `R`, `f.smulRight z` is the
continuous alternating map sending `m` to `f m • z`.
|
@[simps]
compContinuousLinearMapₗ (f : M →L[R] M') : (M' [⋀^ι]→L[R] N) →ₗ[R] (M [⋀^ι]→L[R] N) where
toFun g := g.compContinuousLinearMap f
map_add' g g' := by ext; simp
map_smul' c g := by ext; simp
variable (R M N N')
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
compContinuousLinearMapₗ
|
`ContinuousAlternatingMap.compContinuousLinearMap` as a bundled `LinearMap`.
|
_root_.ContinuousLinearMap.compContinuousAlternatingMapₗ :
(N →L[R] N') →ₗ[R] (M [⋀^ι]→L[R] N) →ₗ[R] (M [⋀^ι]→L[R] N') :=
LinearMap.mk₂ R ContinuousLinearMap.compContinuousAlternatingMap (fun _ _ _ => rfl)
(fun _ _ _ => rfl) (fun f g₁ g₂ => by ext1; apply f.map_add) fun c f g => by ext1; simp
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
_root_.ContinuousLinearMap.compContinuousAlternatingMapₗ
|
`ContinuousLinearMap.compContinuousAlternatingMap` as a bundled bilinear map.
|
@[simps -isSimp apply_toContinuousMultilinearMap]
alternatization : ContinuousMultilinearMap R (fun _ : ι => M) N →+ M [⋀^ι]→L[R] N where
toFun f :=
{ toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • f.domDomCongr σ
map_eq_zero_of_eq' := fun v i j hv hne => by
simpa [MultilinearMap.alternatization_apply]
using f.1.alternatization.map_eq_zero_of_eq' v i j hv hne }
map_zero' := by ext; simp
map_add' _ _ := by ext; simp [Finset.sum_add_distrib]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
alternatization
|
Alternatization of a continuous multilinear map.
|
alternatization_apply_apply (v : ι → M) :
alternatization f v = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • f (v ∘ σ) := by
simp [alternatization, Function.comp_def]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
alternatization_apply_apply
| null |
alternatization_apply_toAlternatingMap :
(alternatization f).toAlternatingMap = MultilinearMap.alternatization f.1 := by
ext v
simp [alternatization_apply_apply, MultilinearMap.alternatization_apply, Function.comp_def]
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
|
alternatization_apply_toAlternatingMap
| null |
instTopologicalSpace : TopologicalSpace (E [⋀^ι]→L[𝕜] F) :=
.induced toContinuousMultilinearMap inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instTopologicalSpace
| null |
isClosed_range_toContinuousMultilinearMap [ContinuousSMul 𝕜 E] [T2Space F] :
IsClosed (Set.range (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F) →
ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) F)) := by
simp only [range_toContinuousMultilinearMap, setOf_forall]
repeat refine isClosed_iInter fun _ ↦ ?_
exact isClosed_singleton.preimage (continuous_eval_const _)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isClosed_range_toContinuousMultilinearMap
| null |
instUniformSpace : UniformSpace (E [⋀^ι]→L[𝕜] F) :=
.comap toContinuousMultilinearMap inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instUniformSpace
| null |
isUniformEmbedding_toContinuousMultilinearMap :
IsUniformEmbedding (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F) → _) where
injective := toContinuousMultilinearMap_injective
comap_uniformity := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isUniformEmbedding_toContinuousMultilinearMap
| null |
uniformContinuous_toContinuousMultilinearMap :
UniformContinuous (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F) → _) :=
isUniformEmbedding_toContinuousMultilinearMap.uniformContinuous
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
uniformContinuous_toContinuousMultilinearMap
| null |
uniformContinuous_coe_fun [ContinuousSMul 𝕜 E] :
UniformContinuous (DFunLike.coe : (E [⋀^ι]→L[𝕜] F) → (ι → E) → F) :=
ContinuousMultilinearMap.uniformContinuous_coe_fun.comp
uniformContinuous_toContinuousMultilinearMap
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
uniformContinuous_coe_fun
| null |
uniformContinuous_eval_const [ContinuousSMul 𝕜 E] (x : ι → E) :
UniformContinuous fun f : E [⋀^ι]→L[𝕜] F ↦ f x :=
uniformContinuous_pi.1 uniformContinuous_coe_fun x
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
uniformContinuous_eval_const
| null |
instIsUniformAddGroup : IsUniformAddGroup (E [⋀^ι]→L[𝕜] F) :=
isUniformEmbedding_toContinuousMultilinearMap.isUniformAddGroup
(toContinuousMultilinearMapLinear (R := ℕ))
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instIsUniformAddGroup
| null |
instUniformContinuousConstSMul {M : Type*}
[Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
UniformContinuousConstSMul M (E [⋀^ι]→L[𝕜] F) :=
isUniformEmbedding_toContinuousMultilinearMap.uniformContinuousConstSMul fun _ _ ↦ rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instUniformContinuousConstSMul
| null |
isUniformInducing_postcomp {G : Type*} [AddCommGroup G] [UniformSpace G]
[IsUniformAddGroup G] [Module 𝕜 G] (g : F →L[𝕜] G) (hg : IsUniformInducing g) :
IsUniformInducing (g.compContinuousAlternatingMap : (E [⋀^ι]→L[𝕜] F) → (E [⋀^ι]→L[𝕜] G)) := by
rw [← isUniformEmbedding_toContinuousMultilinearMap.1.of_comp_iff]
exact (ContinuousMultilinearMap.isUniformInducing_postcomp g hg).comp
isUniformEmbedding_toContinuousMultilinearMap.1
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isUniformInducing_postcomp
| null |
completeSpace (h : IsCoherentWith {s : Set (ι → E) | IsVonNBounded 𝕜 s}) :
CompleteSpace (E [⋀^ι]→L[𝕜] F) := by
wlog hF : T2Space F generalizing F
· rw [(isUniformInducing_postcomp (SeparationQuotient.mkCLM _ _)
SeparationQuotient.isUniformInducing_mk).completeSpace_congr]
· exact this inferInstance
· intro f
use (SeparationQuotient.outCLM _ _).compContinuousAlternatingMap f
ext
simp
have := ContinuousMultilinearMap.completeSpace (F := F) h
rw [completeSpace_iff_isComplete_range
isUniformEmbedding_toContinuousMultilinearMap.isUniformInducing]
apply isClosed_range_toContinuousMultilinearMap.isComplete
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
completeSpace
| null |
instCompleteSpace [IsTopologicalAddGroup E] [SequentialSpace (ι → E)] :
CompleteSpace (E [⋀^ι]→L[𝕜] F) :=
completeSpace <| .of_seq fun _u x hux ↦ (hux.isVonNBounded_range 𝕜).insert x
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instCompleteSpace
| null |
isUniformEmbedding_restrictScalars :
IsUniformEmbedding (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) := by
rw [← isUniformEmbedding_toContinuousMultilinearMap.of_comp_iff]
exact (ContinuousMultilinearMap.isUniformEmbedding_restrictScalars 𝕜').comp
isUniformEmbedding_toContinuousMultilinearMap
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isUniformEmbedding_restrictScalars
| null |
uniformContinuous_restrictScalars :
UniformContinuous (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) :=
(isUniformEmbedding_restrictScalars 𝕜').uniformContinuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
uniformContinuous_restrictScalars
| null |
isEmbedding_toContinuousMultilinearMap :
IsEmbedding (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F → _)) :=
letI := IsTopologicalAddGroup.toUniformSpace F
haveI := isUniformAddGroup_of_addCommGroup (G := F)
isUniformEmbedding_toContinuousMultilinearMap.isEmbedding
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isEmbedding_toContinuousMultilinearMap
| null |
instIsTopologicalAddGroup : IsTopologicalAddGroup (E [⋀^ι]→L[𝕜] F) :=
isEmbedding_toContinuousMultilinearMap.topologicalAddGroup
(toContinuousMultilinearMapLinear (R := ℕ))
@[continuity, fun_prop]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instIsTopologicalAddGroup
| null |
continuous_toContinuousMultilinearMap :
Continuous (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F → _)) :=
isEmbedding_toContinuousMultilinearMap.continuous
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
continuous_toContinuousMultilinearMap
| null |
instContinuousConstSMul
{M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
ContinuousConstSMul M (E [⋀^ι]→L[𝕜] F) :=
isEmbedding_toContinuousMultilinearMap.continuousConstSMul id rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instContinuousConstSMul
| null |
instContinuousSMul [ContinuousSMul 𝕜 F] : ContinuousSMul 𝕜 (E [⋀^ι]→L[𝕜] F) :=
isEmbedding_toContinuousMultilinearMap.continuousSMul continuous_id rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instContinuousSMul
| null |
hasBasis_nhds_zero_of_basis {ι' : Type*} {p : ι' → Prop} {b : ι' → Set F}
(h : (𝓝 (0 : F)).HasBasis p b) :
(𝓝 (0 : E [⋀^ι]→L[𝕜] F)).HasBasis
(fun Si : Set (ι → E) × ι' => IsVonNBounded 𝕜 Si.1 ∧ p Si.2)
fun Si => { f | MapsTo f Si.1 (b Si.2) } := by
rw [nhds_induced]
exact (ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis h).comap _
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
hasBasis_nhds_zero_of_basis
| null |
hasBasis_nhds_zero :
(𝓝 (0 : E [⋀^ι]→L[𝕜] F)).HasBasis
(fun SV : Set (ι → E) × Set F => IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0)
fun SV => { f | MapsTo f SV.1 SV.2 } :=
hasBasis_nhds_zero_of_basis (Filter.basis_sets _)
variable [ContinuousSMul 𝕜 E]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
hasBasis_nhds_zero
| null |
isClosedEmbedding_toContinuousMultilinearMap [T2Space F] :
IsClosedEmbedding (toContinuousMultilinearMap :
(E [⋀^ι]→L[𝕜] F) → ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) F) :=
⟨isEmbedding_toContinuousMultilinearMap, isClosed_range_toContinuousMultilinearMap⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isClosedEmbedding_toContinuousMultilinearMap
| null |
instContinuousEvalConst : ContinuousEvalConst (E [⋀^ι]→L[𝕜] F) (ι → E) F :=
.of_continuous_forget continuous_toContinuousMultilinearMap
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instContinuousEvalConst
| null |
instT2Space [T2Space F] : T2Space (E [⋀^ι]→L[𝕜] F) :=
.of_injective_continuous DFunLike.coe_injective continuous_coeFun
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instT2Space
| null |
instT3Space [T2Space F] : T3Space (E [⋀^ι]→L[𝕜] F) :=
inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
instT3Space
| null |
@[simps! -fullyApplied]
toContinuousMultilinearMapCLM
(R : Type*) [Semiring R] [Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜 R F] :
E [⋀^ι]→L[𝕜] F →L[R] ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) F :=
⟨toContinuousMultilinearMapLinear, continuous_induced_dom⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
toContinuousMultilinearMapCLM
|
The inclusion of *alternating* continuous multi-linear maps into continuous multi-linear maps
as a continuous linear map.
|
isEmbedding_restrictScalars :
IsEmbedding (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) :=
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
(isUniformEmbedding_restrictScalars _).isEmbedding
@[continuity, fun_prop]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
isEmbedding_restrictScalars
| null |
continuous_restrictScalars :
Continuous (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) :=
isEmbedding_restrictScalars.continuous
variable (𝕜') in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
continuous_restrictScalars
| null |
@[simps -fullyApplied apply]
restrictScalarsCLM [ContinuousConstSMul 𝕜' F] :
E [⋀^ι]→L[𝕜] F →L[𝕜'] E [⋀^ι]→L[𝕜'] F where
toFun := restrictScalars 𝕜'
map_add' _ _ := rfl
map_smul' _ _ := rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
restrictScalarsCLM
|
`ContinuousMultilinearMap.restrictScalars` as a `ContinuousLinearMap`.
|
apply [ContinuousConstSMul 𝕜 F] (m : ι → E) : E [⋀^ι]→L[𝕜] F →L[𝕜] F where
toFun c := c m
map_add' _ _ := rfl
map_smul' _ _ := rfl
cont := continuous_eval_const m
variable {𝕜 E F}
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
apply
|
The application of a multilinear map as a `ContinuousLinearMap`.
|
apply_apply [ContinuousConstSMul 𝕜 F] {m : ι → E} {c : E [⋀^ι]→L[𝕜] F} :
apply 𝕜 E F m c = c m := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
apply_apply
| null |
hasSum_eval {α : Type*} {p : α → E [⋀^ι]→L[𝕜] F}
{q : E [⋀^ι]→L[𝕜] F} (h : HasSum p q) (m : ι → E) :
HasSum (fun a => p a m) (q m) :=
h.map (applyAddHom m) (continuous_eval_const m)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
hasSum_eval
| null |
tsum_eval [T2Space F] {α : Type*} {p : α → E [⋀^ι]→L[𝕜] F} (hp : Summable p)
(m : ι → E) : (∑' a, p a) m = ∑' a, p a m :=
(hasSum_eval hp.hasSum m).tsum_eq.symm
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Topology",
"Mathlib.Topology.Algebra.Module.Alternating.Basic"
] |
Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
|
tsum_eval
| null |
ContinuousMultilinearMap (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) (M₂ : Type w₂)
[Semiring R] [∀ i, AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [∀ i, Module R (M₁ i)] [Module R M₂]
[∀ i, TopologicalSpace (M₁ i)] [TopologicalSpace M₂] extends MultilinearMap R M₁ M₂ where
cont : Continuous toFun
attribute [inherit_doc ContinuousMultilinearMap] ContinuousMultilinearMap.cont
@[inherit_doc]
notation:25 M " [×" n "]→L[" R "] " M' => ContinuousMultilinearMap R (fun i : Fin n => M) M'
|
structure
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
ContinuousMultilinearMap
|
Continuous multilinear maps over the ring `R`, from `∀ i, M₁ i` to `M₂` where `M₁ i` and `M₂`
are modules over `R` with a topological structure. In applications, there will be compatibility
conditions between the algebraic and the topological structures, but this is not needed for the
definition.
|
toMultilinearMap_injective :
Function.Injective
(ContinuousMultilinearMap.toMultilinearMap :
ContinuousMultilinearMap R M₁ M₂ → MultilinearMap R M₁ M₂)
| ⟨f, hf⟩, ⟨g, hg⟩, h => by subst h; rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
toMultilinearMap_injective
| null |
funLike : FunLike (ContinuousMultilinearMap R M₁ M₂) (∀ i, M₁ i) M₂ where
coe f := f.toFun
coe_injective' _ _ h := toMultilinearMap_injective <| MultilinearMap.coe_injective h
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
funLike
| null |
continuousMapClass :
ContinuousMapClass (ContinuousMultilinearMap R M₁ M₂) (∀ i, M₁ i) M₂ where
map_continuous := ContinuousMultilinearMap.cont
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
continuousMapClass
| null |
Simps.apply (L₁ : ContinuousMultilinearMap R M₁ M₂) (v : ∀ i, M₁ i) : M₂ :=
L₁ v
initialize_simps_projections ContinuousMultilinearMap (-toMultilinearMap,
toMultilinearMap_toFun → apply)
@[continuity]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.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.
|
coe_continuous : Continuous (f : (∀ i, M₁ i) → M₂) :=
f.cont
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
coe_continuous
| null |
coe_coe : (f.toMultilinearMap : (∀ i, M₁ i) → M₂) = f :=
rfl
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
coe_coe
| null |
ext {f f' : ContinuousMultilinearMap R M₁ M₂} (H : ∀ x, f x = f' x) : f = f' :=
DFunLike.ext _ _ H
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
ext
| null |
map_update_add [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i) :
f (update m i (x + y)) = f (update m i x) + f (update m i y) :=
f.map_update_add' m i x y
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_update_add
| null |
map_update_smul [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (c : R) (x : M₁ i) :
f (update m i (c • x)) = c • f (update m i x) :=
f.map_update_smul' m i c x
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_update_smul
| null |
map_coord_zero {m : ∀ i, M₁ i} (i : ι) (h : m i = 0) : f m = 0 :=
f.toMultilinearMap.map_coord_zero i h
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_coord_zero
| null |
map_zero [Nonempty ι] : f 0 = 0 :=
f.toMultilinearMap.map_zero
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_zero
| null |
@[simp]
zero_apply (m : ∀ i, M₁ i) : (0 : ContinuousMultilinearMap R M₁ M₂) m = 0 :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
zero_apply
| null |
toMultilinearMap_zero : (0 : ContinuousMultilinearMap R M₁ M₂).toMultilinearMap = 0 :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
toMultilinearMap_zero
| null |
@[simp]
smul_apply (f : ContinuousMultilinearMap A M₁ M₂) (c : R') (m : ∀ i, M₁ i) :
(c • f) m = c • f m :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
smul_apply
| null |
toMultilinearMap_smul (c : R') (f : ContinuousMultilinearMap A M₁ M₂) :
(c • f).toMultilinearMap = c • f.toMultilinearMap :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
toMultilinearMap_smul
| null |
@[simp]
add_apply (m : ∀ i, M₁ i) : (f + f') m = f m + f' m :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
add_apply
| null |
toMultilinearMap_add (f g : ContinuousMultilinearMap R M₁ M₂) :
(f + g).toMultilinearMap = f.toMultilinearMap + g.toMultilinearMap :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
toMultilinearMap_add
| null |
addCommMonoid : AddCommMonoid (ContinuousMultilinearMap R M₁ M₂) :=
toMultilinearMap_injective.addCommMonoid _ rfl (fun _ _ => rfl) fun _ _ => rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
addCommMonoid
| null |
applyAddHom (m : ∀ i, M₁ i) : ContinuousMultilinearMap R M₁ M₂ →+ M₂ where
toFun f := f m
map_zero' := rfl
map_add' _ _ := rfl
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
applyAddHom
|
Evaluation of a `ContinuousMultilinearMap` at a vector as an `AddMonoidHom`.
|
sum_apply {α : Type*} (f : α → ContinuousMultilinearMap R M₁ M₂) (m : ∀ i, M₁ i)
{s : Finset α} : (∑ a ∈ s, f a) m = ∑ a ∈ s, f a m :=
map_sum (applyAddHom m) f s
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
sum_apply
| null |
@[simps!] toContinuousLinearMap [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) : M₁ i →L[R] M₂ :=
{ f.toMultilinearMap.toLinearMap m i with
cont := f.cont.comp (continuous_const.update i continuous_id) }
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
toContinuousLinearMap
|
If `f` is a continuous multilinear map, then `f.toContinuousLinearMap m i` is the continuous
linear map obtained by fixing all coordinates but `i` equal to those of `m`, and varying the
`i`-th coordinate.
|
prod (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃) :
ContinuousMultilinearMap R M₁ (M₂ × M₃) :=
{ f.toMultilinearMap.prod g.toMultilinearMap with cont := f.cont.prodMk g.cont }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
prod
|
The Cartesian product of two continuous multilinear maps, as a continuous multilinear map.
|
prod_apply (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃)
(m : ∀ i, M₁ i) : (f.prod g) m = (f m, g m) :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
prod_apply
| null |
pi {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)] [∀ i, TopologicalSpace (M' i)]
[∀ i, Module R (M' i)] (f : ∀ i, ContinuousMultilinearMap R M₁ (M' i)) :
ContinuousMultilinearMap R M₁ (∀ i, M' i) where
cont := continuous_pi fun i => (f i).coe_continuous
toMultilinearMap := MultilinearMap.pi fun i => (f i).toMultilinearMap
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
pi
|
Combine a family of continuous multilinear maps with the same domain and codomains `M' i` into a
continuous multilinear map taking values in the space of functions `∀ i, M' i`.
|
coe_pi {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
[∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)]
(f : ∀ i, ContinuousMultilinearMap R M₁ (M' i)) : ⇑(pi f) = fun m j => f j m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
coe_pi
| null |
pi_apply {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
[∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)]
(f : ∀ i, ContinuousMultilinearMap R M₁ (M' i)) (m : ∀ i, M₁ i) (j : ι') : pi f m j = f j m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
pi_apply
| null |
@[simps! toMultilinearMap apply_coe]
codRestrict (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ v, f v ∈ p) :
ContinuousMultilinearMap R M₁ p :=
⟨f.1.codRestrict p h, f.cont.subtype_mk _⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
codRestrict
|
Restrict the codomain of a continuous multilinear map to a submodule.
|
@[simps! apply_toMultilinearMap apply_apply symm_apply_apply]
ofSubsingleton [Subsingleton ι] (i : ι) :
(M₂ →L[R] M₃) ≃ ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ where
toFun f := ⟨MultilinearMap.ofSubsingleton R M₂ M₃ i f,
(map_continuous f).comp (continuous_apply i)⟩
invFun f := ⟨(MultilinearMap.ofSubsingleton R M₂ M₃ i).symm f.toMultilinearMap,
(map_continuous f).comp <| continuous_pi fun _ ↦ continuous_id⟩
right_inv f := toMultilinearMap_injective <|
(MultilinearMap.ofSubsingleton R M₂ M₃ i).apply_symm_apply f.toMultilinearMap
variable (M₁) {M₂}
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
ofSubsingleton
|
The natural equivalence between continuous linear maps from `M₂` to `M₃`
and continuous 1-multilinear maps from `M₂` to `M₃`.
|
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