fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
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| symbolic_name
stringlengths 1
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| docstring
stringlengths 7
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|---|---|---|---|---|---|---|
@[simps! toMultilinearMap apply]
constOfIsEmpty [IsEmpty ι] (m : M₂) : ContinuousMultilinearMap R M₁ M₂ where
toMultilinearMap := MultilinearMap.constOfIsEmpty R _ m
cont := continuous_const
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
constOfIsEmpty
|
The constant map is multilinear when `ι` is empty.
|
compContinuousLinearMap (g : ContinuousMultilinearMap R M₁' M₄)
(f : ∀ i : ι, M₁ i →L[R] M₁' i) : ContinuousMultilinearMap R M₁ M₄ :=
{ g.toMultilinearMap.compLinearMap fun i => (f i).toLinearMap with
cont := g.cont.comp <| continuous_pi fun j => (f j).cont.comp <| continuous_apply _ }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
compContinuousLinearMap
|
If `g` is continuous multilinear and `f` is a collection of continuous linear maps,
then `g (f₁ m₁, ..., fₙ mₙ)` is again a continuous multilinear map, that we call
`g.compContinuousLinearMap f`.
|
compContinuousLinearMap_apply (g : ContinuousMultilinearMap R M₁' M₄)
(f : ∀ i : ι, M₁ i →L[R] M₁' i) (m : ∀ i, M₁ i) :
g.compContinuousLinearMap f m = g fun i => f i <| m i :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
compContinuousLinearMap_apply
| null |
_root_.ContinuousLinearMap.compContinuousMultilinearMap (g : M₂ →L[R] M₃)
(f : ContinuousMultilinearMap R M₁ M₂) : ContinuousMultilinearMap R M₁ M₃ :=
{ g.toLinearMap.compMultilinearMap f.toMultilinearMap with cont := g.cont.comp f.cont }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
_root_.ContinuousLinearMap.compContinuousMultilinearMap
|
Composing a continuous multilinear map with a continuous linear map gives again a
continuous multilinear map.
|
_root_.ContinuousLinearMap.compContinuousMultilinearMap_coe (g : M₂ →L[R] M₃)
(f : ContinuousMultilinearMap R M₁ M₂) :
(g.compContinuousMultilinearMap f : (∀ i, M₁ i) → M₃) =
(g : M₂ → M₃) ∘ (f : (∀ i, M₁ i) → M₂) := by
ext m
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
_root_.ContinuousLinearMap.compContinuousMultilinearMap_coe
| null |
@[simps apply symm_apply_fst symm_apply_snd, simps -isSimp symm_apply]
prodEquiv :
(ContinuousMultilinearMap R M₁ M₂ × ContinuousMultilinearMap R M₁ M₃) ≃
ContinuousMultilinearMap R M₁ (M₂ × M₃) where
toFun f := f.1.prod f.2
invFun f := ((ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap f,
(ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap f)
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
prodEquiv
|
`ContinuousMultilinearMap.prod` as an `Equiv`.
|
prod_ext_iff {f g : ContinuousMultilinearMap R M₁ (M₂ × M₃)} :
f = g ↔ (ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap f =
(ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap g ∧
(ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap f =
(ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap g := by
rw [← Prod.mk_inj, ← prodEquiv_symm_apply, ← prodEquiv_symm_apply, Equiv.apply_eq_iff_eq]
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
prod_ext_iff
| null |
prod_ext {f g : ContinuousMultilinearMap R M₁ (M₂ × M₃)}
(h₁ : (ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap f =
(ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap g)
(h₂ : (ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap f =
(ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap g) : f = g :=
prod_ext_iff.mpr ⟨h₁, h₂⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
prod_ext
| null |
eq_prod_iff {f : ContinuousMultilinearMap R M₁ (M₂ × M₃)}
{g : ContinuousMultilinearMap R M₁ M₂} {h : ContinuousMultilinearMap R M₁ M₃} :
f = g.prod h ↔ (ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap f = g ∧
(ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap f = h :=
prod_ext_iff
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
eq_prod_iff
| null |
add_prod_add [ContinuousAdd M₂] [ContinuousAdd M₃]
(f₁ f₂ : ContinuousMultilinearMap R M₁ M₂) (g₁ g₂ : ContinuousMultilinearMap R M₁ M₃) :
(f₁ + f₂).prod (g₁ + g₂) = f₁.prod g₁ + f₂.prod g₂ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
add_prod_add
| null |
smul_prod_smul {S : Type*} [Monoid S] [DistribMulAction S M₂] [DistribMulAction S M₃]
[ContinuousConstSMul S M₂] [SMulCommClass R S M₂]
[ContinuousConstSMul S M₃] [SMulCommClass R S M₃]
(c : S) (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃) :
(c • f).prod (c • g) = c • f.prod g :=
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_prod_smul
| null |
zero_prod_zero :
(0 : ContinuousMultilinearMap R M₁ M₂).prod (0 : ContinuousMultilinearMap R M₁ M₃) = 0 :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
zero_prod_zero
| null |
@[simps]
piEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
[∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] :
(∀ i, ContinuousMultilinearMap R M₁ (M' i)) ≃ ContinuousMultilinearMap R M₁ (∀ i, M' i) where
toFun := ContinuousMultilinearMap.pi
invFun f i := (ContinuousLinearMap.proj i : _ →L[R] M' i).compContinuousMultilinearMap f
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
piEquiv
|
`ContinuousMultilinearMap.pi` as an `Equiv`.
|
@[simps]
domDomCongrEquiv {ι' : Type*} (e : ι ≃ ι') :
ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ ≃
ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃ where
toFun := domDomCongr e
invFun := domDomCongr e.symm
left_inv _ := ext fun _ => by simp
right_inv _ := ext fun _ => by simp
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
domDomCongrEquiv
|
An equivalence of the index set defines an equivalence between the spaces of continuous
multilinear maps. This is the forward map of this equivalence. -/
@[simps! toMultilinearMap apply]
nonrec def domDomCongr {ι' : Type*} (e : ι ≃ ι')
(f : ContinuousMultilinearMap R (fun _ : ι => M₂) M₃) :
ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃ where
toMultilinearMap := f.domDomCongr e
cont := f.cont.comp <| continuous_pi fun _ => continuous_apply _
/-- An equivalence of the index set defines an equivalence between the spaces of continuous
multilinear maps. In case of normed spaces, this is a linear isometric equivalence, see
`ContinuousMultilinearMap.domDomCongrₗᵢ`.
|
linearDeriv : (∀ i, M₁ i) →L[R] M₂ := ∑ i : ι, (f.toContinuousLinearMap x i).comp (.proj i)
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
linearDeriv
|
The derivative of a continuous multilinear map, as a continuous linear map
from `∀ i, M₁ i` to `M₂`; see `ContinuousMultilinearMap.hasFDerivAt`.
|
linearDeriv_apply : f.linearDeriv x y = ∑ i, f (Function.update x i (y i)) := by
unfold linearDeriv toContinuousLinearMap
simp only [ContinuousLinearMap.coe_sum', ContinuousLinearMap.coe_comp',
ContinuousLinearMap.coe_mk', Finset.sum_apply]
rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
linearDeriv_apply
| null |
cons_add (f : ContinuousMultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (x y : M 0) :
f (cons (x + y) m) = f (cons x m) + f (cons y m) :=
f.toMultilinearMap.cons_add m x y
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
cons_add
|
In the specific case of continuous multilinear maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `(i : Fin (n+1)) → M i` using `cons`, one can express directly the
additivity of a multilinear map along the first variable.
|
cons_smul (f : ContinuousMultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (c : R)
(x : M 0) : f (cons (c • x) m) = c • f (cons x m) :=
f.toMultilinearMap.cons_smul m c x
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
cons_smul
|
In the specific case of continuous multilinear maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `(i : Fin (n+1)) → M i` using `cons`, one can express directly the
multiplicativity of a multilinear map along the first variable.
|
map_piecewise_add [DecidableEq ι] (m m' : ∀ i, M₁ i) (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.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_piecewise_add
| null |
map_add_univ [DecidableEq ι] [Fintype ι] (m m' : ∀ i, M₁ i) :
f (m + m') = ∑ s : Finset ι, f (s.piecewise m m') :=
f.toMultilinearMap.map_add_univ _ _
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_add_univ
|
Additivity of a continuous multilinear 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 [DecidableEq ι] :
(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.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_sum_finset
|
If `f` is continuous multilinear, 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 [DecidableEq ι] [∀ 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.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_sum
|
If `f` is continuous multilinear, 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 : ContinuousMultilinearMap A M₁ M₂) : ContinuousMultilinearMap R M₁ M₂ where
toMultilinearMap := f.toMultilinearMap.restrictScalars R
cont := f.cont
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
restrictScalars
|
Reinterpret an `A`-multilinear map as an `R`-multilinear 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 : ContinuousMultilinearMap A M₁ M₂) : ⇑(f.restrictScalars R) = f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
coe_restrictScalars
| null |
@[simp]
map_update_sub [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.toMultilinearMap.map_update_sub _ _ _ _
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_update_sub
| null |
@[simp]
neg_apply (m : ∀ i, M₁ i) : (-f) m = -f m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
neg_apply
| null |
@[simp]
sub_apply (m : ∀ i, M₁ i) : (f - f') m = f m - f' m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
sub_apply
| null |
neg_prod_neg [AddCommGroup M₃] [Module R M₃] [TopologicalSpace M₃]
[IsTopologicalAddGroup M₃] (f : ContinuousMultilinearMap R M₁ M₂)
(g : ContinuousMultilinearMap R M₁ M₃) : (-f).prod (-g) = - f.prod g :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
neg_prod_neg
| null |
sub_prod_sub [AddCommGroup M₃] [Module R M₃] [TopologicalSpace M₃]
[IsTopologicalAddGroup M₃] (f₁ f₂ : ContinuousMultilinearMap R M₁ M₂)
(g₁ g₂ : ContinuousMultilinearMap R M₁ M₃) :
(f₁ - f₂).prod (g₁ - g₂) = f₁.prod g₁ - f₂.prod g₂ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
sub_prod_sub
| null |
map_piecewise_smul [DecidableEq ι] (c : ι → R) (m : ∀ i, M₁ i) (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.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_piecewise_smul
| null |
map_smul_univ [Fintype ι] (c : ι → R) (m : ∀ i, M₁ i) :
(f fun i => c i • m i) = (∏ i, c i) • f m :=
f.toMultilinearMap.map_smul_univ _ _
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
map_smul_univ
|
Multiplicativity of a continuous multilinear 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 : ContinuousMultilinearMap R (fun _ : ι => R) M₂⦄
(h : f (fun _ ↦ 1) = g (fun _ ↦ 1)) : f = g :=
toMultilinearMap_injective <| MultilinearMap.ext_ring h
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
ext_ring
|
If two continuous `R`-multilinear maps from `R` are equal on 1, then they are equal.
This is the multilinear version of `ContinuousLinearMap.ext_ring`.
|
@[simps]
toMultilinearMapLinear : ContinuousMultilinearMap A M₁ M₂ →ₗ[R'] MultilinearMap A M₁ M₂ where
toFun := toMultilinearMap
map_add' := toMultilinearMap_add
map_smul' := toMultilinearMap_smul
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
toMultilinearMapLinear
|
The space of continuous multilinear maps over an algebra over `R` is a module over `R`, for the
pointwise addition and scalar multiplication. -/
instance : Module R' (ContinuousMultilinearMap A M₁ M₂) :=
Function.Injective.module _
{ toFun := toMultilinearMap,
map_zero' := toMultilinearMap_zero,
map_add' := toMultilinearMap_add }
toMultilinearMap_injective fun _ _ => rfl
/-- Linear map version of the map `toMultilinearMap` associating to a continuous multilinear map
the corresponding multilinear 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, ContinuousMultilinearMap A M₁ (M' i)) ≃ₗ[R'] ContinuousMultilinearMap A M₁ (∀ i, M' i) :=
{ piEquiv with
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl }
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
piLinearEquiv
|
`ContinuousMultilinearMap.pi` as a `LinearEquiv`.
|
protected mkPiAlgebraFin : A[×n]→L[R] A where
cont := by
change Continuous fun m => (List.ofFn m).prod
simp_rw [List.ofFn_eq_map]
exact continuous_list_prod _ fun i _ => continuous_apply _
toMultilinearMap := MultilinearMap.mkPiAlgebraFin R n A
variable {R n A}
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiAlgebraFin
|
The continuous multilinear map on `A^n`, where `A` is a normed algebra over `𝕜`, associating to
`m` the product of all the `m i`.
See also: `ContinuousMultilinearMap.mkPiAlgebra`.
|
mkPiAlgebraFin_apply (m : Fin n → A) :
ContinuousMultilinearMap.mkPiAlgebraFin R n A m = (List.ofFn m).prod :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiAlgebraFin_apply
| null |
protected mkPiAlgebra : ContinuousMultilinearMap R (fun _ : ι => A) A where
cont := continuous_finset_prod _ fun _ _ => continuous_apply _
toMultilinearMap := MultilinearMap.mkPiAlgebra R ι A
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiAlgebra
|
The continuous multilinear map on `A^ι`, where `A` is a normed commutative algebra
over `𝕜`, associating to `m` the product of all the `m i`.
See also `ContinuousMultilinearMap.mkPiAlgebraFin`.
|
mkPiAlgebra_apply (m : ι → A) : ContinuousMultilinearMap.mkPiAlgebra R ι A m = ∏ i, m i :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiAlgebra_apply
| null |
mkPiAlgebra_eq_mkPiAlgebraFin {n : ℕ} : ContinuousMultilinearMap.mkPiAlgebra R (Fin n) A
= ContinuousMultilinearMap.mkPiAlgebraFin R n A := by
ext
simp [List.prod_ofFn]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiAlgebra_eq_mkPiAlgebraFin
| null |
@[simps! toMultilinearMap apply]
smulRight : ContinuousMultilinearMap R M₁ M₂ where
toMultilinearMap := f.toMultilinearMap.smulRight z
cont := f.cont.smul continuous_const
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
smulRight
|
Given a continuous `R`-multilinear map `f` taking values in `R`, `f.smulRight z` is the
continuous multilinear map sending `m` to `f m • z`.
|
protected mkPiRing (z : M) : ContinuousMultilinearMap R (fun _ : ι => R) M :=
(ContinuousMultilinearMap.mkPiAlgebra R ι R).smulRight z
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiRing
|
The canonical continuous multilinear map on `R^ι`, associating to `m` the product of all the
`m i` (multiplied by a fixed reference element `z` in the target module)
|
mkPiRing_apply (z : M) (m : ι → R) :
(ContinuousMultilinearMap.mkPiRing R ι z : (ι → R) → M) m = (∏ i, m i) • z :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiRing_apply
| null |
mkPiRing_apply_one_eq_self (f : ContinuousMultilinearMap R (fun _ : ι => R) M) :
ContinuousMultilinearMap.mkPiRing R ι (f fun _ => 1) = f :=
toMultilinearMap_injective f.toMultilinearMap.mkPiRing_apply_one_eq_self
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiRing_apply_one_eq_self
| null |
mkPiRing_eq_iff {z₁ z₂ : M} :
ContinuousMultilinearMap.mkPiRing R ι z₁ = ContinuousMultilinearMap.mkPiRing R ι z₂ ↔
z₁ = z₂ := by
rw [← toMultilinearMap_injective.eq_iff]
exact MultilinearMap.mkPiRing_eq_iff
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiRing_eq_iff
| null |
mkPiRing_zero : ContinuousMultilinearMap.mkPiRing R ι (0 : M) = 0 := by
ext; rw [mkPiRing_apply, smul_zero, ContinuousMultilinearMap.zero_apply]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiRing_zero
| null |
mkPiRing_eq_zero_iff (z : M) : ContinuousMultilinearMap.mkPiRing R ι z = 0 ↔ z = 0 := by
rw [← mkPiRing_zero, mkPiRing_eq_iff]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.Algebra.BigOperators.Fin"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
|
mkPiRing_eq_zero_iff
| null |
image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by
classical
if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then
exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩
else
let _ : NontriviallyNormedField 𝕜 := ⟨by simpa using h₁⟩
obtain ⟨I, t, ht₀, hft⟩ :
∃ (I : Finset ι) (t : ∀ i, Set (E i)), (∀ i, t i ∈ 𝓝 0) ∧ Set.pi I t ⊆ f ⁻¹' V := by
have hfV : f ⁻¹' V ∈ 𝓝 0 := (map_continuous f).tendsto' _ _ f.map_zero hV
rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV
have : ∀ i, ∃ c : 𝕜, c ≠ 0 ∧ ∀ c' : 𝕜, ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i := fun i ↦ by
rw [isVonNBounded_pi_iff] at hs
have := (hs i).tendsto_smallSets_nhds.eventually (mem_lift' (ht₀ i))
rcases NormedAddCommGroup.nhds_zero_basis_norm_lt.eventually_iff.1 this with ⟨r, hr₀, hr⟩
rcases NormedField.exists_norm_lt 𝕜 hr₀ with ⟨c, hc₀, hc⟩
refine ⟨c, norm_pos_iff.1 hc₀, fun c' hle x hx ↦ ?_⟩
exact hr (hle.trans_lt hc) ⟨_, ⟨x, hx, rfl⟩, rfl⟩
choose c hc₀ hc using this
rw [absorbs_iff_eventually_nhds_zero (mem_of_mem_nhds hV),
NormedAddCommGroup.nhds_zero_basis_norm_lt.eventually_iff]
have hc₀' : ∏ i ∈ I, c i ≠ 0 := Finset.prod_ne_zero_iff.2 fun i _ ↦ hc₀ i
refine ⟨‖∏ i ∈ I, c i‖, norm_pos_iff.2 hc₀', fun a ha ↦ mapsTo_image_iff.2 fun x hx ↦ ?_⟩
let ⟨i₀⟩ := ‹Nonempty ι›
set y := I.piecewise (fun i ↦ c i • x i) x
calc
f (update y i₀ ((a / ∏ i ∈ I, c i) • y i₀)) ∈ V := hft fun i hi => by
rcases eq_or_ne i i₀ with rfl | hne
· simp_rw [update_self, y, I.piecewise_eq_of_mem _ _ hi, smul_smul]
refine hc _ _ ?_ _ hx
calc
‖(a / ∏ i ∈ I, c i) * c i‖ ≤ (‖∏ i ∈ I, c i‖ / ‖∏ i ∈ I, c i‖) * ‖c i‖ := by
rw [norm_mul, norm_div]; gcongr; exact ha.out.le
_ ≤ 1 * ‖c i‖ := by gcongr; apply div_self_le_one
_ = ‖c i‖ := one_mul _
· simp_rw [update_of_ne hne, y, I.piecewise_eq_of_mem _ _ hi]
exact hc _ _ le_rfl _ hx
_ = a • f x := by
rw [f.map_update_smul, update_eq_self, f.map_piecewise_smul, div_eq_mul_inv, mul_smul,
inv_smul_smul₀ hc₀']
|
theorem
|
Topology
|
[
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
|
image_multilinear'
|
The image of a von Neumann bounded set under a continuous multilinear map
is von Neumann bounded.
This version does not assume that the topologies on the domain and on the codomain
agree with the vector space structure in any way
but it assumes that `ι` is nonempty.
|
image_multilinear [ContinuousSMul 𝕜 F] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := by
cases isEmpty_or_nonempty ι with
| inl h =>
exact (isBounded_iff_isVonNBounded _).1 <|
@Set.Finite.isBounded _ (vonNBornology 𝕜 F) _ (s.toFinite.image _)
| inr h => exact hs.image_multilinear' f
|
theorem
|
Topology
|
[
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
|
image_multilinear
|
The image of a von Neumann bounded set under a continuous multilinear map
is von Neumann bounded.
This version assumes that the codomain is a topological vector space.
|
toUniformOnFun [TopologicalSpace F] (f : ContinuousMultilinearMap 𝕜 E F) :
(Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F :=
UniformOnFun.ofFun _ f
open UniformOnFun in
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
toUniformOnFun
|
An auxiliary definition used to define topology on `ContinuousMultilinearMap 𝕜 E F`.
|
range_toUniformOnFun [DecidableEq ι] [TopologicalSpace F] :
range toUniformOnFun =
{f : (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F |
Continuous (toFun _ f) ∧
(∀ (m : Π i, E i) i x y,
toFun _ f (update m i (x + y)) = toFun _ f (update m i x) + toFun _ f (update m i y)) ∧
(∀ (m : Π i, E i) i (c : 𝕜) x,
toFun _ f (update m i (c • x)) = c • toFun _ f (update m i x))} := by
ext f
constructor
· rintro ⟨f, rfl⟩
exact ⟨f.cont, f.map_update_add, f.map_update_smul⟩
· rintro ⟨hcont, hadd, hsmul⟩
exact ⟨⟨⟨f, by intro; convert hadd, by intro; convert hsmul⟩, hcont⟩, rfl⟩
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
range_toUniformOnFun
| null |
toUniformOnFun_toFun [TopologicalSpace F] (f : ContinuousMultilinearMap 𝕜 E F) :
UniformOnFun.toFun _ f.toUniformOnFun = f :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
toUniformOnFun_toFun
| null |
instTopologicalSpace [TopologicalSpace F] [IsTopologicalAddGroup F] :
TopologicalSpace (ContinuousMultilinearMap 𝕜 E F) :=
.induced toUniformOnFun <|
@UniformOnFun.topologicalSpace _ _ (IsTopologicalAddGroup.toUniformSpace F) _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instTopologicalSpace
| null |
instUniformSpace [UniformSpace F] [IsUniformAddGroup F] :
UniformSpace (ContinuousMultilinearMap 𝕜 E F) :=
.replaceTopology (.comap toUniformOnFun <| UniformOnFun.uniformSpace _ _ _) <| by
rw [instTopologicalSpace, IsUniformAddGroup.toUniformSpace_eq]; rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instUniformSpace
| null |
isUniformInducing_toUniformOnFun :
IsUniformInducing (toUniformOnFun :
ContinuousMultilinearMap 𝕜 E F → ((Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F)) := ⟨rfl⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
isUniformInducing_toUniformOnFun
| null |
isUniformEmbedding_toUniformOnFun :
IsUniformEmbedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) :=
⟨isUniformInducing_toUniformOnFun, DFunLike.coe_injective⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
isUniformEmbedding_toUniformOnFun
| null |
isEmbedding_toUniformOnFun :
IsEmbedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F →
((Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F)) :=
isUniformEmbedding_toUniformOnFun.isEmbedding
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
isEmbedding_toUniformOnFun
| null |
uniformContinuous_coe_fun [∀ i, ContinuousSMul 𝕜 (E i)] :
UniformContinuous (DFunLike.coe : ContinuousMultilinearMap 𝕜 E F → (Π i, E i) → F) :=
(UniformOnFun.uniformContinuous_toFun isVonNBounded_covers).comp
isUniformEmbedding_toUniformOnFun.uniformContinuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
uniformContinuous_coe_fun
| null |
uniformContinuous_eval_const [∀ i, ContinuousSMul 𝕜 (E i)] (x : Π i, E i) :
UniformContinuous fun f : ContinuousMultilinearMap 𝕜 E F ↦ f x :=
uniformContinuous_pi.1 uniformContinuous_coe_fun x
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
uniformContinuous_eval_const
| null |
instIsUniformAddGroup : IsUniformAddGroup (ContinuousMultilinearMap 𝕜 E F) :=
let φ : ContinuousMultilinearMap 𝕜 E F →+ (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F :=
{ toFun := toUniformOnFun, map_add' := fun _ _ ↦ rfl, map_zero' := rfl }
isUniformEmbedding_toUniformOnFun.isUniformAddGroup φ
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instIsUniformAddGroup
| null |
instUniformContinuousConstSMul {M : Type*}
[Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
UniformContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F) :=
haveI := uniformContinuousConstSMul_of_continuousConstSMul M F
isUniformEmbedding_toUniformOnFun.uniformContinuousConstSMul fun _ _ ↦ rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/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.compContinuousMultilinearMap :
ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜 E G) := by
rw [← isUniformInducing_toUniformOnFun.of_comp_iff]
exact (UniformOnFun.postcomp_isUniformInducing hg).comp isUniformInducing_toUniformOnFun
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
isUniformInducing_postcomp
| null |
completeSpace (h : IsCoherentWith {s : Set (Π i, E i) | IsVonNBounded 𝕜 s}) :
CompleteSpace (ContinuousMultilinearMap 𝕜 E F) := by
classical
wlog hF : T2Space F generalizing F
· rw [(isUniformInducing_postcomp (SeparationQuotient.mkCLM _ _)
SeparationQuotient.isUniformInducing_mk).completeSpace_congr]
· exact this inferInstance
· intro f
use (SeparationQuotient.outCLM _ _).compContinuousMultilinearMap f
simp [DFunLike.ext_iff]
have H : ∀ {m : Π i, E i},
Continuous fun f : (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F ↦ toFun _ f m :=
(uniformContinuous_eval (isVonNBounded_covers) _).continuous
rw [completeSpace_iff_isComplete_range isUniformInducing_toUniformOnFun, range_toUniformOnFun]
simp only [setOf_and, setOf_forall]
apply_rules [IsClosed.isComplete, IsClosed.inter]
· exact UniformOnFun.isClosed_setOf_continuous h
· exact isClosed_iInter fun m ↦ isClosed_iInter fun i ↦
isClosed_iInter fun x ↦ isClosed_iInter fun y ↦ isClosed_eq H (H.add H)
· exact isClosed_iInter fun m ↦ isClosed_iInter fun i ↦
isClosed_iInter fun c ↦ isClosed_iInter fun x ↦ isClosed_eq H (H.const_smul _)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
completeSpace
| null |
instCompleteSpace [∀ i, IsTopologicalAddGroup (E i)] [SequentialSpace (Π i, E i)] :
CompleteSpace (ContinuousMultilinearMap 𝕜 E F) :=
completeSpace <| .of_seq fun _u x hux ↦ (hux.isVonNBounded_range 𝕜).insert x
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instCompleteSpace
| null |
isUniformEmbedding_restrictScalars :
IsUniformEmbedding
(restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) := by
letI : NontriviallyNormedField 𝕜 :=
⟨let ⟨x, hx⟩ := @NontriviallyNormedField.non_trivial 𝕜' _; ⟨algebraMap 𝕜' 𝕜 x, by simpa⟩⟩
rw [← isUniformEmbedding_toUniformOnFun.of_comp_iff]
convert isUniformEmbedding_toUniformOnFun using 4 with s
exact ⟨fun h ↦ h.extend_scalars _, fun h ↦ h.restrict_scalars _⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
isUniformEmbedding_restrictScalars
| null |
uniformContinuous_restrictScalars :
UniformContinuous
(restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) :=
(isUniformEmbedding_restrictScalars 𝕜').uniformContinuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
uniformContinuous_restrictScalars
| null |
instIsTopologicalAddGroup : IsTopologicalAddGroup (ContinuousMultilinearMap 𝕜 E F) :=
letI := IsTopologicalAddGroup.toUniformSpace F
haveI := isUniformAddGroup_of_addCommGroup (G := F)
inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instIsTopologicalAddGroup
| null |
instContinuousConstSMul
{M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
ContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F) := by
letI := IsTopologicalAddGroup.toUniformSpace F
haveI := isUniformAddGroup_of_addCommGroup (G := F)
infer_instance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instContinuousConstSMul
| null |
instContinuousSMul [ContinuousSMul 𝕜 F] :
ContinuousSMul 𝕜 (ContinuousMultilinearMap 𝕜 E F) :=
letI := IsTopologicalAddGroup.toUniformSpace F
haveI := isUniformAddGroup_of_addCommGroup (G := F)
let φ : ContinuousMultilinearMap 𝕜 E F →ₗ[𝕜] (Π i, E i) → F :=
{ toFun := (↑), map_add' := fun _ _ ↦ rfl, map_smul' := fun _ _ ↦ rfl }
UniformOnFun.continuousSMul_induced_of_image_bounded _ _ _ _ φ
isEmbedding_toUniformOnFun.isInducing fun _ _ hu ↦ hu.image_multilinear _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/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 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
(fun Si : Set (Π i, E i) × ι => IsVonNBounded 𝕜 Si.1 ∧ p Si.2)
fun Si => { f | MapsTo f Si.1 (b Si.2) } := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
rw [nhds_induced]
refine (UniformOnFun.hasBasis_nhds_zero_of_basis _ ?_ ?_ h).comap DFunLike.coe
· exact ⟨∅, isVonNBounded_empty _ _⟩
· exact directedOn_of_sup_mem fun _ _ => Bornology.IsVonNBounded.union
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
hasBasis_nhds_zero_of_basis
| null |
hasBasis_nhds_zero :
(𝓝 (0 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
(fun SV : Set (Π i, E i) × 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 [∀ i, ContinuousSMul 𝕜 (E i)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
hasBasis_nhds_zero
| null |
instT2Space [T2Space F] : T2Space (ContinuousMultilinearMap 𝕜 E F) :=
.of_injective_continuous DFunLike.coe_injective continuous_coeFun
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instT2Space
| null |
instT3Space [T2Space F] : T3Space (ContinuousMultilinearMap 𝕜 E F) :=
inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
instT3Space
| null |
isEmbedding_restrictScalars :
IsEmbedding
(restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E 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.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
isEmbedding_restrictScalars
| null |
continuous_restrictScalars :
Continuous
(restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) :=
isEmbedding_restrictScalars.continuous
variable (𝕜') in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
continuous_restrictScalars
| null |
@[simps -fullyApplied apply]
restrictScalarsLinear [ContinuousConstSMul 𝕜' F] :
ContinuousMultilinearMap 𝕜 E F →L[𝕜'] ContinuousMultilinearMap 𝕜' E F where
toFun := restrictScalars 𝕜'
map_add' _ _ := rfl
map_smul' _ _ := rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
restrictScalarsLinear
|
`ContinuousMultilinearMap.restrictScalars` as a `ContinuousLinearMap`.
|
apply [ContinuousConstSMul 𝕜 F] (m : Π i, E i) : ContinuousMultilinearMap 𝕜 E 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.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
apply
|
The application of a multilinear map as a `ContinuousLinearMap`.
|
apply_apply [ContinuousConstSMul 𝕜 F] {m : Π i, E i} {c : ContinuousMultilinearMap 𝕜 E F} :
apply 𝕜 E F m c = c m := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
apply_apply
| null |
hasSum_eval {α : Type*} {p : α → ContinuousMultilinearMap 𝕜 E F}
{q : ContinuousMultilinearMap 𝕜 E F} (h : HasSum p q) (m : Π i, E i) :
HasSum (fun a => p a m) (q m) :=
h.map (applyAddHom m) (continuous_eval_const m)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
hasSum_eval
| null |
tsum_eval [T2Space F] {α : Type*} {p : α → ContinuousMultilinearMap 𝕜 E F} (hp : Summable p)
(m : Π i, E i) : (∑' a, p a) m = ∑' a, p a m :=
(hasSum_eval hp.hasSum m).tsum_eq.symm
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Multilinear.Bounded",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.Algebra.InfiniteSum.Basic"
] |
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
tsum_eval
| null |
making it a full subcategory of `TopCat`.
The fully faithful functor `CompHaus ⥤ TopCat` is denoted `compHausToTop`.
**Note:** The file `Mathlib/Topology/Category/Compactum.lean` provides the equivalence between
`Compactum`, which is defined as the category of algebras for the ultrafilter monad, and `CompHaus`.
`CompactumToCompHaus` is the functor from `Compactum` to `CompHaus` which is proven to be an
equivalence of categories in `CompactumToCompHaus.isEquivalence`.
See `Mathlib/Topology/Category/Compactum.lean` for a more detailed discussion where these
definitions are introduced.
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
making
| null |
CompHaus := CompHausLike (fun _ ↦ True)
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
CompHaus
|
The category of compact Hausdorff spaces.
|
of : CompHaus := CompHausLike.of _ X
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
of
|
A constructor for objects of the category `CompHaus`,
taking a type, and bundling the compact Hausdorff topology
found by typeclass inference.
|
compHausToTop : CompHaus.{u} ⥤ TopCat.{u} :=
CompHausLike.compHausLikeToTop _
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
compHausToTop
|
The fully faithful embedding of `CompHaus` in `TopCat`.
|
@[simps!]
stoneCechObj (X : TopCat) : CompHaus :=
CompHaus.of (StoneCech X)
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
stoneCechObj
|
(Implementation) The object part of the compactification functor from topological spaces to
compact Hausdorff spaces.
|
noncomputable stoneCechEquivalence (X : TopCat.{u}) (Y : CompHaus.{u}) :
(stoneCechObj X ⟶ Y) ≃ (X ⟶ compHausToTop.obj Y) where
toFun f := TopCat.ofHom
{ toFun := f ∘ stoneCechUnit
continuous_toFun := f.hom.2.comp (@continuous_stoneCechUnit X _) }
invFun f := CompHausLike.ofHom _
{ toFun := stoneCechExtend f.hom.2
continuous_toFun := continuous_stoneCechExtend f.hom.2 }
left_inv := by
rintro ⟨f : StoneCech X ⟶ Y, hf : Continuous f⟩
ext x
refine congr_fun ?_ x
apply Continuous.ext_on denseRange_stoneCechUnit (continuous_stoneCechExtend _) hf
· rintro _ ⟨y, rfl⟩
apply congr_fun (stoneCechExtend_extends (hf.comp _)) y
apply continuous_stoneCechUnit
right_inv := by
rintro ⟨f : (X : Type _) ⟶ Y, hf : Continuous f⟩
ext
exact congr_fun (stoneCechExtend_extends hf) _
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
stoneCechEquivalence
|
(Implementation) The bijection of homsets to establish the reflective adjunction of compact
Hausdorff spaces in topological spaces.
|
noncomputable topToCompHaus : TopCat.{u} ⥤ CompHaus.{u} :=
Adjunction.leftAdjointOfEquiv stoneCechEquivalence.{u} fun _ _ _ _ _ => rfl
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
topToCompHaus
|
The Stone-Cech compactification functor from topological spaces to compact Hausdorff spaces,
left adjoint to the inclusion functor.
|
topToCompHaus_obj (X : TopCat) : ↥(topToCompHaus.obj X) = StoneCech X :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
topToCompHaus_obj
| null |
noncomputable compHausToTop.reflective : Reflective compHausToTop where
L := topToCompHaus
adj := Adjunction.adjunctionOfEquivLeft _ _
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
compHausToTop.reflective
|
The category of compact Hausdorff spaces is reflective in the category of topological spaces.
|
noncomputable compHausToTop.createsLimits : CreatesLimits compHausToTop :=
monadicCreatesLimits _
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
compHausToTop.createsLimits
| null |
CompHaus.hasLimits : Limits.HasLimits CompHaus :=
hasLimits_of_hasLimits_createsLimits compHausToTop
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
CompHaus.hasLimits
| null |
CompHaus.hasColimits : Limits.HasColimits CompHaus :=
hasColimits_of_reflective compHausToTop
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
CompHaus.hasColimits
| null |
limitCone {J : Type v} [SmallCategory J] (F : J ⥤ CompHaus.{max v u}) : Limits.Cone F :=
letI FF : J ⥤ TopCat := F ⋙ compHausToTop
{ pt := {
toTop := (TopCat.limitCone FF).pt
is_compact := by
change CompactSpace { u : ∀ j, F.obj j | ∀ {i j : J} (f : i ⟶ j), (F.map f) (u i) = u j }
rw [← isCompact_iff_compactSpace]
apply IsClosed.isCompact
have :
{ u : ∀ j, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f (u i) = u j } =
⋂ (i : J) (j : J) (f : i ⟶ j), { u | F.map f (u i) = u j } := by
ext1
simp only [Set.mem_iInter, Set.mem_setOf_eq]
rw [this]
apply isClosed_iInter
intro i
apply isClosed_iInter
intro j
apply isClosed_iInter
intro f
apply isClosed_eq
· exact ((F.map f).hom.continuous).comp (continuous_apply i)
· exact continuous_apply j
is_hausdorff :=
show T2Space { u : ∀ j, F.obj j | ∀ {i j : J} (f : i ⟶ j), (F.map f) (u i) = u j } from
inferInstance
prop := trivial }
π := {
app := fun j => (TopCat.limitCone FF).π.app j
naturality := by
intro _ _ f
ext ⟨x, hx⟩
simp only [Functor.const_obj_map]
exact (hx f).symm } }
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
limitCone
|
An explicit limit cone for a functor `F : J ⥤ CompHaus`, defined in terms of
`TopCat.limitCone`.
|
limitConeIsLimit {J : Type v} [SmallCategory J] (F : J ⥤ CompHaus.{max v u}) :
Limits.IsLimit.{v} (limitCone.{v,u} F) :=
letI FF : J ⥤ TopCat := F ⋙ compHausToTop
{ lift := fun S => (TopCat.limitConeIsLimit FF).lift (compHausToTop.mapCone S)
fac := fun S => (TopCat.limitConeIsLimit FF).fac (compHausToTop.mapCone S)
uniq := fun S => (TopCat.limitConeIsLimit FF).uniq (compHausToTop.mapCone S) }
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
limitConeIsLimit
|
The limit cone `CompHaus.limitCone F` is indeed a limit cone.
|
epi_iff_surjective {X Y : CompHaus.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by
constructor
· dsimp [Function.Surjective]
contrapose!
rintro ⟨y, hy⟩ hf
let C := Set.range f
have hC : IsClosed C := (isCompact_range f.hom.continuous).isClosed
let D := ({y} : Set Y)
have hD : IsClosed D := isClosed_singleton
have hCD : Disjoint C D := by
rw [Set.disjoint_singleton_right]
rintro ⟨y', hy'⟩
exact hy y' hy'
obtain ⟨φ, hφ0, hφ1, hφ01⟩ := exists_continuous_zero_one_of_isClosed hC hD hCD
haveI : CompactSpace (ULift.{u} <| Set.Icc (0 : ℝ) 1) := Homeomorph.ulift.symm.compactSpace
haveI : T2Space (ULift.{u} <| Set.Icc (0 : ℝ) 1) := Homeomorph.ulift.symm.t2Space
let Z := of (ULift.{u} <| Set.Icc (0 : ℝ) 1)
let g : Y ⟶ Z := ofHom _
⟨fun y' => ⟨⟨φ y', hφ01 y'⟩⟩,
continuous_uliftUp.comp (φ.continuous.subtype_mk fun y' => hφ01 y')⟩
let h : Y ⟶ Z := ofHom _
⟨fun _ => ⟨⟨0, Set.left_mem_Icc.mpr zero_le_one⟩⟩, continuous_const⟩
have H : h = g := by
rw [← cancel_epi f]
ext x : 4
simp [g, h, Z, hφ0 (Set.mem_range_self x)]
apply_fun fun e => (e y).down.1 at H
dsimp [g, h, Z] at H
simp only [hφ1 (Set.mem_singleton y), Pi.one_apply] at H
exact zero_ne_one H
· rw [← CategoryTheory.epi_iff_surjective]
apply (forget CompHaus).epi_of_epi_map
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
epi_iff_surjective
| null |
compHausLikeToCompHaus (P : TopCat → Prop) : CompHausLike P ⥤ CompHaus :=
CompHausLike.toCompHausLike (by simp only [implies_true])
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.Category.CompHausLike.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
compHausLikeToCompHaus
|
Every `CompHausLike` admits a functor to `CompHaus`.
|
effectiveEpi_tfae
{B X : CompHaus.{u}} (π : X ⟶ B) :
TFAE
[ EffectiveEpi π
, Epi π
, Function.Surjective π
] := by
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 2 ↔ 3 := epi_iff_surjective π
tfae_have 3 → 1 := fun hπ ↦ ⟨⟨effectiveEpiStruct π hπ⟩⟩
tfae_finish
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.CompHaus.Limits",
"Mathlib.Topology.Category.CompHausLike.EffectiveEpi"
] |
Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
|
effectiveEpi_tfae
| null |
effectiveEpiFamily_tfae
{α : Type} [Finite α] {B : CompHaus.{u}}
(X : α → CompHaus.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] := by
tfae_have 2 → 1
| _ => by
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2
| _ => inferInstance
tfae_have 3 → 2
| e => by
rw [epi_iff_surjective]
intro b
obtain ⟨t, x, h⟩ := e b
refine ⟨Sigma.ι X t x, ?_⟩
change (Sigma.ι X t ≫ Sigma.desc π) x = _
simpa using h
tfae_have 2 → 3
| e => by
rw [epi_iff_surjective] at e
let i : ∐ X ≅ finiteCoproduct X :=
(colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
intro b
obtain ⟨t, rfl⟩ := e b
let q := i.hom t
refine ⟨q.1,q.2,?_⟩
have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
rw [this]
change _ = (i.inv ≫ Sigma.desc π) (i.hom t)
suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
rw [this]; rfl
rw [Iso.inv_comp_eq]
apply colimit.hom_ext
rintro ⟨a⟩
simp only [i, Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
colimit.comp_coconePointUniqueUpToIso_hom_assoc]
ext; rfl
tfae_finish
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.CompHaus.Limits",
"Mathlib.Topology.Category.CompHausLike.EffectiveEpi"
] |
Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
|
effectiveEpiFamily_tfae
| null |
effectiveEpiFamily_of_jointly_surjective
{α : Type} [Finite α] {B : CompHaus.{u}}
(X : α → CompHaus.{u}) (π : (a : α) → (X a ⟶ B))
(surj : ∀ b : B, ∃ (a : α) (x : X a), π a x = b) :
EffectiveEpiFamily X π :=
((effectiveEpiFamily_tfae X π).out 2 0).mp surj
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.CompHaus.Limits",
"Mathlib.Topology.Category.CompHausLike.EffectiveEpi"
] |
Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
|
effectiveEpiFamily_of_jointly_surjective
| null |
@[simps]
topCatOpToFrm : TopCatᵒᵖ ⥤ Frm where
obj X := Frm.of (Opens (unop X : TopCat))
map f := Frm.ofHom <| Opens.comap <| (Quiver.Hom.unop f).hom
|
def
|
Topology
|
[
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/CompHaus/Frm.lean
|
topCatOpToFrm
|
The forgetful functor from `TopCatᵒᵖ` to `Frm`.
|
CompHausOpToFrame.faithful : (compHausToTop.op ⋙ topCatOpToFrm.{u}).Faithful :=
⟨fun {X _ _ _} h => Quiver.Hom.unop_inj <| ConcreteCategory.ext <|
Opens.comap_injective (β := (unop X).toTop) <| FrameHom.ext <|
CategoryTheory.congr_fun h⟩
|
instance
|
Topology
|
[
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/CompHaus/Frm.lean
|
CompHausOpToFrame.faithful
| null |
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