Topology on continuous alternating maps #
In this file we define UniformSpace
and TopologicalSpace
structures
on the space of continuous alternating maps between topological vector spaces.
The structures are induced by those on ContinuousMultilinearMap
s,
and most of the lemmas follow from the corresponding lemmas about ContinuousMultilinearMap
s.
instance
ContinuousAlternatingMap.instTopologicalSpace
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
:
TopologicalSpace (E [⋀^ι]→L[𝕜] F)
theorem
ContinuousAlternatingMap.isClosed_range_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[T2Space F]
:
instance
ContinuousAlternatingMap.instUniformSpace
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
:
UniformSpace (E [⋀^ι]→L[𝕜] F)
theorem
ContinuousAlternatingMap.isUniformEmbedding_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
:
theorem
ContinuousAlternatingMap.uniformContinuous_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
:
theorem
ContinuousAlternatingMap.uniformContinuous_coe_fun
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
[ContinuousSMul 𝕜 E]
:
theorem
ContinuousAlternatingMap.uniformContinuous_eval_const
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
[ContinuousSMul 𝕜 E]
(x : ι → E)
:
UniformContinuous fun (f : E [⋀^ι]→L[𝕜] F) => f x
instance
ContinuousAlternatingMap.instIsUniformAddGroup
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
:
IsUniformAddGroup (E [⋀^ι]→L[𝕜] F)
instance
ContinuousAlternatingMap.instUniformContinuousConstSMul
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
{M : Type u_5}
[Monoid M]
[DistribMulAction M F]
[SMulCommClass 𝕜 M F]
[ContinuousConstSMul M F]
:
UniformContinuousConstSMul M (E [⋀^ι]→L[𝕜] F)
theorem
ContinuousAlternatingMap.isUniformInducing_postcomp
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
{G : Type u_5}
[AddCommGroup G]
[UniformSpace G]
[IsUniformAddGroup G]
[Module 𝕜 G]
(g : F →L[𝕜] G)
(hg : IsUniformInducing ⇑g)
:
theorem
ContinuousAlternatingMap.completeSpace
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
[ContinuousSMul 𝕜 E]
[ContinuousConstSMul 𝕜 F]
[CompleteSpace F]
(h : Topology.IsCoherentWith {s : Set (ι → E) | Bornology.IsVonNBounded 𝕜 s})
:
CompleteSpace (E [⋀^ι]→L[𝕜] F)
instance
ContinuousAlternatingMap.instCompleteSpace
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
[ContinuousSMul 𝕜 E]
[ContinuousConstSMul 𝕜 F]
[CompleteSpace F]
[IsTopologicalAddGroup E]
[SequentialSpace (ι → E)]
:
CompleteSpace (E [⋀^ι]→L[𝕜] F)
theorem
ContinuousAlternatingMap.isUniformEmbedding_restrictScalars
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
(𝕜' : Type u_5)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
[ContinuousSMul 𝕜 E]
:
theorem
ContinuousAlternatingMap.uniformContinuous_restrictScalars
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[UniformSpace F]
[IsUniformAddGroup F]
(𝕜' : Type u_5)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
[ContinuousSMul 𝕜 E]
:
theorem
ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
:
@[deprecated ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap (since := "2024-10-26")]
theorem
ContinuousAlternatingMap.embedding_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
:
Alias of ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap
.
instance
ContinuousAlternatingMap.instIsTopologicalAddGroup
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
:
IsTopologicalAddGroup (E [⋀^ι]→L[𝕜] F)
theorem
ContinuousAlternatingMap.continuous_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
:
instance
ContinuousAlternatingMap.instContinuousConstSMul
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
{M : Type u_5}
[Monoid M]
[DistribMulAction M F]
[SMulCommClass 𝕜 M F]
[ContinuousConstSMul M F]
:
ContinuousConstSMul M (E [⋀^ι]→L[𝕜] F)
instance
ContinuousAlternatingMap.instContinuousSMul
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 F]
:
ContinuousSMul 𝕜 (E [⋀^ι]→L[𝕜] F)
theorem
ContinuousAlternatingMap.hasBasis_nhds_zero_of_basis
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
{ι' : Type u_5}
{p : ι' → Prop}
{b : ι' → Set F}
(h : (nhds 0).HasBasis p b)
:
theorem
ContinuousAlternatingMap.hasBasis_nhds_zero
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
:
theorem
ContinuousAlternatingMap.isClosedEmbedding_toContinuousMultilinearMap
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[T2Space F]
:
instance
ContinuousAlternatingMap.instContinuousEvalConst
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
:
ContinuousEvalConst (E [⋀^ι]→L[𝕜] F) (ι → E) F
instance
ContinuousAlternatingMap.instT2Space
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[T2Space F]
:
instance
ContinuousAlternatingMap.instT3Space
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[T2Space F]
:
theorem
ContinuousAlternatingMap.isEmbedding_restrictScalars
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
{𝕜' : Type u_5}
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
:
@[deprecated ContinuousAlternatingMap.isEmbedding_restrictScalars (since := "2024-10-26")]
theorem
ContinuousAlternatingMap.embedding_restrictScalars
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
{𝕜' : Type u_5}
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
:
Alias of ContinuousAlternatingMap.isEmbedding_restrictScalars
.
theorem
ContinuousAlternatingMap.continuous_restrictScalars
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
{𝕜' : Type u_5}
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
:
def
ContinuousAlternatingMap.restrictScalarsCLM
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
(𝕜' : Type u_5)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
[ContinuousConstSMul 𝕜' F]
:
ContinuousMultilinearMap.restrictScalars
as a ContinuousLinearMap
.
Equations
- ContinuousAlternatingMap.restrictScalarsCLM 𝕜' = { toFun := ContinuousAlternatingMap.restrictScalars 𝕜', map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }
Instances For
@[simp]
theorem
ContinuousAlternatingMap.restrictScalarsCLM_apply
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
(𝕜' : Type u_5)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[Module 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[Module 𝕜' F]
[IsScalarTower 𝕜' 𝕜 F]
[ContinuousConstSMul 𝕜' F]
:
def
ContinuousAlternatingMap.apply
(𝕜 : Type u_1)
(E : Type u_2)
(F : Type u_3)
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[ContinuousConstSMul 𝕜 F]
(m : ι → E)
:
The application of a multilinear map as a ContinuousLinearMap
.
Equations
Instances For
@[simp]
theorem
ContinuousAlternatingMap.apply_apply
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[ContinuousConstSMul 𝕜 F]
{m : ι → E}
{c : E [⋀^ι]→L[𝕜] F}
:
theorem
ContinuousAlternatingMap.hasSum_eval
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
{α : Type u_5}
{p : α → E [⋀^ι]→L[𝕜] F}
{q : E [⋀^ι]→L[𝕜] F}
(h : HasSum p q)
(m : ι → E)
:
HasSum (fun (a : α) => (p a) m) (q m)
theorem
ContinuousAlternatingMap.tsum_eval
{𝕜 : Type u_1}
{E : Type u_2}
{F : Type u_3}
{ι : Type u_4}
[NormedField 𝕜]
[AddCommGroup E]
[Module 𝕜 E]
[TopologicalSpace E]
[AddCommGroup F]
[Module 𝕜 F]
[TopologicalSpace F]
[IsTopologicalAddGroup F]
[ContinuousSMul 𝕜 E]
[T2Space F]
{α : Type u_5}
{p : α → E [⋀^ι]→L[𝕜] F}
(hp : Summable p)
(m : ι → E)
: