fact
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stringclasses 32
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topologicalSpace_eq [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
instTopologicalSpace σ F 𝔖 = TopologicalSpace.induced (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe)
(UniformOnFun.topologicalSpace E F 𝔖) := by
rw [instTopologicalSpace]
congr
exact IsUniformAddGroup.toUniformSpace_eq
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
topologicalSpace_eq
| null |
instUniformSpace [UniformSpace F] [IsUniformAddGroup F]
(𝔖 : Set (Set E)) : UniformSpace (UniformConvergenceCLM σ F 𝔖) :=
UniformSpace.replaceTopology
((UniformOnFun.uniformSpace E F 𝔖).comap (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe))
(by rw [UniformConvergenceCLM.instTopologicalSpace, IsUniformAddGroup.toUniformSpace_eq]; rfl)
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instUniformSpace
|
The uniform structure associated with `ContinuousLinearMap.strongTopology`. We make sure
that this has nice definitional properties.
|
uniformSpace_eq [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
instUniformSpace σ F 𝔖 =
UniformSpace.comap (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe)
(UniformOnFun.uniformSpace E F 𝔖) := by
rw [instUniformSpace, UniformSpace.replaceTopology_eq]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
uniformSpace_eq
| null |
uniformity_toTopologicalSpace_eq [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
(UniformConvergenceCLM.instUniformSpace σ F 𝔖).toTopologicalSpace =
UniformConvergenceCLM.instTopologicalSpace σ F 𝔖 :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
uniformity_toTopologicalSpace_eq
| null |
isUniformInducing_coeFn [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
IsUniformInducing (α := UniformConvergenceCLM σ F 𝔖) (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) :=
⟨rfl⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isUniformInducing_coeFn
| null |
isUniformEmbedding_coeFn [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
IsUniformEmbedding (α := UniformConvergenceCLM σ F 𝔖) (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) :=
⟨isUniformInducing_coeFn .., DFunLike.coe_injective⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isUniformEmbedding_coeFn
| null |
isEmbedding_coeFn [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
IsEmbedding (X := UniformConvergenceCLM σ F 𝔖) (Y := E →ᵤ[𝔖] F)
(UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) :=
IsUniformEmbedding.isEmbedding (isUniformEmbedding_coeFn _ _ _)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isEmbedding_coeFn
| null |
instAddCommGroup [TopologicalSpace F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E)) :
AddCommGroup (UniformConvergenceCLM σ F 𝔖) := ContinuousLinearMap.addCommGroup
@[simp]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instAddCommGroup
| null |
coe_zero [TopologicalSpace F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E)) :
⇑(0 : UniformConvergenceCLM σ F 𝔖) = 0 :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
coe_zero
| null |
instIsUniformAddGroup [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
IsUniformAddGroup (UniformConvergenceCLM σ F 𝔖) := by
let φ : (UniformConvergenceCLM σ F 𝔖) →+ E →ᵤ[𝔖] F :=
⟨⟨(DFunLike.coe : (UniformConvergenceCLM σ F 𝔖) → E →ᵤ[𝔖] F), rfl⟩, fun _ _ => rfl⟩
exact (isUniformEmbedding_coeFn _ _ _).isUniformAddGroup φ
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instIsUniformAddGroup
| null |
instIsTopologicalAddGroup [TopologicalSpace F] [IsTopologicalAddGroup F]
(𝔖 : Set (Set E)) : IsTopologicalAddGroup (UniformConvergenceCLM σ F 𝔖) := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
infer_instance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instIsTopologicalAddGroup
| null |
continuousEvalConst [TopologicalSpace F] [IsTopologicalAddGroup F]
(𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = Set.univ) :
ContinuousEvalConst (UniformConvergenceCLM σ F 𝔖) E F where
continuous_eval_const x := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
exact (UniformOnFun.uniformContinuous_eval h𝔖 x).continuous.comp
(isEmbedding_coeFn σ F 𝔖).continuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
continuousEvalConst
| null |
t2Space [TopologicalSpace F] [IsTopologicalAddGroup F] [T2Space F]
(𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = univ) : T2Space (UniformConvergenceCLM σ F 𝔖) := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
haveI : T2Space (E →ᵤ[𝔖] F) := UniformOnFun.t2Space_of_covering h𝔖
exact (isEmbedding_coeFn σ F 𝔖).t2Space
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
t2Space
| null |
instDistribMulAction (M : Type*) [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
[TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousConstSMul M F] (𝔖 : Set (Set E)) :
DistribMulAction M (UniformConvergenceCLM σ F 𝔖) := ContinuousLinearMap.distribMulAction
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instDistribMulAction
| null |
instModule (R : Type*) [Semiring R] [Module R F] [SMulCommClass 𝕜₂ R F]
[TopologicalSpace F] [ContinuousConstSMul R F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E)) :
Module R (UniformConvergenceCLM σ F 𝔖) := ContinuousLinearMap.module
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instModule
| null |
continuousSMul [RingHomSurjective σ] [RingHomIsometric σ]
[TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : Set (Set E))
(h𝔖₃ : ∀ S ∈ 𝔖, IsVonNBounded 𝕜₁ S) :
ContinuousSMul 𝕜₂ (UniformConvergenceCLM σ F 𝔖) := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
let φ : (UniformConvergenceCLM σ F 𝔖) →ₗ[𝕜₂] E → F :=
⟨⟨DFunLike.coe, fun _ _ => rfl⟩, fun _ _ => rfl⟩
exact UniformOnFun.continuousSMul_induced_of_image_bounded 𝕜₂ E F (UniformConvergenceCLM σ F 𝔖) φ
⟨rfl⟩ fun u s hs => (h𝔖₃ s hs).image u
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
continuousSMul
| null |
hasBasis_nhds_zero_of_basis [TopologicalSpace F] [IsTopologicalAddGroup F]
{ι : Type*} (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop}
{b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
(𝓝 (0 : UniformConvergenceCLM σ F 𝔖)).HasBasis
(fun Si : Set E × ι => Si.1 ∈ 𝔖 ∧ p Si.2)
fun Si => { f : E →SL[σ] F | ∀ x ∈ Si.1, f x ∈ b Si.2 } := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
rw [(isEmbedding_coeFn σ F 𝔖).isInducing.nhds_eq_comap]
exact (UniformOnFun.hasBasis_nhds_zero_of_basis 𝔖 h𝔖₁ h𝔖₂ h).comap DFunLike.coe
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
hasBasis_nhds_zero_of_basis
| null |
hasBasis_nhds_zero [TopologicalSpace F] [IsTopologicalAddGroup F]
(𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
(𝓝 (0 : UniformConvergenceCLM σ F 𝔖)).HasBasis
(fun SV : Set E × Set F => SV.1 ∈ 𝔖 ∧ SV.2 ∈ (𝓝 0 : Filter F)) fun SV =>
{ f : UniformConvergenceCLM σ F 𝔖 | ∀ x ∈ SV.1, f x ∈ SV.2 } :=
hasBasis_nhds_zero_of_basis σ F 𝔖 h𝔖₁ h𝔖₂ (𝓝 0).basis_sets
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
hasBasis_nhds_zero
| null |
nhds_zero_eq_of_basis [TopologicalSpace F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E))
{ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
𝓝 (0 : UniformConvergenceCLM σ F 𝔖) =
⨅ (s : Set E) (_ : s ∈ 𝔖) (i : ι) (_ : p i),
𝓟 {f : UniformConvergenceCLM σ F 𝔖 | MapsTo f s (b i)} := by
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
rw [(isEmbedding_coeFn σ F 𝔖).isInducing.nhds_eq_comap,
UniformOnFun.nhds_eq_of_basis _ _ h.uniformity_of_nhds_zero]
simp [MapsTo]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
nhds_zero_eq_of_basis
| null |
nhds_zero_eq [TopologicalSpace F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E)) :
𝓝 (0 : UniformConvergenceCLM σ F 𝔖) =
⨅ s ∈ 𝔖, ⨅ t ∈ 𝓝 (0 : F),
𝓟 {f : UniformConvergenceCLM σ F 𝔖 | MapsTo f s t} :=
nhds_zero_eq_of_basis _ _ _ (𝓝 0).basis_sets
variable {F} in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
nhds_zero_eq
| null |
eventually_nhds_zero_mapsTo [TopologicalSpace F] [IsTopologicalAddGroup F]
{𝔖 : Set (Set E)} {s : Set E} (hs : s ∈ 𝔖) {U : Set F} (hu : U ∈ 𝓝 0) :
∀ᶠ f : UniformConvergenceCLM σ F 𝔖 in 𝓝 0, MapsTo f s U := by
rw [nhds_zero_eq]
apply_rules [mem_iInf_of_mem, mem_principal_self]
variable {σ F} in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
eventually_nhds_zero_mapsTo
| null |
isVonNBounded_image2_apply {R : Type*} [SeminormedRing R]
[TopologicalSpace F] [IsTopologicalAddGroup F]
[DistribMulAction R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
{𝔖 : Set (Set E)} {S : Set (UniformConvergenceCLM σ F 𝔖)} (hS : IsVonNBounded R S)
{s : Set E} (hs : s ∈ 𝔖) : IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) := by
intro U hU
filter_upwards [hS (eventually_nhds_zero_mapsTo σ hs hU)] with c hc
rw [image2_subset_iff]
intro f hf x hx
rcases hc hf with ⟨g, hg, rfl⟩
exact smul_mem_smul_set (hg hx)
variable {σ F} in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isVonNBounded_image2_apply
| null |
isVonNBounded_iff {R : Type*} [NormedDivisionRing R]
[TopologicalSpace F] [IsTopologicalAddGroup F]
[Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
{𝔖 : Set (Set E)} {S : Set (UniformConvergenceCLM σ F 𝔖)} :
IsVonNBounded R S ↔ ∀ s ∈ 𝔖, IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) := by
refine ⟨fun hS s hs ↦ isVonNBounded_image2_apply hS hs, fun h ↦ ?_⟩
simp_rw [isVonNBounded_iff_absorbing_le, nhds_zero_eq, le_iInf_iff, le_principal_iff]
intro s hs U hU
rw [Filter.mem_absorbing, Absorbs]
filter_upwards [h s hs hU, eventually_ne_cobounded 0] with c hc hc₀ f hf
rw [mem_smul_set_iff_inv_smul_mem₀ hc₀]
intro x hx
simpa only [mem_smul_set_iff_inv_smul_mem₀ hc₀] using hc (mem_image2_of_mem hf hx)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isVonNBounded_iff
|
A set `S` of continuous linear maps with topology of uniform convergence on sets `s ∈ 𝔖`
is von Neumann bounded iff for any `s ∈ 𝔖`,
the set `{f x | (f ∈ S) (x ∈ s)}` is von Neumann bounded.
|
instUniformContinuousConstSMul (M : Type*)
[Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
[UniformSpace F] [IsUniformAddGroup F] [UniformContinuousConstSMul M F] (𝔖 : Set (Set E)) :
UniformContinuousConstSMul M (UniformConvergenceCLM σ F 𝔖) :=
(isUniformInducing_coeFn σ F 𝔖).uniformContinuousConstSMul fun _ _ ↦ by rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instUniformContinuousConstSMul
| null |
instContinuousConstSMul (M : Type*)
[Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
[TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousConstSMul M F] (𝔖 : Set (Set E)) :
ContinuousConstSMul M (UniformConvergenceCLM σ F 𝔖) :=
let _ := IsTopologicalAddGroup.toUniformSpace F
have _ : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
have _ := uniformContinuousConstSMul_of_continuousConstSMul M F
inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instContinuousConstSMul
| null |
tendsto_iff_tendstoUniformlyOn {ι : Type*} {p : Filter ι} [UniformSpace F]
[IsUniformAddGroup F] (𝔖 : Set (Set E)) {a : ι → UniformConvergenceCLM σ F 𝔖}
{a₀ : UniformConvergenceCLM σ F 𝔖} :
Filter.Tendsto a p (𝓝 a₀) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (a · ·) a₀ p s := by
rw [(isEmbedding_coeFn σ F 𝔖).tendsto_nhds_iff, UniformOnFun.tendsto_iff_tendstoUniformlyOn]
rfl
variable {F} in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
tendsto_iff_tendstoUniformlyOn
| null |
isUniformInducing_postcomp
{G : Type*} [AddCommGroup G] [UniformSpace G] [IsUniformAddGroup G]
{𝕜₃ : Type*} [NormedField 𝕜₃] [Module 𝕜₃ G]
{τ : 𝕜₂ →+* 𝕜₃} {ρ : 𝕜₁ →+* 𝕜₃} [RingHomCompTriple σ τ ρ] [UniformSpace F] [IsUniformAddGroup F]
(g : F →SL[τ] G) (hg : IsUniformInducing g) (𝔖 : Set (Set E)) :
IsUniformInducing (α := UniformConvergenceCLM σ F 𝔖) (β := UniformConvergenceCLM ρ G 𝔖)
g.comp := by
rw [← (isUniformInducing_coeFn _ _ _).of_comp_iff]
exact (UniformOnFun.postcomp_isUniformInducing hg).comp (isUniformInducing_coeFn _ _ _)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isUniformInducing_postcomp
| null |
completeSpace [UniformSpace F] [IsUniformAddGroup F] [ContinuousSMul 𝕜₂ F] [CompleteSpace F]
{𝔖 : Set (Set E)} (h𝔖 : IsCoherentWith 𝔖) (h𝔖U : ⋃₀ 𝔖 = univ) :
CompleteSpace (UniformConvergenceCLM σ F 𝔖) := by
wlog hF : T2Space F generalizing F
· rw [(isUniformInducing_postcomp σ (SeparationQuotient.mkCLM 𝕜₂ F)
SeparationQuotient.isUniformInducing_mk _).completeSpace_congr]
exacts [this _ inferInstance, SeparationQuotient.postcomp_mkCLM_surjective F σ E]
rw [completeSpace_iff_isComplete_range (isUniformInducing_coeFn _ _ _)]
apply IsClosed.isComplete
have H₁ : IsClosed {f : E →ᵤ[𝔖] F | Continuous ((UniformOnFun.toFun 𝔖) f)} :=
UniformOnFun.isClosed_setOf_continuous h𝔖
convert H₁.inter <| (LinearMap.isClosed_range_coe E F σ).preimage
(UniformOnFun.uniformContinuous_toFun h𝔖U).continuous
exact ContinuousLinearMap.range_coeFn_eq
variable {𝔖₁ 𝔖₂ : Set (Set E)}
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
completeSpace
| null |
uniformSpace_mono [UniformSpace F] [IsUniformAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁) :
instUniformSpace σ F 𝔖₁ ≤ instUniformSpace σ F 𝔖₂ := by
simp_rw [uniformSpace_eq]
exact UniformSpace.comap_mono (UniformOnFun.mono (le_refl _) h)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
uniformSpace_mono
| null |
topologicalSpace_mono [TopologicalSpace F] [IsTopologicalAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁) :
instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂ := by
letI := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
simp_rw [← uniformity_toTopologicalSpace_eq]
exact UniformSpace.toTopologicalSpace_mono (uniformSpace_mono σ F h)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
topologicalSpace_mono
| null |
topologicalSpace [TopologicalSpace F] [IsTopologicalAddGroup F] :
TopologicalSpace (E →SL[σ] F) :=
UniformConvergenceCLM.instTopologicalSpace σ F { S | IsVonNBounded 𝕜₁ S }
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
topologicalSpace
|
The topology of bounded convergence on `E →L[𝕜] F`. This coincides with the topology induced by
the operator norm when `E` and `F` are normed spaces.
|
topologicalAddGroup [TopologicalSpace F] [IsTopologicalAddGroup F] :
IsTopologicalAddGroup (E →SL[σ] F) :=
UniformConvergenceCLM.instIsTopologicalAddGroup σ F _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
topologicalAddGroup
| null |
continuousSMul [RingHomSurjective σ] [RingHomIsometric σ] [TopologicalSpace F]
[IsTopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] : ContinuousSMul 𝕜₂ (E →SL[σ] F) :=
UniformConvergenceCLM.continuousSMul σ F { S | IsVonNBounded 𝕜₁ S } fun _ hs => hs
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
continuousSMul
| null |
uniformSpace [UniformSpace F] [IsUniformAddGroup F] : UniformSpace (E →SL[σ] F) :=
UniformConvergenceCLM.instUniformSpace σ F { S | IsVonNBounded 𝕜₁ S }
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
uniformSpace
| null |
isUniformAddGroup [UniformSpace F] [IsUniformAddGroup F] :
IsUniformAddGroup (E →SL[σ] F) :=
UniformConvergenceCLM.instIsUniformAddGroup σ F _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isUniformAddGroup
| null |
instContinuousEvalConst [TopologicalSpace F] [IsTopologicalAddGroup F]
[ContinuousSMul 𝕜₁ E] : ContinuousEvalConst (E →SL[σ] F) E F :=
UniformConvergenceCLM.continuousEvalConst σ F _ Bornology.isVonNBounded_covers
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instContinuousEvalConst
| null |
instT2Space [TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜₁ E]
[T2Space F] : T2Space (E →SL[σ] F) :=
UniformConvergenceCLM.t2Space σ F _ Bornology.isVonNBounded_covers
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instT2Space
| null |
protected hasBasis_nhds_zero_of_basis [TopologicalSpace F] [IsTopologicalAddGroup F]
{ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
(𝓝 (0 : E →SL[σ] F)).HasBasis (fun Si : Set E × ι => IsVonNBounded 𝕜₁ Si.1 ∧ p Si.2)
fun Si => { f : E →SL[σ] F | ∀ x ∈ Si.1, f x ∈ b Si.2 } :=
UniformConvergenceCLM.hasBasis_nhds_zero_of_basis σ F { S | IsVonNBounded 𝕜₁ S }
⟨∅, isVonNBounded_empty 𝕜₁ E⟩
(directedOn_of_sup_mem fun _ _ => IsVonNBounded.union) h
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
hasBasis_nhds_zero_of_basis
| null |
protected hasBasis_nhds_zero [TopologicalSpace F] [IsTopologicalAddGroup F] :
(𝓝 (0 : E →SL[σ] F)).HasBasis
(fun SV : Set E × Set F => IsVonNBounded 𝕜₁ SV.1 ∧ SV.2 ∈ (𝓝 0 : Filter F))
fun SV => { f : E →SL[σ] F | ∀ x ∈ SV.1, f x ∈ SV.2 } :=
ContinuousLinearMap.hasBasis_nhds_zero_of_basis (𝓝 0).basis_sets
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
hasBasis_nhds_zero
| null |
isUniformEmbedding_toUniformOnFun [UniformSpace F] [IsUniformAddGroup F] :
IsUniformEmbedding
fun f : E →SL[σ] F ↦ UniformOnFun.ofFun {s | Bornology.IsVonNBounded 𝕜₁ s} f :=
UniformConvergenceCLM.isUniformEmbedding_coeFn ..
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isUniformEmbedding_toUniformOnFun
| null |
uniformContinuousConstSMul
{M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
[UniformSpace F] [IsUniformAddGroup F] [UniformContinuousConstSMul M F] :
UniformContinuousConstSMul M (E →SL[σ] F) :=
UniformConvergenceCLM.instUniformContinuousConstSMul σ F _ _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
uniformContinuousConstSMul
| null |
continuousConstSMul {M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
[TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousConstSMul M F] :
ContinuousConstSMul M (E →SL[σ] F) :=
UniformConvergenceCLM.instContinuousConstSMul σ F _ _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
continuousConstSMul
| null |
protected nhds_zero_eq_of_basis [TopologicalSpace F] [IsTopologicalAddGroup F]
{ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
𝓝 (0 : E →SL[σ] F) =
⨅ (s : Set E) (_ : IsVonNBounded 𝕜₁ s) (i : ι) (_ : p i),
𝓟 {f : E →SL[σ] F | MapsTo f s (b i)} :=
UniformConvergenceCLM.nhds_zero_eq_of_basis _ _ _ h
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
nhds_zero_eq_of_basis
| null |
protected nhds_zero_eq [TopologicalSpace F] [IsTopologicalAddGroup F] :
𝓝 (0 : E →SL[σ] F) =
⨅ (s : Set E) (_ : IsVonNBounded 𝕜₁ s) (U : Set F) (_ : U ∈ 𝓝 0),
𝓟 {f : E →SL[σ] F | MapsTo f s U} :=
UniformConvergenceCLM.nhds_zero_eq ..
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
nhds_zero_eq
| null |
eventually_nhds_zero_mapsTo [TopologicalSpace F] [IsTopologicalAddGroup F]
{s : Set E} (hs : IsVonNBounded 𝕜₁ s) {U : Set F} (hu : U ∈ 𝓝 0) :
∀ᶠ f : E →SL[σ] F in 𝓝 0, MapsTo f s U :=
UniformConvergenceCLM.eventually_nhds_zero_mapsTo _ hs hu
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
eventually_nhds_zero_mapsTo
|
If `s` is a von Neumann bounded set and `U` is a neighbourhood of zero,
then sufficiently small continuous linear maps map `s` to `U`.
|
isVonNBounded_image2_apply {R : Type*} [SeminormedRing R]
[TopologicalSpace F] [IsTopologicalAddGroup F]
[DistribMulAction R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
{S : Set (E →SL[σ] F)} (hS : IsVonNBounded R S) {s : Set E} (hs : IsVonNBounded 𝕜₁ s) :
IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) :=
UniformConvergenceCLM.isVonNBounded_image2_apply hS hs
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isVonNBounded_image2_apply
|
If `S` is a von Neumann bounded set of continuous linear maps `f : E →SL[σ] F`
and `s` is a von Neumann bounded set in the domain,
then the set `{f x | (f ∈ S) (x ∈ s)}` is von Neumann bounded.
See also `isVonNBounded_iff` for an `Iff` version with stronger typeclass assumptions.
|
isVonNBounded_iff {R : Type*} [NormedDivisionRing R]
[TopologicalSpace F] [IsTopologicalAddGroup F]
[Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
{S : Set (E →SL[σ] F)} :
IsVonNBounded R S ↔
∀ s, IsVonNBounded 𝕜₁ s → IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) :=
UniformConvergenceCLM.isVonNBounded_iff
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isVonNBounded_iff
|
A set `S` of continuous linear maps is von Neumann bounded
iff for any von Neumann bounded set `s`,
the set `{f x | (f ∈ S) (x ∈ s)}` is von Neumann bounded.
For the forward implication with weaker typeclass assumptions, see `isVonNBounded_image2_apply`.
|
completeSpace [UniformSpace F] [IsUniformAddGroup F] [ContinuousSMul 𝕜₂ F] [CompleteSpace F]
[ContinuousSMul 𝕜₁ E] (h : IsCoherentWith {s : Set E | IsVonNBounded 𝕜₁ s}) :
CompleteSpace (E →SL[σ] F) :=
UniformConvergenceCLM.completeSpace _ _ h isVonNBounded_covers
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
completeSpace
| null |
instCompleteSpace [IsTopologicalAddGroup E] [ContinuousSMul 𝕜₁ E] [SequentialSpace E]
[UniformSpace F] [IsUniformAddGroup F] [ContinuousSMul 𝕜₂ F] [CompleteSpace F] :
CompleteSpace (E →SL[σ] F) :=
completeSpace <| .of_seq fun _ _ h ↦ (h.isVonNBounded_range 𝕜₁).insert _
variable (G) [TopologicalSpace F] [TopologicalSpace G]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
instCompleteSpace
| null |
@[simps]
precomp [IsTopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G] [RingHomSurjective σ]
[RingHomIsometric σ] (L : E →SL[σ] F) : (F →SL[τ] G) →L[𝕜₃] E →SL[ρ] G where
toFun f := f.comp L
map_add' f g := add_comp f g L
map_smul' a f := smul_comp a f L
cont := by
letI : UniformSpace G := IsTopologicalAddGroup.toUniformSpace G
haveI : IsUniformAddGroup G := isUniformAddGroup_of_addCommGroup
rw [(UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous_iff]
apply (UniformOnFun.precomp_uniformContinuous fun S hS => hS.image L).continuous.comp
(UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous
variable (E) {G}
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
precomp
|
Pre-composition by a *fixed* continuous linear map as a continuous linear map.
Note that in non-normed space it is not always true that composition is continuous
in both variables, so we have to fix one of them.
|
@[simps]
postcomp [IsTopologicalAddGroup F] [IsTopologicalAddGroup G] [ContinuousConstSMul 𝕜₃ G]
[ContinuousConstSMul 𝕜₂ F] (L : F →SL[τ] G) : (E →SL[σ] F) →SL[τ] E →SL[ρ] G where
toFun f := L.comp f
map_add' := comp_add L
map_smul' := comp_smulₛₗ L
cont := by
letI : UniformSpace G := IsTopologicalAddGroup.toUniformSpace G
haveI : IsUniformAddGroup G := isUniformAddGroup_of_addCommGroup
letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
rw [(UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous_iff]
exact
(UniformOnFun.postcomp_uniformContinuous L.uniformContinuous).continuous.comp
(UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
postcomp
|
Post-composition by a *fixed* continuous linear map as a continuous linear map.
Note that in non-normed space it is not always true that composition is continuous
in both variables, so we have to fix one of them.
|
toLinearMap₁₂ (L : E →SL[σ₁₃] F →SL[σ₂₃] G) : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G :=
(coeLMₛₗ σ₂₃).comp L.toLinearMap
@[deprecated (since := "2025-07-28")] alias toLinearMap₂ := toLinearMap₁₂
@[simp] lemma toLinearMap₁₂_apply (L : E →SL[σ₁₃] F →SL[σ₂₃] G) (v : E) (w : F) :
L.toLinearMap₁₂ v w = L v w := rfl
@[deprecated (since := "2025-07-28")] alias toLinearMap₂_apply := toLinearMap₁₂_apply
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
toLinearMap₁₂
|
Send a continuous bilinear map to an abstract bilinear map (forgetting continuity).
|
isUniformEmbedding_restrictScalars :
IsUniformEmbedding (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) := by
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.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isUniformEmbedding_restrictScalars
| null |
uniformContinuous_restrictScalars :
UniformContinuous (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) :=
(isUniformEmbedding_restrictScalars 𝕜').uniformContinuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
uniformContinuous_restrictScalars
| null |
isEmbedding_restrictScalars :
IsEmbedding (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →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.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
isEmbedding_restrictScalars
| null |
continuous_restrictScalars :
Continuous (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) :=
(isEmbedding_restrictScalars _).continuous
variable (𝕜 E F)
variable (𝕜'' : Type*) [Ring 𝕜'']
[Module 𝕜'' F] [ContinuousConstSMul 𝕜'' F] [SMulCommClass 𝕜 𝕜'' F] [SMulCommClass 𝕜' 𝕜'' F]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
continuous_restrictScalars
| null |
restrictScalarsL : (E →L[𝕜] F) →L[𝕜''] E →L[𝕜'] F :=
.mk <| restrictScalarsₗ 𝕜 E F 𝕜' 𝕜''
variable {𝕜 E F 𝕜' 𝕜''}
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
restrictScalarsL
|
`ContinuousLinearMap.restrictScalars` as a `ContinuousLinearMap`.
|
coe_restrictScalarsL : (restrictScalarsL 𝕜 E F 𝕜' 𝕜'' : (E →L[𝕜] F) →ₗ[𝕜''] E →L[𝕜'] F) =
restrictScalarsₗ 𝕜 E F 𝕜' 𝕜'' :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
coe_restrictScalarsL
| null |
coe_restrict_scalarsL' : ⇑(restrictScalarsL 𝕜 E F 𝕜' 𝕜'') = restrictScalars 𝕜' :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
coe_restrict_scalarsL'
| null |
@[simps apply symm_apply toLinearEquiv_apply toLinearEquiv_symm_apply]
arrowCongrSL (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) :
(E →SL[σ₁₄] H) ≃SL[σ₄₃] F →SL[σ₂₃] G :=
{ e₁₂.arrowCongrEquiv e₄₃ with
toFun := fun L => (e₄₃ : H →SL[σ₄₃] G).comp (L.comp (e₁₂.symm : F →SL[σ₂₁] E))
invFun := fun L => (e₄₃.symm : G →SL[σ₃₄] H).comp (L.comp (e₁₂ : E →SL[σ₁₂] F))
map_add' := fun f g => by simp only [add_comp, comp_add]
map_smul' := fun t f => by simp only [smul_comp, comp_smulₛₗ]
continuous_toFun := ((postcomp F e₄₃.toContinuousLinearMap).comp
(precomp H e₁₂.symm.toContinuousLinearMap)).continuous
continuous_invFun := ((precomp H e₁₂.toContinuousLinearMap).comp
(postcomp F e₄₃.symm.toContinuousLinearMap)).continuous }
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
arrowCongrSL
|
A pair of continuous (semi)linear equivalences generates a (semi)linear equivalence between the
spaces of continuous (semi)linear maps.
|
arrowCongr (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) : (E →L[𝕜] H) ≃L[𝕜] F →L[𝕜] G :=
e₁.arrowCongrSL e₂
@[simp] lemma arrowCongr_apply (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) (f : E →L[𝕜] H) (x : F) :
e₁.arrowCongr e₂ f x = e₂ (f (e₁.symm x)) := rfl
@[simp] lemma arrowCongr_symm (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) :
(e₁.arrowCongr e₂).symm = e₁.symm.arrowCongr e₂.symm := rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
arrowCongr
|
A pair of continuous linear equivalences generates a continuous linear equivalence between
the spaces of continuous linear maps.
|
UniformFun.continuousSMul_induced_of_range_bounded (φ : hom)
(hφ : IsInducing (ofFun ∘ φ)) (h : ∀ u : H, Bornology.IsVonNBounded 𝕜 (Set.range (φ u))) :
ContinuousSMul 𝕜 H := by
have : IsTopologicalAddGroup H :=
let ofFun' : (α → E) →+ (α →ᵤ E) := AddMonoidHom.id _
IsInducing.topologicalAddGroup (ofFun'.comp (φ : H →+ (α → E))) hφ
have hb : (𝓝 (0 : H)).HasBasis (· ∈ 𝓝 (0 : E)) fun V ↦ {u | ∀ x, φ u x ∈ V} := by
simp only [hφ.nhds_eq_comap, Function.comp_apply, map_zero]
exact UniformFun.hasBasis_nhds_zero.comap _
apply ContinuousSMul.of_basis_zero hb
· intro U hU
have : Tendsto (fun x : 𝕜 × E ↦ x.1 • x.2) (𝓝 0) (𝓝 0) :=
continuous_smul.tendsto' _ _ (zero_smul _ _)
rcases ((Filter.basis_sets _).prod_nhds (Filter.basis_sets _)).tendsto_left_iff.1 this U hU
with ⟨⟨V, W⟩, ⟨hV, hW⟩, hVW⟩
refine ⟨V, hV, W, hW, Set.smul_subset_iff.2 fun a ha u hu x ↦ ?_⟩
rw [map_smul]
exact hVW (Set.mk_mem_prod ha (hu x))
· intro c U hU
have : Tendsto (c • · : E → E) (𝓝 0) (𝓝 0) :=
(continuous_const_smul c).tendsto' _ _ (smul_zero _)
refine ⟨_, this hU, fun u hu x ↦ ?_⟩
simpa only [map_smul] using hu x
· intro u U hU
simp only [Set.mem_setOf_eq, map_smul, Pi.smul_apply]
simpa only [Set.mapsTo_range_iff] using (h u hU).eventually_nhds_zero (mem_of_mem_nhds hU)
|
lemma
|
Topology
|
[
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence"
] |
Mathlib/Topology/Algebra/Module/UniformConvergence.lean
|
UniformFun.continuousSMul_induced_of_range_bounded
|
Let `E` be a topological vector space over a normed field `𝕜`, let `α` be any type.
Let `H` be a submodule of `α →ᵤ E` such that the range of each `f ∈ H` is von Neumann bounded.
Then `H` is a topological vector space over `𝕜`,
i.e., the pointwise scalar multiplication is continuous in both variables.
For convenience we require that `H` is a vector space over `𝕜`
with a topology induced by `UniformFun.ofFun ∘ φ`, where `φ : H →ₗ[𝕜] (α → E)`.
|
UniformOnFun.continuousSMul_induced_of_image_bounded (φ : hom) (hφ : IsInducing (ofFun 𝔖 ∘ φ))
(h : ∀ u : H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 ((φ u : α → E) '' s)) :
ContinuousSMul 𝕜 H := by
obtain rfl := hφ.eq_induced; clear hφ
simp only [induced_iInf, UniformOnFun.topologicalSpace_eq, induced_compose]
refine continuousSMul_iInf fun s ↦ continuousSMul_iInf fun hs ↦ ?_
letI : TopologicalSpace H :=
.induced (UniformFun.ofFun ∘ s.restrict ∘ φ) (UniformFun.topologicalSpace s E)
set φ' : H →ₗ[𝕜] (s → E) :=
{ toFun := s.restrict ∘ φ,
map_smul' := fun c x ↦ by exact congr_arg s.restrict (map_smul φ c x),
map_add' := fun x y ↦ by exact congr_arg s.restrict (map_add φ x y) }
refine UniformFun.continuousSMul_induced_of_range_bounded 𝕜 s E H φ' ⟨rfl⟩ fun u ↦ ?_
simpa only [Set.image_eq_range] using h u s hs
|
lemma
|
Topology
|
[
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence"
] |
Mathlib/Topology/Algebra/Module/UniformConvergence.lean
|
UniformOnFun.continuousSMul_induced_of_image_bounded
|
Let `E` be a TVS, `𝔖 : Set (Set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `Bornology.IsVonNBounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.
For convenience, we don't literally ask for `H : Submodule (α →ᵤ[𝔖] E)`. Instead, we prove the
result for any vector space `H` equipped with a linear inducing to `α →ᵤ[𝔖] E`, which is often
easier to use. We also state the `Submodule` version as
`UniformOnFun.continuousSMul_submodule_of_image_bounded`.
|
UniformOnFun.continuousSMul_submodule_of_image_bounded (H : Submodule 𝕜 (α →ᵤ[𝔖] E))
(h : ∀ u ∈ H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (u '' s)) :
@ContinuousSMul 𝕜 H _ _ ((UniformOnFun.topologicalSpace α E 𝔖).induced ((↑) : H → α →ᵤ[𝔖] E)) :=
UniformOnFun.continuousSMul_induced_of_image_bounded 𝕜 α E H
(LinearMap.id.domRestrict H : H →ₗ[𝕜] α → E) IsInducing.subtypeVal fun ⟨u, hu⟩ => h u hu
|
theorem
|
Topology
|
[
"Mathlib.Analysis.LocallyConvex.Bounded",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.UniformConvergence"
] |
Mathlib/Topology/Algebra/Module/UniformConvergence.lean
|
UniformOnFun.continuousSMul_submodule_of_image_bounded
|
Let `E` be a TVS, `𝔖 : Set (Set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `Bornology.IsVonNBounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.
If you have a hard time using this lemma, try the one above instead.
|
@[nolint unusedArguments]
WeakBilin [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F]
(_ : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) := E
deriving AddCommMonoid, Module 𝕜
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
WeakBilin
|
The space `E` equipped with the weak topology induced by the bilinear form `B`.
|
instAddCommGroup [CommSemiring 𝕜] [a : AddCommGroup E] [Module 𝕜 E] [AddCommMonoid F]
[Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : AddCommGroup (WeakBilin B) := a
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
instAddCommGroup
| null |
instIsScalarTower [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E] [Module 𝕜 E]
[AddCommMonoid F] [Module 𝕜 F] [SMul 𝕝 𝕜] [Module 𝕝 E] [s : IsScalarTower 𝕝 𝕜 E]
(B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : IsScalarTower 𝕝 𝕜 (WeakBilin B) := s
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
instIsScalarTower
| null |
instTopologicalSpace : TopologicalSpace (WeakBilin B) :=
TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
instTopologicalSpace
| null |
coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
continuous_induced_dom
@[fun_prop]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
coeFn_continuous
|
The coercion `(fun x y => B x y) : E → (F → 𝕜)` is continuous.
|
eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
eval_continuous
| null |
continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakBilin B}
(h : ∀ y, Continuous fun a => B (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
continuous_of_continuous_eval
| null |
isEmbedding {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
IsEmbedding fun (x : WeakBilin B) y => B x y :=
Function.Injective.isEmbedding_induced <| LinearMap.coe_injective.comp hB
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
isEmbedding
|
The coercion `(fun x y => B x y) : E → (F → 𝕜)` is an embedding.
|
tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by
rw [← tendsto_pi_nhds, (isEmbedding hB).tendsto_nhds_iff]
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
tendsto_iff_forall_eval_tendsto
| null |
instContinuousAdd [ContinuousAdd 𝕜] : ContinuousAdd (WeakBilin B) := by
refine ⟨continuous_induced_rng.2 ?_⟩
refine
cast (congr_arg _ ?_)
(((coeFn_continuous B).comp continuous_fst).add ((coeFn_continuous B).comp continuous_snd))
ext
simp only [Function.comp_apply, Pi.add_apply, map_add, LinearMap.add_apply]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
instContinuousAdd
|
Addition in `WeakBilin B` is continuous.
|
instContinuousSMul [ContinuousSMul 𝕜 𝕜] : ContinuousSMul 𝕜 (WeakBilin B) := by
refine ⟨continuous_induced_rng.2 ?_⟩
refine cast (congr_arg _ ?_) (continuous_fst.smul ((coeFn_continuous B).comp continuous_snd))
ext
simp only [Function.comp_apply, Pi.smul_apply, LinearMap.map_smulₛₗ, RingHom.id_apply,
LinearMap.smul_apply]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
instContinuousSMul
|
Scalar multiplication by `𝕜` on `WeakBilin B` is continuous.
|
eval [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] :
F →ₗ[𝕜] StrongDual 𝕜 (WeakBilin B) where
toFun f := ⟨B.flip f, by fun_prop⟩
map_add' _ _ := by ext; simp
map_smul' _ _ := by ext; simp
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
eval
|
Map `F` into the topological dual of `E` with the weak topology induced by `F`
|
instIsTopologicalAddGroup [ContinuousAdd 𝕜] : IsTopologicalAddGroup (WeakBilin B) where
toContinuousAdd := by infer_instance
continuous_neg := by
refine continuous_induced_rng.2 (continuous_pi_iff.mpr fun y => ?_)
refine cast (congr_arg _ ?_) (eval_continuous B (-y))
ext x
simp only [map_neg, Function.comp_apply, LinearMap.neg_apply]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.LinearAlgebra.BilinearMap"
] |
Mathlib/Topology/Algebra/Module/WeakBilin.lean
|
instIsTopologicalAddGroup
|
`WeakBilin B` is a `IsTopologicalAddGroup`, meaning that addition and negation are
continuous.
|
WeakDual (𝕜 E : Type*) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
[ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] :=
WeakBilin (topDualPairing 𝕜 E)
deriving AddCommMonoid, Module 𝕜, TopologicalSpace, ContinuousAdd, Inhabited,
FunLike, ContinuousLinearMapClass
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakDual
|
The weak star topology is the topology coarsest topology on `E →L[𝕜] 𝕜` such that all
functionals `fun v => v x` are continuous.
|
instMulAction (M) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜]
[ContinuousConstSMul M 𝕜] : MulAction M (WeakDual 𝕜 E) :=
ContinuousLinearMap.mulAction
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instMulAction
|
If a monoid `M` distributively continuously acts on `𝕜` and this action commutes with
multiplication on `𝕜`, then it acts on `WeakDual 𝕜 E`.
|
instDistribMulAction (M) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜]
[ContinuousConstSMul M 𝕜] : DistribMulAction M (WeakDual 𝕜 E) :=
ContinuousLinearMap.distribMulAction
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instDistribMulAction
|
If a monoid `M` distributively continuously acts on `𝕜` and this action commutes with
multiplication on `𝕜`, then it acts distributively on `WeakDual 𝕜 E`.
|
instModule' (R) [Semiring R] [Module R 𝕜] [SMulCommClass 𝕜 R 𝕜] [ContinuousConstSMul R 𝕜] :
Module R (WeakDual 𝕜 E) :=
ContinuousLinearMap.module
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instModule'
|
If `𝕜` is a topological module over a semiring `R` and scalar multiplication commutes with the
multiplication on `𝕜`, then `WeakDual 𝕜 E` is a module over `R`.
|
instContinuousConstSMul (M) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜]
[ContinuousConstSMul M 𝕜] : ContinuousConstSMul M (WeakDual 𝕜 E) :=
⟨fun m =>
continuous_induced_rng.2 <| (WeakBilin.coeFn_continuous (topDualPairing 𝕜 E)).const_smul m⟩
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instContinuousConstSMul
| null |
instContinuousSMul (M) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜]
[TopologicalSpace M] [ContinuousSMul M 𝕜] : ContinuousSMul M (WeakDual 𝕜 E) :=
⟨continuous_induced_rng.2 <|
continuous_fst.smul ((WeakBilin.coeFn_continuous (topDualPairing 𝕜 E)).comp continuous_snd)⟩
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instContinuousSMul
|
If a monoid `M` distributively continuously acts on `𝕜` and this action commutes with
multiplication on `𝕜`, then it continuously acts on `WeakDual 𝕜 E`.
|
coeFn_continuous : Continuous fun (x : WeakDual 𝕜 E) y => x y :=
continuous_induced_dom
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
coeFn_continuous
| null |
eval_continuous (y : E) : Continuous fun x : WeakDual 𝕜 E => x y :=
continuous_pi_iff.mp coeFn_continuous y
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
eval_continuous
| null |
continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakDual 𝕜 E}
(h : ∀ y, Continuous fun a => (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
continuous_of_continuous_eval
| null |
instT2Space [T2Space 𝕜] : T2Space (WeakDual 𝕜 E) :=
(WeakBilin.isEmbedding ContinuousLinearMap.coe_injective).t2Space
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instT2Space
| null |
instAddCommGroup : AddCommGroup (WeakDual 𝕜 E) :=
WeakBilin.instAddCommGroup (topDualPairing 𝕜 E)
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instAddCommGroup
| null |
instIsTopologicalAddGroup : IsTopologicalAddGroup (WeakDual 𝕜 E) :=
WeakBilin.instIsTopologicalAddGroup (topDualPairing 𝕜 E)
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instIsTopologicalAddGroup
| null |
WeakSpace (𝕜 E) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
[ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] :=
WeakBilin (topDualPairing 𝕜 E).flip
deriving AddCommMonoid, Module 𝕜, TopologicalSpace, ContinuousAdd
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakSpace
|
The weak topology is the topology coarsest topology on `E` such that all functionals
`fun x => v x` are continuous.
|
instModule' [CommSemiring 𝕝] [Module 𝕝 E] : Module 𝕝 (WeakSpace 𝕜 E) :=
WeakBilin.instModule' (topDualPairing 𝕜 E).flip
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instModule'
| null |
instIsScalarTower [CommSemiring 𝕝] [Module 𝕝 𝕜] [Module 𝕝 E] [IsScalarTower 𝕝 𝕜 E] :
IsScalarTower 𝕝 𝕜 (WeakSpace 𝕜 E) :=
WeakBilin.instIsScalarTower (topDualPairing 𝕜 E).flip
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instIsScalarTower
| null |
instContinuousSMul [ContinuousSMul 𝕜 𝕜] : ContinuousSMul 𝕜 (WeakSpace 𝕜 E) :=
WeakBilin.instContinuousSMul _
variable [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F]
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instContinuousSMul
| null |
map (f : E →L[𝕜] F) : WeakSpace 𝕜 E →L[𝕜] WeakSpace 𝕜 F :=
{ f with
cont :=
WeakBilin.continuous_of_continuous_eval _ fun l => WeakBilin.eval_continuous _ (l ∘L f) }
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
map
|
A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
their weak topologies.
|
map_apply (f : E →L[𝕜] F) (x : E) : WeakSpace.map f x = f x :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
map_apply
| null |
coe_map (f : E →L[𝕜] F) : (WeakSpace.map f : E → F) = f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
coe_map
| null |
toWeakSpace : E ≃ₗ[𝕜] WeakSpace 𝕜 E := LinearEquiv.refl 𝕜 E
variable (𝕜 E) in
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
toWeakSpace
|
There is a canonical map `E → WeakSpace 𝕜 E` (the "identity"
mapping). It is a linear equivalence.
|
toWeakSpaceCLM : E →L[𝕜] WeakSpace 𝕜 E where
__ := toWeakSpace 𝕜 E
cont := by
apply WeakBilin.continuous_of_continuous_eval
exact ContinuousLinearMap.continuous
variable (𝕜 E) in
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
toWeakSpaceCLM
|
For a topological vector space `E`, "identity mapping" `E → WeakSpace 𝕜 E` is continuous.
This definition implements it as a continuous linear map.
|
toWeakSpaceCLM_eq_toWeakSpace (x : E) :
toWeakSpaceCLM 𝕜 E x = toWeakSpace 𝕜 E x := by rfl
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
toWeakSpaceCLM_eq_toWeakSpace
| null |
toWeakSpaceCLM_bijective :
Function.Bijective (toWeakSpaceCLM 𝕜 E) :=
(toWeakSpace 𝕜 E).bijective
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
toWeakSpaceCLM_bijective
| null |
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