fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
idealOf_compl_singleton_isMaximal (x : X) : (idealOfSet 𝕜 ({x}ᶜ : Set X)).IsMaximal :=
(idealOfSet_isMaximal_iff 𝕜 (Closeds.singleton x).compl).mpr <| Opens.isCoatom_iff.mpr ⟨x, rfl⟩
variable {𝕜} | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOf_compl_singleton_isMaximal | null |
setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal] :
∃ x : X, setOfIdeal I = {x}ᶜ := by
have h : (idealOfSet 𝕜 (setOfIdeal I)).IsMaximal :=
(idealOfSet_ofIdeal_isClosed (inferInstance : IsClosed (I : Set C(X, 𝕜)))).symm ▸ hI
obtain ⟨x, hx⟩ := Opens.isCoatom_iff.1 ((idealOfSet_isMaximal_... | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | setOfIdeal_eq_compl_singleton | null |
ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :
I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 {x}ᶜ = I := by
refine
⟨?_, fun h =>
let ⟨x, hx⟩ := h
hx ▸ idealOf_compl_singleton_isMaximal 𝕜 x⟩
intro hI'
obtain ⟨x, hx⟩ := setOfIdeal_eq_compl_singleton I
exact
⟨x, by
... | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | ideal_isMaximal_iff | null |
continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜)) where
toFun x :=
⟨{ toFun := fun f => f x
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl
cont := continuous_eval_const x }, by
rw [CharacterSpace.eq_set_map_one_map_mul]; exact ⟨rfl, fun f g => rfl⟩⟩
continuous_toFun :... | def | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | continuousMapEval | The natural continuous map from a locally compact topological space `X` to the
`WeakDual.characterSpace 𝕜 C(X, 𝕜)` which sends `x : X` to point evaluation at `x`. |
continuousMapEval_apply_apply (x : X) (f : C(X, 𝕜)) : continuousMapEval X 𝕜 x f = f x :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | continuousMapEval_apply_apply | null |
continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
refine ⟨fun x y hxy => ?_, fun φ => ?_⟩
· contrapose! hxy
rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.IsClosed {x})
(isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
... | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | continuousMapEval_bijective | null |
noncomputable homeoEval : X ≃ₜ characterSpace 𝕜 C(X, 𝕜) :=
@Continuous.homeoOfEquivCompactToT2 _ _ _ _ _ _
{ Equiv.ofBijective _ (continuousMapEval_bijective X 𝕜) with toFun := continuousMapEval X 𝕜 }
(map_continuous (continuousMapEval X 𝕜)) | def | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | homeoEval | This is the natural homeomorphism between a compact Hausdorff space `X` and the
`WeakDual.characterSpace 𝕜 C(X, 𝕜)`. |
IccInclusionLeft : C(Icc a b, Icc a c) :=
.inclusion <| Icc_subset_Icc le_rfl Fact.out | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | IccInclusionLeft | The embedding into an interval from a sub-interval lying on the left, as a `ContinuousMap`. |
IccInclusionRight : C(Icc b c, Icc a c) :=
.inclusion <| Icc_subset_Icc Fact.out le_rfl | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | IccInclusionRight | The embedding into an interval from a sub-interval lying on the right, as a `ContinuousMap`. |
projIccCM : C(α, Icc a b) :=
⟨projIcc a b Fact.out, continuous_projIcc⟩ | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | projIccCM | The map `projIcc` from `α` onto an interval in `α`, as a `ContinuousMap`. |
IccExtendCM : C(C(Icc a b, E), C(α, E)) where
toFun f := f.comp projIccCM
continuous_toFun := continuous_precomp projIccCM
@[simp] | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | IccExtendCM | The extension operation from continuous maps on an interval to continuous maps on the whole
type, as a `ContinuousMap`. |
IccExtendCM_of_mem {f : C(Icc a b, E)} {x : α} (hx : x ∈ Icc a b) :
IccExtendCM f x = f ⟨x, hx⟩ := by
simp [IccExtendCM, projIccCM, projIcc, hx.1, hx.2] | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | IccExtendCM_of_mem | null |
noncomputable concat (f : C(Icc a b, E)) (g : C(Icc b c, E)) :
C(Icc a c, E) := by
by_cases hb : f ⊤ = g ⊥
· let h (t : α) : E := if t ≤ b then IccExtendCM f t else IccExtendCM g t
suffices Continuous h from ⟨fun t => h t, by fun_prop⟩
apply Continuous.if_le (by fun_prop) (by fun_prop) continuous_id con... | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concat | The concatenation of two continuous maps defined on adjacent intervals. If the values of the
functions on the common bound do not agree, this is defined as an arbitrarily chosen constant
map. See `concatCM` for the corresponding map on the subtype of compatible function pairs. |
concat_comp_IccInclusionLeft (hb : f ⊤ = g ⊥) :
(concat f g).comp IccInclusionLeft = f := by
ext x
simp [concat, IccExtendCM, hb, IccInclusionLeft, projIccCM, inclusion, x.2.2] | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concat_comp_IccInclusionLeft | null |
concat_comp_IccInclusionRight (hb : f ⊤ = g ⊥) :
(concat f g).comp IccInclusionRight = g := by
ext ⟨x, hx⟩
obtain rfl | hxb := eq_or_ne x b
· simpa [concat, IccInclusionRight, IccExtendCM, projIccCM, inclusion, hb]
· have h : ¬ x ≤ b := lt_of_le_of_ne hx.1 (Ne.symm hxb) |>.not_ge
simp [concat, hb, IccIn... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concat_comp_IccInclusionRight | null |
concat_left (hb : f ⊤ = g ⊥) {t : Icc a c} (ht : t ≤ b) :
concat f g t = f ⟨t, t.2.1, ht⟩ := by
nth_rewrite 2 [← concat_comp_IccInclusionLeft hb]
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concat_left | null |
concat_right (hb : f ⊤ = g ⊥) {t : Icc a c} (ht : b ≤ t) :
concat f g t = g ⟨t, ht, t.2.2⟩ := by
nth_rewrite 2 [← concat_comp_IccInclusionRight hb]
rfl | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concat_right | null |
tendsto_concat {ι : Type*} {p : Filter ι} {F : ι → C(Icc a b, E)} {G : ι → C(Icc b c, E)}
(hfg : ∀ᶠ i in p, (F i) ⊤ = (G i) ⊥) (hfg' : f ⊤ = g ⊥)
(hf : Tendsto F p (𝓝 f)) (hg : Tendsto G p (𝓝 g)) :
Tendsto (fun i => concat (F i) (G i)) p (𝓝 (concat f g)) := by
rw [tendsto_nhds_compactOpen] at hf hg ⊢
... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | tendsto_concat | null |
noncomputable concatCM :
C({fg : C(Icc a b, E) × C(Icc b c, E) // fg.1 ⊤ = fg.2 ⊥}, C(Icc a c, E)) where
toFun fg := concat fg.val.1 fg.val.2
continuous_toFun := by
let S : Set (C(Icc a b, E) × C(Icc b c, E)) := {fg | fg.1 ⊤ = fg.2 ⊥}
change Continuous (S.restrict concat.uncurry)
refine continuousOn... | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concatCM | The concatenation of compatible pairs of continuous maps on adjacent intervals, defined as a
`ContinuousMap` on a subtype of the product. |
concatCM_left {x : Icc a c} (hx : x ≤ b)
{fg : {fg : C(Icc a b, E) × C(Icc b c, E) // fg.1 ⊤ = fg.2 ⊥}} :
concatCM fg x = fg.1.1 ⟨x.1, x.2.1, hx⟩ := by
exact concat_left fg.2 hx
@[simp] | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concatCM_left | null |
concatCM_right {x : Icc a c} (hx : b ≤ x)
{fg : {fg : C(Icc a b, E) × C(Icc b c, E) // fg.1 ⊤ = fg.2 ⊥}} :
concatCM fg x = fg.1.2 ⟨x.1, hx, x.2.2⟩ :=
concat_right fg.2 hx | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Order.ProjIcc"
] | Mathlib/Topology/ContinuousMap/Interval.lean | concatCM_right | null |
@[to_additive (attr := simp, norm_cast)]
coe_mabs (f : C(α, β)) : ⇑|f|ₘ = |⇑f|ₘ := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Ordered"
] | Mathlib/Topology/ContinuousMap/Lattice.lean | coe_mabs | null |
mabs_apply (f : C(α, β)) (x : α) : |f|ₘ x = |f x|ₘ := rfl | lemma | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Ordered"
] | Mathlib/Topology/ContinuousMap/Lattice.lean | mabs_apply | null |
@[to_additive (attr := simps) /-- The inclusion of locally-constant functions into continuous
functions as an additive monoid hom. -/]
toContinuousMapMonoidHom [Monoid Y] [ContinuousMul Y] : LocallyConstant X Y →* C(X, Y) where
toFun := (↑)
map_one' := by
ext
simp
map_mul' x y := by
ext
simp | def | Topology | [
"Mathlib.Topology.LocallyConstant.Algebra",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/LocallyConstant.lean | toContinuousMapMonoidHom | The inclusion of locally-constant functions into continuous functions as a multiplicative
monoid hom. |
@[simps]
toContinuousMapLinearMap (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y]
[ContinuousAdd Y] [ContinuousConstSMul R Y] : LocallyConstant X Y →ₗ[R] C(X, Y) where
toFun := (↑)
map_add' x y := by
ext
simp
map_smul' x y := by
ext
simp | def | Topology | [
"Mathlib.Topology.LocallyConstant.Algebra",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/LocallyConstant.lean | toContinuousMapLinearMap | The inclusion of locally-constant functions into continuous functions as a linear map. |
@[simps]
toContinuousMapAlgHom (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y]
[IsTopologicalSemiring Y] : LocallyConstant X Y →ₐ[R] C(X, Y) where
toFun := (↑)
map_one' := by
ext
simp
map_mul' x y := by
ext
simp
map_zero' := by
ext
simp
map_add' x y := by
ext
simp
... | def | Topology | [
"Mathlib.Topology.LocallyConstant.Algebra",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/LocallyConstant.lean | toContinuousMapAlgHom | The inclusion of locally-constant functions into continuous functions as an algebra map. |
ContinuousMap.instLocallyConvexSpace {X 𝕜 E : Type*}
[TopologicalSpace X]
[Semiring 𝕜] [PartialOrder 𝕜]
[AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [LocallyConvexSpace 𝕜 E]
[IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] :
LocallyConvexSpace 𝕜 C(X, E) :=
.ofBasisZero _ _ _ _ (Loc... | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.Algebra.Module.LocallyConvex"
] | Mathlib/Topology/ContinuousMap/LocallyConvex.lean | ContinuousMap.instLocallyConvexSpace | null |
partialOrder [PartialOrder β] : PartialOrder C(α, β) :=
PartialOrder.lift (fun f => f.toFun) (fun f g _ => by aesop) | instance | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | partialOrder | null |
le_def [PartialOrder β] {f g : C(α, β)} : f ≤ g ↔ ∀ a, f a ≤ g a :=
Pi.le_def | theorem | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | le_def | null |
lt_def [PartialOrder β] {f g : C(α, β)} : f < g ↔ (∀ a, f a ≤ g a) ∧ ∃ a, f a < g a :=
Pi.lt_def | theorem | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | lt_def | null |
sup : Max C(α, β) where max f g := { toFun := fun a ↦ f a ⊔ g a }
@[simp, norm_cast] lemma coe_sup (f g : C(α, β)) : ⇑(f ⊔ g) = ⇑f ⊔ g := rfl
@[simp] lemma sup_apply (f g : C(α, β)) (a : α) : (f ⊔ g) a = f a ⊔ g a := rfl | instance | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | sup | null |
semilatticeSup : SemilatticeSup C(α, β) :=
DFunLike.coe_injective.semilatticeSup _ fun _ _ ↦ rfl | instance | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | semilatticeSup | null |
sup'_apply {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) (a : α) :
s.sup' H f a = s.sup' H fun i ↦ f i a :=
Finset.comp_sup'_eq_sup'_comp H (fun g : C(α, β) ↦ g a) fun _ _ ↦ rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | sup'_apply | null |
coe_sup' {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) :
⇑(s.sup' H f) = s.sup' H fun i ↦ ⇑(f i) := by ext; simp [sup'_apply] | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | coe_sup' | null |
inf : Min C(α, β) where min f g := { toFun := fun a ↦ f a ⊓ g a }
@[simp, norm_cast] lemma coe_inf (f g : C(α, β)) : ⇑(f ⊓ g) = ⇑f ⊓ g := rfl
@[simp] lemma inf_apply (f g : C(α, β)) (a : α) : (f ⊓ g) a = f a ⊓ g a := rfl | instance | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | inf | null |
semilatticeInf : SemilatticeInf C(α, β) :=
DFunLike.coe_injective.semilatticeInf _ fun _ _ ↦ rfl | instance | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | semilatticeInf | null |
inf'_apply {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) (a : α) :
s.inf' H f a = s.inf' H fun i ↦ f i a :=
Finset.comp_inf'_eq_inf'_comp H (fun g : C(α, β) ↦ g a) fun _ _ ↦ rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | inf'_apply | null |
coe_inf' {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) :
⇑(s.inf' H f) = s.inf' H fun i ↦ ⇑(f i) := by ext; simp [inf'_apply] | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | coe_inf' | null |
IccExtend (f : C(Set.Icc a b, β)) : C(α, β) where
toFun := Set.IccExtend h f
@[simp] | def | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | IccExtend | Extend a continuous function `f : C(Set.Icc a b, β)` to a function `f : C(α, β)`. |
coe_IccExtend (f : C(Set.Icc a b, β)) :
((IccExtend h f : C(α, β)) : α → β) = Set.IccExtend h f :=
rfl | theorem | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/ContinuousMap/Ordered.lean | coe_IccExtend | null |
periodic_tsum_comp_add_zsmul [AddCommGroup X] [ContinuousAdd X] [AddCommMonoid Y]
[ContinuousAdd Y] [T2Space Y] (f : C(X, Y)) (p : X) :
Function.Periodic (⇑(∑' n : ℤ, f.comp (ContinuousMap.addRight (n • p)))) p := by
intro x
by_cases h : Summable fun n : ℤ => f.comp (ContinuousMap.addRight (n • p))
· conv... | theorem | Topology | [
"Mathlib.Algebra.Ring.Periodic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Periodic.lean | periodic_tsum_comp_add_zsmul | Summing the translates of `f` by `ℤ • p` gives a map which is periodic with period `p`.
(This is true without any convergence conditions, since if the sum doesn't converge it is taken to
be the zero map, which is periodic.) |
@[simps]
toContinuousMap (p : R[X]) : C(R, R) :=
⟨fun x : R => p.eval x, by fun_prop⟩
open ContinuousMap in | def | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | toContinuousMap | Every polynomial with coefficients in a topological semiring gives a (bundled) continuous function. |
toContinuousMap_X_eq_id : X.toContinuousMap = .id R := by
ext; simp | lemma | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | toContinuousMap_X_eq_id | null |
@[simps]
toContinuousMapOn (p : R[X]) (X : Set R) : C(X, R) :=
⟨fun x : X => p.toContinuousMap x, by fun_prop⟩
open ContinuousMap in | def | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | toContinuousMapOn | A polynomial as a continuous function,
with domain restricted to some subset of the semiring of coefficients.
(This is particularly useful when restricting to compact sets, e.g. `[0,1]`.) |
toContinuousMapOn_X_eq_restrict_id (s : Set R) :
X.toContinuousMapOn s = restrict s (.id R) := by
ext; simp | lemma | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | toContinuousMapOn_X_eq_restrict_id | null |
@[simp]
aeval_continuousMap_apply (g : R[X]) (f : C(α, R)) (x : α) :
((Polynomial.aeval f) g) x = g.eval (f x) := by
refine Polynomial.induction_on' g ?_ ?_
· intro p q hp hq
simp [hp, hq]
· intro n a
simp | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | aeval_continuousMap_apply | null |
@[simps]
toContinuousMapAlgHom : R[X] →ₐ[R] C(R, R) where
toFun p := p.toContinuousMap
map_zero' := by
ext
simp
map_add' _ _ := by
ext
simp
map_one' := by
ext
simp
map_mul' _ _ := by
ext
simp
commutes' _ := by
ext
simp [Algebra.algebraMap_eq_smul_one] | def | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | toContinuousMapAlgHom | The algebra map from `R[X]` to continuous functions `C(R, R)`. |
@[simps]
toContinuousMapOnAlgHom (X : Set R) : R[X] →ₐ[R] C(X, R) where
toFun p := p.toContinuousMapOn X
map_zero' := by
ext
simp
map_add' _ _ := by
ext
simp
map_one' := by
ext
simp
map_mul' _ _ := by
ext
simp
commutes' _ := by
ext
simp [Algebra.algebraMap_eq_smul_one... | def | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | toContinuousMapOnAlgHom | The algebra map from `R[X]` to continuous functions `C(X, R)`, for any subset `X` of `R`. |
noncomputable
polynomialFunctions (X : Set R) : Subalgebra R C(X, R) :=
(⊤ : Subalgebra R R[X]).map (Polynomial.toContinuousMapOnAlgHom X)
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions | The subalgebra of polynomial functions in `C(X, R)`, for `X` a subset of some topological semiring
`R`. |
polynomialFunctions_coe (X : Set R) :
(polynomialFunctions X : Set C(X, R)) = Set.range (Polynomial.toContinuousMapOnAlgHom X) := by
ext
simp [polynomialFunctions] | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions_coe | null |
polynomialFunctions_separatesPoints (X : Set R) : (polynomialFunctions X).SeparatesPoints :=
fun x y h => by
refine ⟨_, ⟨⟨_, ⟨⟨Polynomial.X, ⟨Algebra.mem_top, rfl⟩⟩, rfl⟩⟩, ?_⟩⟩
dsimp; simp only [Polynomial.eval_X]
exact fun h' => h (Subtype.ext h')
open unitInterval
open ContinuousMap | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions_separatesPoints | null |
polynomialFunctions.comap_compRightAlgHom_iccHomeoI (a b : ℝ) (h : a < b) :
(polynomialFunctions I).comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm) =
polynomialFunctions (Set.Icc a b) := by
ext f
fconstructor
· rintro ⟨p, ⟨-, w⟩⟩
rw [DFunLike.ext_iff] at w
dsimp at w
let q := p.comp ((b -... | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions.comap_compRightAlgHom_iccHomeoI | The preimage of polynomials on `[0,1]` under the pullback map by `x ↦ (b-a) * x + a`
is the polynomials on `[a,b]`. |
polynomialFunctions.eq_adjoin_X (s : Set R) :
polynomialFunctions s = Algebra.adjoin R {toContinuousMapOnAlgHom s X} := by
refine le_antisymm ?_
(Algebra.adjoin_le fun _ h => ⟨X, trivial, (Set.mem_singleton_iff.1 h).symm⟩)
rintro - ⟨p, -, rfl⟩
rw [AlgHom.coe_toRingHom]
refine p.induction_on (fun r => ?_... | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions.eq_adjoin_X | null |
polynomialFunctions.le_equalizer {A : Type*} [Semiring A] [Algebra R A] (s : Set R)
(φ ψ : C(s, R) →ₐ[R] A)
(h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) :
polynomialFunctions s ≤ AlgHom.equalizer φ ψ := by
rw [polynomialFunctions.eq_adjoin_X s]
exact φ.adjoin_le_equalizer ψ fu... | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions.le_equalizer | null |
polynomialFunctions.starClosure_eq_adjoin_X [StarRing R] [ContinuousStar R] (s : Set R) :
(polynomialFunctions s).starClosure = adjoin R {toContinuousMapOnAlgHom s X} := by
rw [polynomialFunctions.eq_adjoin_X s, adjoin_eq_starClosure_adjoin] | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions.starClosure_eq_adjoin_X | null |
polynomialFunctions.starClosure_le_equalizer {A : Type*} [StarRing R] [ContinuousStar R]
[Semiring A] [StarRing A] [Algebra R A] (s : Set R) (φ ψ : C(s, R) →⋆ₐ[R] A)
(h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) :
(polynomialFunctions s).starClosure ≤ StarAlgHom.equalizer φ ψ := by... | theorem | Topology | [
"Mathlib.Topology.Algebra.Polynomial",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UnitInterval",
"Mathlib.Algebra.Star.Subalgebra"
] | Mathlib/Topology/ContinuousMap/Polynomial.lean | polynomialFunctions.starClosure_le_equalizer | null |
compactOpen_eq_generateFrom {S : Set (Set X)} {T : Set (Set Y)}
(hS₁ : ∀ K ∈ S, IsCompact K) (hT : IsTopologicalBasis T)
(hS₂ : ∀ f : C(X, Y), ∀ x, ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo f K V) :
compactOpen = .generateFrom (.image2 (fun K t ↦
{f : C(X, Y) | MapsTo f K (⋃₀ t)}) S {t : Set (Set ... | theorem | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/SecondCountableSpace.lean | compactOpen_eq_generateFrom | null |
secondCountableTopology [SecondCountableTopology Y]
(hX : ∃ S : Set (Set X), S.Countable ∧ (∀ K ∈ S, IsCompact K) ∧
∀ f : C(X, Y), ∀ V, IsOpen V → ∀ x ∈ f ⁻¹' V, ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo f K V) :
SecondCountableTopology C(X, Y) where
is_open_generated_countable := by
rcases hX with ⟨S, hScount, hS... | theorem | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/SecondCountableSpace.lean | secondCountableTopology | A version of `instSecondCountableTopology` with a technical assumption
instead of `[SecondCountableTopology X] [LocallyCompactSpace X]`.
It is here as a reminder of what could be an intermediate goal,
if someone tries to weaken the assumptions in the instance
(e.g., from `[LocallyCompactSpace X]` to `[LocallyCompactPai... |
instSecondCountableTopology [SecondCountableTopology X] [LocallyCompactSpace X]
[SecondCountableTopology Y] : SecondCountableTopology C(X, Y) := by
apply secondCountableTopology
have (U : countableBasis X) : LocallyCompactSpace U.1 :=
(isOpen_of_mem_countableBasis U.2).locallyCompactSpace
set K := fun U :... | instance | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/SecondCountableSpace.lean | instSecondCountableTopology | null |
instSeparableSpace [SecondCountableTopology X] [LocallyCompactSpace X]
[SecondCountableTopology Y] : SeparableSpace C(X, Y) :=
inferInstance | instance | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/SecondCountableSpace.lean | instSeparableSpace | null |
isEmbedding_sigmaMk_comp [Nonempty X] :
IsEmbedding (fun g : Σ i, C(X, Y i) ↦ (sigmaMk g.1).comp g.2) where
toIsInducing := inducing_sigma.2
⟨fun i ↦ (sigmaMk i).isInducing_postcomp IsEmbedding.sigmaMk.isInducing, fun i ↦
let ⟨x⟩ := ‹Nonempty X›
⟨_, (isOpen_sigma_fst_preimage {i}).preimage (contin... | theorem | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/Sigma.lean | isEmbedding_sigmaMk_comp | null |
exists_lift_sigma (f : C(X, Σ i, Y i)) : ∃ i g, f = (sigmaMk i).comp g :=
let ⟨i, g, hg, hfg⟩ := (map_continuous f).exists_lift_sigma
⟨i, ⟨g, hg⟩, DFunLike.ext' hfg⟩
variable (X Y) | theorem | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/Sigma.lean | exists_lift_sigma | Every continuous map from a connected topological space to the disjoint union of a family of
topological spaces is a composition of the embedding `ContinuousMap.sigmaMk i : C(Y i, Σ i, Y i)`
for some `i` and a continuous map `g : C(X, Y i)`. See also `Continuous.exists_lift_sigma` for a
version with unbundled functions... |
@[simps! symm_apply]
sigmaCodHomeomorph : C(X, Σ i, Y i) ≃ₜ Σ i, C(X, Y i) :=
.symm <| Equiv.toHomeomorphOfIsInducing
(.ofBijective _ ⟨isEmbedding_sigmaMk_comp.injective, fun f ↦
let ⟨i, g, hg⟩ := f.exists_lift_sigma; ⟨⟨i, g⟩, hg.symm⟩⟩)
isEmbedding_sigmaMk_comp.isInducing | def | Topology | [
"Mathlib.Topology.CompactOpen"
] | Mathlib/Topology/ContinuousMap/Sigma.lean | sigmaCodHomeomorph | Homeomorphism between the type `C(X, Σ i, Y i)` of continuous maps from a connected topological
space to the disjoint union of a family of topological spaces and the disjoint union of the types of
continuous maps `C(X, Y i)`.
The inverse map sends `⟨i, g⟩` to `ContinuousMap.comp (ContinuousMap.sigmaMk i) g`. |
@[simp]
coe_star (f : C(α, β)) : ⇑(star f) = star (⇑f) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | coe_star | null |
star_apply (f : C(α, β)) (x : α) : star f x = star (f x) :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | star_apply | null |
instTrivialStar [TrivialStar β] : TrivialStar C(α, β) where
star_trivial _ := ext fun _ => star_trivial _ | instance | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | instTrivialStar | null |
starAddMonoid [AddMonoid β] [ContinuousAdd β] [StarAddMonoid β] [ContinuousStar β] :
StarAddMonoid C(α, β) where
star_add _ _ := ext fun _ => star_add _ _ | instance | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | starAddMonoid | null |
starMul [Mul β] [ContinuousMul β] [StarMul β] [ContinuousStar β] :
StarMul C(α, β) where
star_mul _ _ := ext fun _ => star_mul _ _ | instance | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | starMul | null |
@[simps]
compStarAlgHom' (f : C(X, Y)) : C(Y, A) →⋆ₐ[𝕜] C(X, A) where
toFun g := g.comp f
map_one' := one_comp _
map_mul' _ _ := rfl
map_zero' := zero_comp f
map_add' _ _ := rfl
commutes' _ := rfl
map_star' _ := rfl | def | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgHom' | The functorial map taking `f : C(X, Y)` to `C(Y, A) →⋆ₐ[𝕜] C(X, A)` given by pre-composition
with the continuous function `f`. See `ContinuousMap.compMonoidHom'` and
`ContinuousMap.compAddMonoidHom'`, `ContinuousMap.compRightAlgHom` for bundlings of
pre-composition into a `MonoidHom`, an `AddMonoidHom` and an `AlgHom`... |
compStarAlgHom'_id : compStarAlgHom' 𝕜 A (ContinuousMap.id X) = StarAlgHom.id 𝕜 C(X, A) :=
StarAlgHom.ext fun _ => ContinuousMap.ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgHom'_id | `ContinuousMap.compStarAlgHom'` sends the identity continuous map to the identity
`StarAlgHom` |
compStarAlgHom'_comp (g : C(Y, Z)) (f : C(X, Y)) :
compStarAlgHom' 𝕜 A (g.comp f) = (compStarAlgHom' 𝕜 A f).comp (compStarAlgHom' 𝕜 A g) :=
StarAlgHom.ext fun _ => ContinuousMap.ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgHom'_comp | `ContinuousMap.compStarAlgHom'` is functorial. |
@[simps]
compStarAlgHom (φ : A →⋆ₐ[𝕜] B) (hφ : Continuous φ) :
C(X, A) →⋆ₐ[𝕜] C(X, B) where
toFun f := (⟨φ, hφ⟩ : C(A, B)).comp f
map_one' := ext fun _ => map_one φ
map_mul' f g := ext fun x => map_mul φ (f x) (g x)
map_zero' := ext fun _ => map_zero φ
map_add' f g := ext fun x => map_add φ (f x) (g x)
... | def | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgHom | Post-composition with a continuous star algebra homomorphism is a star algebra homomorphism
between spaces of continuous maps. |
compStarAlgHom_id : compStarAlgHom X (.id 𝕜 A) continuous_id = .id 𝕜 C(X, A) := rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgHom_id | `ContinuousMap.compStarAlgHom` sends the identity `StarAlgHom` on `A` to the identity
`StarAlgHom` on `C(X, A)`. |
compStarAlgHom_comp (φ : A →⋆ₐ[𝕜] B) (ψ : B →⋆ₐ[𝕜] C) (hφ : Continuous φ)
(hψ : Continuous ψ) : compStarAlgHom X (ψ.comp φ) (hψ.comp hφ) =
(compStarAlgHom X ψ hψ).comp (compStarAlgHom X φ hφ) :=
rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgHom_comp | `ContinuousMap.compStarAlgHom` is functorial. |
@[simps]
compStarAlgEquiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) :=
{ (f : C(X, Y)).compStarAlgHom' 𝕜 A with
toFun := (f : C(X, Y)).compStarAlgHom' 𝕜 A
invFun := (f.symm : C(Y, X)).compStarAlgHom' 𝕜 A
left_inv := fun g => by
simp only [ContinuousMap.compStarAlgHom'_apply, ContinuousMap.comp_asso... | def | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | compStarAlgEquiv' | `ContinuousMap.compStarAlgHom'` as a `StarAlgEquiv` when the continuous map `f` is
actually a homeomorphism. |
@[simps!]
ContinuousMap.evalStarAlgHom [StarRing R] [ContinuousStar R] (x : X) :
C(X, R) →⋆ₐ[S] R :=
{ ContinuousMap.evalAlgHom S R x with
map_star' := fun _ => rfl } | def | Topology | [
"Mathlib.Topology.Algebra.Star",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Star.lean | ContinuousMap.evalStarAlgHom | Evaluation of continuous maps at a point, bundled as a star algebra homomorphism. |
ContinuousSqrt (R : Type*) [LE R] [NonUnitalSemiring R] [TopologicalSpace R] where
/-- `sqrt (a, b)` returns a value `s` such that `b = a + s * s` when `a ≤ b`. -/
protected sqrt : R × R → R
protected continuousOn_sqrt : ContinuousOn sqrt {x | x.1 ≤ x.2}
protected sqrt_nonneg (x : R × R) : x.1 ≤ x.2 → 0 ≤ sqrt ... | class | Topology | [
"Mathlib.Algebra.Order.Star.Basic",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Ordered"
] | Mathlib/Topology/ContinuousMap/StarOrdered.lean | ContinuousSqrt | A type class encoding the property that there is a continuous square root function on
nonnegative elements. This holds for `ℝ≥0`, `ℝ` and `ℂ` (as well as any C⋆-algebra), and this
allows us to derive an instance of `StarOrderedRing C(α, R)` under appropriate hypotheses.
In order for this to work on `ℝ≥0`, we actually m... |
instStarOrderedRing {R : Type*}
[TopologicalSpace R] [CommSemiring R] [PartialOrder R] [NoZeroDivisors R] [StarRing R]
[StarOrderedRing R] [IsTopologicalSemiring R] [ContinuousStar R] [StarOrderedRing C(α, R)] :
StarOrderedRing C(α, R)₀ where
le_iff f g := by
constructor
· rw [le_def, ← Continuous... | instance | Topology | [
"Mathlib.Algebra.Order.Star.Basic",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Ordered"
] | Mathlib/Topology/ContinuousMap/StarOrdered.lean | instStarOrderedRing | null |
attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖) where
toFun x := ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩
@[simp] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | attachBound | Turn a function `f : C(X, ℝ)` into a continuous map into `Set.Icc (-‖f‖) (‖f‖)`,
thereby explicitly attaching bounds. |
attachBound_apply_coe (f : C(X, ℝ)) (x : X) : ((attachBound f) x : ℝ) = f x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | attachBound_apply_coe | null |
polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =
Polynomial.aeval f g := by
ext
simp only [Polynomial.aeval_subalgebra_coe, Polynomial.aeval_continuousMap_apply]
simp | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | polynomial_comp_attachBound | null |
polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A := by
rw [polynomial_comp_attachBound]
apply SetLike.coe_mem | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | polynomial_comp_attachBound_mem | Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial
gives another function in `A`.
This lemma proves something slightly more subtle than this:
we take `f`, and think of it as a function into the restricted target `Set.Icc (-‖f‖) ‖f‖)`,
and then postcompose with a polynomial func... |
comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
(p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure := by
have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure :=
continuousMap_mem_polynomialFunctions_closure _ _ p
have fr... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | comp_attachBound_mem_closure | null |
abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
|(f : C(X, ℝ))| ∈ A.topologicalClosure := by
let f' := attachBound (f : C(X, ℝ))
let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => |(x : ℝ)| }
change abs.comp f' ∈ A.topologicalClosure
apply comp_attachBound_mem_clo... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | abs_mem_subalgebra_closure | null |
inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by
rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.sub_mem
(A.topologicalClosure.add_mem (A.le_topologicalClosure f.prop... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | inf_mem_subalgebra_closure | null |
inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A := by
convert inf_mem_subalgebra_closure A f g
apply SetLike.ext'
symm
rw [Subalgebra.topologicalClosure_coe, closure_eq_iff_isClosed]
exact h | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | inf_mem_closed_subalgebra | null |
sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure := by
rw [sup_eq_half_smul_add_add_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.add_mem
(A.topologicalClosure.add_mem (A.le_topologicalClosure f.prop... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | sup_mem_subalgebra_closure | null |
sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A := by
convert sup_mem_subalgebra_closure A f g
apply SetLike.ext'
symm
dsimp
rw [closure_eq_iff_isClosed]
exact h
open scoped Topology | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | sup_mem_closed_subalgebra | null |
sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
(inf_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊓ g ∈ L)
(sup_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly) :
closure L = ⊤ := by
rw [eq_top_iff]
rintro f -
refine
Filter.Frequently.mem_closure
((Filter.HasBasis.frequent... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | sublattice_closure_eq_top | null |
subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by
apply SetLike.ext'
let L := A.topologicalClosure
have n : Set.Nonempty (L : Set C(X, ℝ)) := ⟨(1 : C(X, ℝ)), A.le_topologicalClosure A.one_mem⟩
convert
sublattice_cl... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | subalgebra_topologicalClosure_eq_top_of_separatesPoints | The **Stone-Weierstrass Approximation Theorem**,
that a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space,
is dense if it separates points. |
continuousMap_mem_subalgebra_closure_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) (f : C(X, ℝ)) : f ∈ A.topologicalClosure := by
rw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w]
simp | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | continuousMap_mem_subalgebra_closure_of_separatesPoints | An alternative statement of the Stone-Weierstrass theorem.
If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
every real-valued continuous function on `X` is a uniform limit of elements of `A`. |
exists_mem_subalgebra_near_continuousMap_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) (f : C(X, ℝ)) (ε : ℝ) (pos : 0 < ε) :
∃ g : A, ‖(g : C(X, ℝ)) - f‖ < ε := by
have w :=
mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f)
rw [Metric.n... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | exists_mem_subalgebra_near_continuousMap_of_separatesPoints | An alternative statement of the Stone-Weierstrass theorem,
for those who like their epsilons.
If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
every real-valued continuous function on `X` is within any `ε > 0` of some element of `A`. |
exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) (f : X → ℝ) (c : Continuous f) (ε : ℝ) (pos : 0 < ε) :
∃ g : A, ∀ x, ‖(g : X → ℝ) x - f x‖ < ε := by
obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuousMap_of_separatesPoints A w ⟨f, c⟩ ε pos
use g... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | exists_mem_subalgebra_near_continuous_of_separatesPoints | An alternative statement of the Stone-Weierstrass theorem,
for those who like their epsilons and don't like bundled continuous functions.
If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
every real-valued continuous function on `X` is within any `ε > 0` of some element of `A`. |
exists_mem_subalgebra_near_continuous_of_isCompact_of_separatesPoints
{X : Type*} [TopologicalSpace X] {A : Subalgebra ℝ C(X, ℝ)} (hA : A.SeparatesPoints)
(f : C(X, ℝ)) {K : Set X} (hK : IsCompact K) {ε : ℝ} (pos : 0 < ε) :
∃ g ∈ A, ∀ x ∈ K, ‖(g : X → ℝ) x - f x‖ < ε := by
let restrict_on_K : C(X, ℝ) →⋆ₐ[... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | exists_mem_subalgebra_near_continuous_of_isCompact_of_separatesPoints | A variant of the Stone-Weierstrass theorem where `X` need not be compact:
If `A` is a subalgebra of `C(X, ℝ)` which separates points, then, for any compact set `K ⊆ X`,
every real-valued continuous function on `X` is within any `ε > 0` of some element of `A` on `K`. |
Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
(hA : A.SeparatesPoints) :
((A.restrictScalars ℝ).comap
(ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints := by
intro x₁ x₂ hx
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx
let F : C(X, 𝕜) := f - const _ (f x₂)... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | Subalgebra.SeparatesPoints.rclike_to_real | If a star subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
of its purely real-valued elements also separates points. |
ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
(A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by
rw [StarSubalgebra.eq_top_iff]
let I : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ofRealCLM.compLeftContinuous ℝ X
have key : LinearMap.range I ≤ (A.toSubmodule.r... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints | The Stone-Weierstrass approximation theorem, `RCLike` version, that a star subalgebra `A` of
`C(X, 𝕜)`, where `X` is a compact topological space and `RCLike 𝕜`, is dense if it separates
points. |
polynomialFunctions.topologicalClosure (s : Set ℝ)
[CompactSpace s] : (polynomialFunctions s).topologicalClosure = ⊤ :=
ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints _
(polynomialFunctions_separatesPoints s) | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | polynomialFunctions.topologicalClosure | Polynomial functions in are dense in `C(s, ℝ)` when `s` is compact.
See `polynomialFunctions_closure_eq_top` for the special case `s = Set.Icc a b` which does not use
the full Stone-Weierstrass theorem. Of course, that version could be used to prove this one as
well. |
polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type*} [RCLike 𝕜] (s : Set 𝕜)
[CompactSpace s] : (polynomialFunctions s).starClosure.topologicalClosure = ⊤ :=
ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints _
(Subalgebra.separatesPoints_monotone le_sup_left (polynomialFunc... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | polynomialFunctions.starClosure_topologicalClosure | The star subalgebra generated by polynomials functions is dense in `C(s, 𝕜)` when `s` is
compact and `𝕜` is either `ℝ` or `ℂ`. |
ContinuousMap.elemental_id_eq_top {𝕜 : Type*} [RCLike 𝕜] (s : Set 𝕜) [CompactSpace s] :
elemental 𝕜 (ContinuousMap.restrict s (.id 𝕜)) = ⊤ := by
rw [StarAlgebra.elemental, ← polynomialFunctions.starClosure_topologicalClosure,
polynomialFunctions.starClosure_eq_adjoin_X]
congr
exact Polynomial.toConti... | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMap.elemental_id_eq_top | null |
@[elab_as_elim]
ContinuousMap.induction_on {𝕜 : Type*} [RCLike 𝕜] {s : Set 𝕜}
{p : C(s, 𝕜) → Prop} (const : ∀ r, p (.const s r)) (id : p (.restrict s <| .id 𝕜))
(star_id : p (star (.restrict s <| .id 𝕜)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(closure : (∀ f ∈ (... | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMap.induction_on | An induction principle for `C(s, 𝕜)`. |
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