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 ⌀ |
|---|---|---|---|---|---|---|
norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | norm_eq_iSup_norm | null |
enorm_eq_iSup_enorm : ‖f‖ₑ = ⨆ x, ‖f x‖ₑ :=
(mkOfCompact f).enorm_eq_iSup_enorm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | enorm_eq_iSup_enorm | null |
norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | norm_restrict_mono_set | null |
normedSpace {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E] : NormedSpace 𝕜 C(α, E) where
norm_smul_le := norm_smul_le | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | normedSpace | null |
linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun _ => rfl } | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | linearIsometryBoundedOfCompact | When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`. |
@[simp]
linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | linearIsometryBoundedOfCompact_symm_apply | null |
linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | linearIsometryBoundedOfCompact_apply_apply | null |
linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | linearIsometryBoundedOfCompact_toIsometryEquiv | null |
linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | linearIsometryBoundedOfCompact_toAddEquiv | null |
linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | linearIsometryBoundedOfCompact_of_compact_toEquiv | null |
@[simp] nnnorm_smul_const {R β : Type*} [SeminormedAddCommGroup β] [SeminormedRing R]
[Module R β] [NormSMulClass R β] (f : C(α, R)) (b : β) :
‖f • const α b‖₊ = ‖f‖₊ * ‖b‖₊ := by
simp only [nnnorm_eq_iSup_nnnorm, smul_apply', const_apply, nnnorm_smul, iSup_mul]
@[simp] lemma norm_smul_const {R β : Type*} [SeminormedAddCommGroup β] [SeminormedRing R]
[Module R β] [NormSMulClass R β] (f : C(α, R)) (b : β) :
‖f • const α b‖ = ‖f‖ * ‖b‖ := by
simp only [← coe_nnnorm, NNReal.coe_mul, nnnorm_smul_const] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | nnnorm_smul_const | null |
uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | uniform_continuity | null |
modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h) | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | modulus | An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. |
modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1 | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | modulus_pos | null |
dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | dist_lt_of_dist_lt_modulus | null |
summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
classical
refine (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => ?_
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict K (∑ i ∈ s, F i) = ∑ i ∈ s, restrict K (F i) := by
intro s
ext1 x
simp [-SetLike.coe_sort_coe]
simpa only [HasSum, A] using (hF K).of_norm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | summable_of_locally_summable_norm | null |
_root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.Star",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Sets.Compacts",
"Mathlib.Analysis.Normed.Group.InfiniteSum"
] | Mathlib/Topology/ContinuousMap/Compact.lean | _root_.BoundedContinuousFunction.mkOfCompact_star | null |
CompactlySupportedContinuousMap (α β : Type*) [TopologicalSpace α] [Zero β]
[TopologicalSpace β] extends ContinuousMap α β where
/-- The function has compact support . -/
hasCompactSupport' : HasCompactSupport toFun
@[inherit_doc]
scoped[CompactlySupported] notation (priority := 2000)
"C_c(" α ", " β ")" => CompactlySupportedContinuousMap α β
@[inherit_doc]
scoped[CompactlySupported] notation α " →C_c " β => CompactlySupportedContinuousMap α β
open CompactlySupported | structure | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | CompactlySupportedContinuousMap | `C_c(α, β)` is the type of continuous functions `α → β` with compact support from a topological
space to a topological space with a zero element.
When possible, instead of parametrizing results over `f : C_c(α, β)`,
you should parametrize over `{F : Type*} [CompactlySupportedContinuousMapClass F α β] (f : F)`.
When you extend this structure, make sure to extend `CompactlySupportedContinuousMapClass`. |
CompactlySupportedContinuousMapClass (F : Type*) (α β : outParam <| Type*)
[TopologicalSpace α] [Zero β] [TopologicalSpace β] [FunLike F α β] : Prop
extends ContinuousMapClass F α β where
/-- Each member of the class has compact support. -/
hasCompactSupport (f : F) : HasCompactSupport f | class | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | CompactlySupportedContinuousMapClass | `CompactlySupportedContinuousMapClass F α β` states that `F` is a type of continuous maps with
compact support.
You should also extend this typeclass when you extend `CompactlySupportedContinuousMap`. |
protected hasCompactSupport (f : C_c(α, β)) : HasCompactSupport f := f.hasCompactSupport' | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | hasCompactSupport | null |
@[simp]
coe_toContinuousMap (f : C_c(α, β)) : (f.toContinuousMap : α → β) = f :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_toContinuousMap | null |
ext {f g : C_c(α, β)} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | ext | null |
coe_mk (f : C(α, β)) (h : HasCompactSupport f) : ⇑(⟨f, h⟩ : C_c(α, β)) = f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_mk | null |
protected copy (f : C_c(α, β)) (f' : α → β) (h : f' = f) : C_c(α, β) where
toFun := f'
continuous_toFun := by
rw [h]
exact f.continuous_toFun
hasCompactSupport' := by
simp_rw [h]
exact f.hasCompactSupport'
@[simp] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | copy | Copy of a `CompactlySupportedContinuousMap` with a new `toFun` equal to the old one. Useful
to fix definitional equalities. |
coe_copy (f : C_c(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_copy | null |
copy_eq (f : C_c(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | copy_eq | null |
eq_of_empty [IsEmpty α] (f g : C_c(α, β)) : f = g :=
ext <| IsEmpty.elim ‹_› | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | eq_of_empty | null |
@[simps]
ContinuousMap.liftCompactlySupported [CompactSpace α] : C(α, β) ≃ C_c(α, β) where
toFun f :=
{ toFun := f
hasCompactSupport' := HasCompactSupport.of_compactSpace f }
invFun f := f
variable {γ : Type*} [TopologicalSpace γ] [Zero γ] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | ContinuousMap.liftCompactlySupported | A continuous function on a compact space automatically has compact support. |
noncomputable compLeft (g : C(β, γ)) (f : C_c(α, β)) : C_c(α, γ) where
toContinuousMap := by classical exact if g 0 = 0 then g.comp f else 0
hasCompactSupport' := by
split_ifs with hg
· exact f.hasCompactSupport'.comp_left hg
· exact .zero | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | compLeft | Composition of a continuous function `f` with compact support with another continuous function
`g` sending `0` to `0` from the left yields another continuous function `g ∘ f` with compact
support.
If `g` doesn't send `0` to `0`, `f.compLeft g` defaults to `0`. |
toContinuousMap_compLeft {g : C(β, γ)} (hg : g 0 = 0) (f : C_c(α, β)) :
(f.compLeft g).toContinuousMap = g.comp f := if_pos hg | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toContinuousMap_compLeft | null |
coe_compLeft {g : C(β, γ)} (hg : g 0 = 0) (f : C_c(α, β)) : f.compLeft g = g ∘ f := by
simp [compLeft, if_pos hg] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_compLeft | null |
compLeft_apply {g : C(β, γ)} (hg : g 0 = 0) (f : C_c(α, β)) (a : α) :
f.compLeft g a = g (f a) := by simp [coe_compLeft hg f] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | compLeft_apply | null |
@[simp]
coe_zero [Zero β] : ⇑(0 : C_c(α, β)) = 0 :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_zero | null |
zero_apply [Zero β] : (0 : C_c(α, β)) x = 0 :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | zero_apply | null |
@[simp]
coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C_c(α, β)) : ⇑(f * g) = f * g :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_mul | null |
mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C_c(α, β)) : (f * g) x = f x * g x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | mul_apply | null |
coeFnMonoidHom [AddMonoid β] [ContinuousAdd β] : C_c(α, β) →+ α → β where
toFun f := f
map_zero' := coe_zero
map_add' := coe_add | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coeFnMonoidHom | the product of `f : F` assuming `ContinuousMapClass F α γ` and `ContinuousSMul γ β` and
`g : C_c(α, β)` is in `C_c(α, β)` -/
instance [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
{F : Type*} [FunLike F α γ] [ContinuousMapClass F α γ] : SMul F C_c(α, β) where
smul f g :=
⟨⟨fun x ↦ f x • g x, (map_continuous f).smul (map_continuous g)⟩, g.hasCompactSupport.smul_left⟩
@[simp]
theorem coe_smulc [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
{F : Type*} [FunLike F α γ] [ContinuousMapClass F α γ] (f : F) (g : C_c(α, β)) :
⇑(f • g) = fun x => f x • g x :=
rfl
theorem smulc_apply [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
{F : Type*} [FunLike F α γ] [ContinuousMapClass F α γ] (f : F) (g : C_c(α, β)) (x : α) :
(f • g) x = f x • g x :=
rfl
instance [MulZeroClass β] [ContinuousMul β] : MulZeroClass C_c(α, β) :=
DFunLike.coe_injective.mulZeroClass _ coe_zero coe_mul
instance [SemigroupWithZero β] [ContinuousMul β] :
SemigroupWithZero C_c(α, β) :=
DFunLike.coe_injective.semigroupWithZero _ coe_zero coe_mul
instance [AddZeroClass β] [ContinuousAdd β] : Add C_c(α, β) :=
⟨fun f g => ⟨f + g, HasCompactSupport.add f.2 g.2⟩⟩
@[simp]
theorem coe_add [AddZeroClass β] [ContinuousAdd β] (f g : C_c(α, β)) : ⇑(f + g) = f + g :=
rfl
theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C_c(α, β)) : (f + g) x = f x + g x :=
rfl
instance [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C_c(α, β) :=
DFunLike.coe_injective.addZeroClass _ coe_zero coe_add
/-- Coercion to a function as a `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`. |
@[simp, norm_cast]
coe_smul [Zero β] {R : Type*} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R)
(f : C_c(α, β)) : ⇑(r • f) = r • ⇑f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_smul | null |
smul_apply [Zero β] {R : Type*} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R)
(f : C_c(α, β)) (x : α) : (r • f) x = r • f x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | smul_apply | null |
@[simp]
coe_sum [AddCommMonoid β] [ContinuousAdd β] {ι : Type*} (s : Finset ι) (f : ι → C_c(α, β)) :
⇑(∑ i ∈ s, f i) = ∑ i ∈ s, (f i : α → β) :=
map_sum coeFnMonoidHom f s | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_sum | null |
sum_apply [AddCommMonoid β] [ContinuousAdd β] {ι : Type*} (s : Finset ι) (f : ι → C_c(α, β))
(a : α) : (∑ i ∈ s, f i) a = ∑ i ∈ s, f i a := by simp | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | sum_apply | null |
@[simp]
coe_neg : ⇑(-f) = -f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_neg | null |
neg_apply : (-f) x = -f x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | neg_apply | null |
@[simp]
coe_sub : ⇑(f - g) = f - g :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_sub | null |
sub_apply : (f - g) x = f x - g x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | sub_apply | null |
@[simp]
coe_star (f : C_c(α, β)) : ⇑(star f) = star (⇑f) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_star | null |
star_apply (f : C_c(α, β)) (x : α) : (star f) x = star (f x) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | star_apply | null |
partialOrder : PartialOrder C_c(α, β) := PartialOrder.lift (⇑) DFunLike.coe_injective | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | partialOrder | null |
le_def {f g : C_c(α, β)} : f ≤ g ↔ ∀ a, f a ≤ g a := Pi.le_def | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | le_def | null |
lt_def {f g : C_c(α, β)} : f < g ↔ (∀ a, f a ≤ g a) ∧ ∃ a, f a < g a := Pi.lt_def | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | lt_def | null |
instSup : Max C_c(α, β) where max f g :=
{ toFun := f ⊔ g
continuous_toFun := Continuous.sup f.continuous g.continuous
hasCompactSupport' := f.hasCompactSupport.sup g.hasCompactSupport }
@[simp, norm_cast] lemma coe_sup (f g : C_c(α, β)) : ⇑(f ⊔ g) = ⇑f ⊔ g := rfl
@[simp] lemma sup_apply (f g : C_c(α, β)) (a : α) : (f ⊔ g) a = f a ⊔ g a := rfl | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | instSup | null |
semilatticeSup : SemilatticeSup C_c(α, β) :=
DFunLike.coe_injective.semilatticeSup _ coe_sup | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | semilatticeSup | null |
finsetSup'_apply {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C_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_c(α, β) ↦ g a) fun _ _ ↦ rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | finsetSup'_apply | null |
coe_finsetSup' {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C_c(α, β)) :
⇑(s.sup' H f) = s.sup' H fun i ↦ ⇑(f i) := by ext; simp [finsetSup'_apply] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_finsetSup' | null |
instInf : Min C_c(α, β) where min f g :=
{ toFun := f ⊓ g
continuous_toFun := Continuous.inf f.continuous g.continuous
hasCompactSupport' := f.hasCompactSupport.inf g.hasCompactSupport }
@[simp, norm_cast] lemma coe_inf (f g : C_c(α, β)) : ⇑(f ⊓ g) = ⇑f ⊓ g := rfl
@[simp] lemma inf_apply (f g : C_c(α, β)) (a : α) : (f ⊓ g) a = f a ⊓ g a := rfl | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | instInf | null |
semilatticeInf : SemilatticeInf C_c(α, β) :=
DFunLike.coe_injective.semilatticeInf _ coe_inf | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | semilatticeInf | null |
finsetInf'_apply {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C_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_c(α, β) ↦ g a) fun _ _ ↦ rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | finsetInf'_apply | null |
coe_finsetInf' {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C_c(α, β)) :
⇑(s.inf' H f) = s.inf' H fun i ↦ ⇑(f i) := by ext; simp [finsetInf'_apply] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_finsetInf' | null |
instMulLeftMono [PartialOrder β] [MulZeroClass β] [ContinuousMul β] [MulLeftMono β] :
MulLeftMono C_c(α, β) :=
⟨fun _ _ _ hg₁₂ x => mul_le_mul_left' (hg₁₂ x) _⟩ | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | instMulLeftMono | null |
instMulRightMono [PartialOrder β] [MulZeroClass β] [ContinuousMul β] [MulRightMono β] :
MulRightMono C_c(α, β) :=
⟨fun _ _ _ hg₁₂ x => mul_le_mul_right' (hg₁₂ x) _⟩ | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | instMulRightMono | null |
instAddLeftMono [PartialOrder β] [AddZeroClass β] [ContinuousAdd β] [AddLeftMono β] :
AddLeftMono C_c(α, β) :=
⟨fun _ _ _ hg₁₂ x => add_le_add_left (hg₁₂ x) _⟩ | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | instAddLeftMono | null |
instAddRightMono [PartialOrder β] [AddZeroClass β] [ContinuousAdd β] [AddRightMono β] :
AddRightMono C_c(α, β) :=
⟨fun _ _ _ hg₁₂ x => add_le_add_right (hg₁₂ x) _⟩ | instance | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | instAddRightMono | null |
comp (f : C_c(γ, δ)) (g : β →co γ) : C_c(β, δ) where
toContinuousMap := (f : C(γ, δ)).comp g
hasCompactSupport' := by
apply IsCompact.of_isClosed_subset (g.isCompact_preimage_of_isClosed f.2 (isClosed_tsupport _))
(isClosed_tsupport (f ∘ g))
intro x hx
rw [tsupport, Set.mem_preimage, _root_.mem_closure_iff]
intro o ho hgxo
rw [tsupport, _root_.mem_closure_iff] at hx
obtain ⟨y, hy⟩ := hx (g ⁻¹' o) (IsOpen.preimage g.1.2 ho) hgxo
exact ⟨g y, hy⟩
@[simp] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | comp | Composition of a continuous function with compact support with a cocompact map
yields another continuous function with compact support. |
coe_comp_to_continuous_fun (f : C_c(γ, δ)) (g : β →co γ) : ((f.comp g) : β → δ) = f ∘ g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_comp_to_continuous_fun | null |
comp_id (f : C_c(γ, δ)) : f.comp (CocompactMap.id γ) = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | comp_id | null |
comp_assoc (f : C_c(γ, δ)) (g : β →co γ) (h : α →co β) :
(f.comp g).comp h = f.comp (g.comp h) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | comp_assoc | null |
zero_comp (g : β →co γ) : (0 : C_c(γ, δ)).comp g = 0 :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | zero_comp | null |
compAddMonoidHom [AddMonoid δ] [ContinuousAdd δ] (g : β →co γ) : C_c(γ, δ) →+ C_c(β, δ) where
toFun f := f.comp g
map_zero' := zero_comp g
map_add' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | compAddMonoidHom | Composition as an additive monoid homomorphism. |
compMulHom [MulZeroClass δ] [ContinuousMul δ] (g : β →co γ) : C_c(γ, δ) →ₙ* C_c(β, δ) where
toFun f := f.comp g
map_mul' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | compMulHom | Composition as a semigroup homomorphism. |
compLinearMap [AddCommMonoid δ] [ContinuousAdd δ] {R : Type*} [Semiring R] [Module R δ]
[ContinuousConstSMul R δ] (g : β →co γ) : C_c(γ, δ) →ₗ[R] C_c(β, δ) where
toFun f := f.comp g
map_add' _ _ := rfl
map_smul' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | compLinearMap | Composition as a linear map. |
compNonUnitalAlgHom {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring δ]
[IsTopologicalSemiring δ] [Module R δ] [ContinuousConstSMul R δ] (g : β →co γ) :
C_c(γ, δ) →ₙₐ[R] C_c(β, δ) where
toFun f := f.comp g
map_smul' _ _ := rfl
map_zero' := rfl
map_add' _ _ := rfl
map_mul' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | compNonUnitalAlgHom | Composition as a non-unital algebra homomorphism. |
of_compactSpace (G : Type*) [FunLike G α β]
[ContinuousMapClass G α β] [CompactSpace α] : CompactlySupportedContinuousMapClass G α β where
map_continuous := map_continuous
hasCompactSupport := by
intro f
exact HasCompactSupport.of_compactSpace f | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | of_compactSpace | A continuous function on a compact space has automatically compact support. This is not an
instance to avoid type class loops. |
uniformContinuous (f : F) : UniformContinuous (f : β → γ) :=
(map_continuous f).uniformContinuous_of_tendsto_cocompact
(HasCompactSupport.is_zero_at_infty (hasCompactSupport f)) | theorem | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | uniformContinuous | null |
protected exists_add_of_le {f₁ f₂ : C_c(α, ℝ≥0)} (h : f₁ ≤ f₂) : ∃ (g : C_c(α, ℝ≥0)),
f₁ + g = f₂ := by
refine ⟨⟨f₂.1 - f₁.1, ?_⟩, ?_⟩
· apply (f₁.hasCompactSupport'.union f₂.hasCompactSupport').of_isClosed_subset isClosed_closure
rw [tsupport, tsupport, ← closure_union]
apply closure_mono
intro x hx
contrapose! hx
simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, Set.mem_union, Function.mem_support,
ne_eq, not_or, Decidable.not_not, ContinuousMap.coe_sub, Pi.sub_apply] at hx ⊢
simp [hx.1, hx.2]
· ext x
simpa [← NNReal.coe_add] using add_tsub_cancel_of_le (h x) | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | exists_add_of_le | null |
noncomputable nnrealPart (f : C_c(α, ℝ)) : C_c(α, ℝ≥0) where
toFun := Real.toNNReal.comp f.toFun
continuous_toFun := Continuous.comp continuous_real_toNNReal f.continuous
hasCompactSupport' := HasCompactSupport.comp_left f.hasCompactSupport' Real.toNNReal_zero
@[simp] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart | The nonnegative part of a bounded continuous `ℝ`-valued function as a bounded
continuous `ℝ≥0`-valued function. |
nnrealPart_apply (f : C_c(α, ℝ)) (x : α) :
f.nnrealPart x = Real.toNNReal (f x) := rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_apply | null |
nnrealPart_neg_eq_zero_of_nonneg {f : C_c(α, ℝ)} (hf : 0 ≤ f) : (-f).nnrealPart = 0 := by
ext x
simpa using hf x | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_neg_eq_zero_of_nonneg | null |
nnrealPart_smul_pos (f : C_c(α, ℝ)) {a : ℝ} (ha : 0 ≤ a) :
(a • f).nnrealPart = a.toNNReal • f.nnrealPart := by
ext x
simp only [nnrealPart_apply, coe_smul, Pi.smul_apply, Real.coe_toNNReal', smul_eq_mul,
NNReal.coe_mul, ha, sup_of_le_left]
rcases le_total 0 (f x) with hfx | hfx
· simp [ha, hfx, mul_nonneg]
· simp [mul_nonpos_iff, ha, hfx] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_smul_pos | null |
nnrealPart_smul_neg (f : C_c(α, ℝ)) {a : ℝ} (ha : a ≤ 0) :
(a • f).nnrealPart = (-a).toNNReal • (-f).nnrealPart := by
ext x
simp only [nnrealPart_apply, coe_smul, Pi.smul_apply, smul_eq_mul, Real.coe_toNNReal', coe_neg,
Pi.neg_apply, NNReal.coe_mul]
rcases le_total 0 (f x) with hfx | hfx
· simp [mul_nonpos_iff, ha, hfx]
· simp [ha, hfx, mul_nonneg_of_nonpos_of_nonpos] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_smul_neg | null |
nnrealPart_add_le_add_nnrealPart (f g : C_c(α, ℝ)) :
(f + g).nnrealPart ≤ f.nnrealPart + g.nnrealPart := by
intro x
simpa using Real.toNNReal_add_le | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_add_le_add_nnrealPart | null |
exists_add_nnrealPart_add_eq (f g : C_c(α, ℝ)) : ∃ (h : C_c(α, ℝ≥0)),
(f + g).nnrealPart + h = f.nnrealPart + g.nnrealPart ∧
(-f + -g).nnrealPart + h = (-f).nnrealPart + (-g).nnrealPart := by
obtain ⟨h, hh⟩ := CompactlySupportedContinuousMap.exists_add_of_le
(nnrealPart_add_le_add_nnrealPart f g)
use h
refine ⟨hh, ?_⟩
ext x
simp only [coe_add, Pi.add_apply, nnrealPart_apply, coe_neg, Pi.neg_apply, NNReal.coe_add,
Real.coe_toNNReal', ← neg_add]
have hhx : (f x + g x) ⊔ 0 + ↑(h x) = f x ⊔ 0 + g x ⊔ 0 := by
rw [← Real.coe_toNNReal', ← Real.coe_toNNReal', ← Real.coe_toNNReal', ← NNReal.coe_add,
← NNReal.coe_add]
have hhx' : ((f + g).nnrealPart + h) x = (f.nnrealPart + g.nnrealPart) x := by congr
simp only [coe_add, Pi.add_apply, nnrealPart_apply] at hhx'
exact congrArg toReal hhx'
rcases le_total 0 (f x) with hfx | hfx
· rcases le_total 0 (g x) with hgx | hgx
· simp only [hfx, hgx, add_nonneg, sup_of_le_left, add_eq_left, coe_eq_zero] at hhx
simp [hhx, hfx, hgx, add_nonpos]
· rcases le_total 0 (f x + g x) with hfgx | hfgx
· simp only [hfgx, sup_of_le_left, add_assoc, hfx, hgx, sup_of_le_right, add_zero,
add_eq_left] at hhx
rw [sup_of_le_right (neg_nonpos.mpr hfx), sup_of_le_left (neg_nonneg.mpr hgx),
sup_of_le_right (neg_nonpos.mpr hfgx)]
linarith
· simp only [hfgx, sup_of_le_right, zero_add, hfx, sup_of_le_left, hgx, add_zero] at hhx
rw [sup_of_le_right (neg_nonpos.mpr hfx), sup_of_le_left (neg_nonneg.mpr hgx),
sup_of_le_left (neg_nonneg.mpr hfgx), hhx]
ring
· rcases le_total 0 (g x) with hgx | hgx
· rcases le_total 0 (f x + g x) with hfgx | hfgx
· simp only [hfgx, sup_of_le_left, add_comm, hfx, sup_of_le_right, hgx, zero_add] at hhx
rw [sup_of_le_left (neg_nonneg.mpr hfx), sup_of_le_right (neg_nonpos.mpr hgx),
sup_of_le_right (neg_nonpos.mpr hfgx), zero_add, add_zero]
linarith
· simp only [hfgx, sup_of_le_right, zero_add, hfx, hgx, sup_of_le_left] at hhx
rw [sup_of_le_left (neg_nonneg.mpr hfx), sup_of_le_right (neg_nonpos.mpr hgx),
sup_of_le_left (neg_nonneg.mpr hfgx), hhx]
ring
· simp only [(add_nonpos hfx hgx), sup_of_le_right, zero_add, hfx, hgx, add_zero,
coe_eq_zero] at hhx
rw [sup_of_le_left (neg_nonneg.mpr hfx),
sup_of_le_left (neg_nonneg.mpr hgx),
sup_of_le_left (neg_nonneg.mpr (add_nonpos hfx hgx)), hhx, neg_add_rev, NNReal.coe_zero,
add_zero]
ring | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | exists_add_nnrealPart_add_eq | null |
noncomputable toReal (f : C_c(α, ℝ≥0)) : C_c(α, ℝ) :=
f.compLeft ContinuousMap.coeNNRealReal
@[simp] lemma toReal_apply (f : C_c(α, ℝ≥0)) (x : α) : f.toReal x = f x := compLeft_apply rfl _ _
@[simp] lemma toReal_nonneg {f : C_c(α, ℝ≥0)} : 0 ≤ f.toReal := fun _ ↦ by simp
@[simp] lemma toReal_add (f g : C_c(α, ℝ≥0)) : (f + g).toReal = f.toReal + g.toReal := by ext; simp
@[simp] lemma toReal_smul (r : ℝ≥0) (f : C_c(α, ℝ≥0)) : (r • f).toReal = r • f.toReal := by
ext; simp [NNReal.smul_def]
@[simp] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toReal | The compactly supported continuous `ℝ≥0`-valued function as a compactly supported `ℝ`-valued
function. |
nnrealPart_sub_nnrealPart_neg (f : C_c(α, ℝ)) :
(nnrealPart f).toReal - (nnrealPart (-f)).toReal = f := by ext x; simp | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_sub_nnrealPart_neg | null |
noncomputable toRealLinearMap : C_c(α, ℝ≥0) →ₗ[ℝ≥0] C_c(α, ℝ) where
toFun := toReal
map_add' f g := by ext x; simp
map_smul' a f := by ext x; simp
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toRealLinearMap | The map `toReal` defined as a `ℝ≥0`-linear map. |
coe_toRealLinearMap : (toRealLinearMap : C_c(α, ℝ≥0) → C_c(α, ℝ)) = toReal := rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | coe_toRealLinearMap | null |
toRealLinearMap_apply (f : C_c(α, ℝ≥0)) : toRealLinearMap f = f.toReal := rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toRealLinearMap_apply | null |
toRealLinearMap_apply_apply (f : C_c(α, ℝ≥0)) (x : α) :
toRealLinearMap f x = (f x).toReal := by simp
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toRealLinearMap_apply_apply | null |
nnrealPart_toReal_eq (f : C_c(α, ℝ≥0)) : nnrealPart (toReal f) = f := by ext x; simp
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_toReal_eq | null |
nnrealPart_neg_toReal_eq (f : C_c(α, ℝ≥0)) : nnrealPart (-toReal f) = 0 := by ext x; simp | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | nnrealPart_neg_toReal_eq | null |
noncomputable toNNRealLinear (Λ : C_c(α, ℝ) →ₚ[ℝ] ℝ) :
C_c(α, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 where
toFun f := ⟨Λ (toRealLinearMap f), Λ.map_nonneg (by simp)⟩
map_add' f g := by ext; simp
map_smul' a f := by ext; simp [NNReal.smul_def]
@[simp] | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toNNRealLinear | For a positive linear functional `Λ : C_c(α, ℝ) → ℝ`, define a `ℝ≥0`-linear map. |
toNNRealLinear_apply (Λ : C_c(α, ℝ) →ₚ[ℝ] ℝ) (f : C_c(α, ℝ≥0)) :
toNNRealLinear Λ f = Λ (toReal f) := rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toNNRealLinear_apply | null |
toNNRealLinear_inj (Λ₁ Λ₂ : C_c(α, ℝ) →ₚ[ℝ] ℝ) :
toNNRealLinear Λ₁ = toNNRealLinear Λ₂ ↔ Λ₁ = Λ₂ := by
refine ⟨fun h ↦ ?_, fun h ↦ by rw [h]⟩
ext f
rw [← nnrealPart_sub_nnrealPart_neg f]
simp only [LinearMap.ext_iff, NNReal.eq_iff, toNNRealLinear_apply] at h
simp_rw [map_sub, h] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toNNRealLinear_inj | null |
noncomputable toRealPositiveLinear (Λ : C_c(α, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) : C_c(α, ℝ) →ₚ[ℝ] ℝ :=
PositiveLinearMap.mk₀
{ toFun := fun f => Λ (nnrealPart f) - Λ (nnrealPart (- f))
map_add' f g := by
simp only [neg_add_rev]
obtain ⟨h, hh⟩ := exists_add_nnrealPart_add_eq f g
rw [← add_zero ((Λ (f + g).nnrealPart).toReal - (Λ (-g + -f).nnrealPart).toReal),
← sub_self (Λ h).toReal, sub_add_sub_comm, ← NNReal.coe_add, ← NNReal.coe_add,
← LinearMap.map_add, ← LinearMap.map_add, hh.1, add_comm (-g) (-f), hh.2]
simp only [map_add, NNReal.coe_add]
ring
map_smul' a f := by
rcases le_total 0 a with ha | ha
· rw [RingHom.id_apply, smul_eq_mul, ← (smul_neg a f), nnrealPart_smul_pos f ha,
nnrealPart_smul_pos (-f) ha]
simp [sup_of_le_left ha, mul_sub]
· simp only [RingHom.id_apply, smul_eq_mul, ← (smul_neg a f),
nnrealPart_smul_neg f ha, nnrealPart_smul_neg (-f) ha, map_smul,
NNReal.coe_mul, Real.coe_toNNReal', neg_neg, sup_of_le_left (neg_nonneg.mpr ha)]
ring }
(fun g hg ↦ by simp [nnrealPart_neg_eq_zero_of_nonneg hg]) | def | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toRealPositiveLinear | For a positive linear functional `Λ : C_c(α, ℝ≥0) → ℝ≥0`, define a positive `ℝ`-linear map. |
toRealPositiveLinear_apply {Λ : C_c(α, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0} (f : C_c(α, ℝ)) :
toRealPositiveLinear Λ f = Λ (nnrealPart f) - Λ (nnrealPart (-f)) := rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toRealPositiveLinear_apply | null |
eq_toRealPositiveLinear_toReal (Λ : C_c(α, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) (f : C_c(α, ℝ≥0)) :
toRealPositiveLinear Λ (toReal f) = Λ f := by
simp [toRealPositiveLinear_apply]
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | eq_toRealPositiveLinear_toReal | null |
eq_toNNRealLinear_toRealPositiveLinear (Λ : C_c(α, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) :
toNNRealLinear (toRealPositiveLinear Λ) = Λ := by
ext f
simp
@[deprecated (since := "2025-08-08")]
alias toRealLinear := toRealPositiveLinear
@[deprecated (since := "2025-08-08")]
alias toRealLinear_apply := toRealPositiveLinear_apply
@[deprecated map_nonneg (since := "2025-08-08")] | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | eq_toNNRealLinear_toRealPositiveLinear | null |
toRealLinear_nonneg (Λ : C_c(α, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) (g : C_c(α, ℝ)) (hg : 0 ≤ g) :
0 ≤ toRealPositiveLinear Λ g := map_nonneg _ hg
@[deprecated (since := "2025-08-08")]
alias eq_toRealLinear_toReal := eq_toRealPositiveLinear_toReal
@[deprecated (since := "2025-08-08")]
alias eq_toNNRealLinear_toRealLinear := eq_toNNRealLinear_toRealPositiveLinear | lemma | Topology | [
"Mathlib.Algebra.Order.Module.PositiveLinearMap",
"Mathlib.Topology.Algebra.Order.Support",
"Mathlib.Topology.ContinuousMap.ZeroAtInfty"
] | Mathlib/Topology/ContinuousMap/CompactlySupported.lean | toRealLinear_nonneg | null |
ContinuousMapZero (X R : Type*) [Zero X] [Zero R] [TopologicalSpace X]
[TopologicalSpace R] extends C(X, R) where
map_zero' : toContinuousMap 0 = 0 | structure | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | ContinuousMapZero | The type of continuous maps which map zero to zero.
Note that one should never use the structure projection `ContinuousMapZero.toContinuousMap` and
instead favor the coercion `(↑) : C(X, R)₀ → C(X, R)` available from the instance of
`ContinuousMapClass`. All the instances on `C(X, R)₀` from `C(X, R)` passes through this coercion,
not the structure projection. Of course, the two are definitionally equal, but not reducibly so. |
instFunLike : FunLike C(X, R)₀ X R where
coe f := f.toFun
coe_injective' _ _ h := congr(⟨⟨$(h), _⟩, _⟩) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instFunLike | null |
instContinuousMapClass : ContinuousMapClass C(X, R)₀ X R where
map_continuous f := f.continuous | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instContinuousMapClass | null |
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