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 ⌀ |
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pi (f : ∀ i, C(A, X i)) : C(A, ∀ i, X i) where
toFun (a : A) (i : I) := f i a
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | pi | Abbreviation for product of continuous maps, which is continuous |
pi_eval (f : ∀ i, C(A, X i)) (a : A) : (pi f) a = fun i : I => (f i) a :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | pi_eval | null |
@[simps -fullyApplied]
eval (i : I) : C(∀ j, X j, X i) where
toFun := Function.eval i
variable (A X) in | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | eval | Evaluation at point as a bundled continuous map. |
@[simps]
piEquiv : (∀ i, C(A, X i)) ≃ C(A, ∀ i, X i) where
toFun := pi
invFun f i := (eval i).comp f | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | piEquiv | Giving a continuous map out of a disjoint union is the same as giving a continuous map out of
each term |
@[simps!]
piMap (f : ∀ i, C(X i, Y i)) : C((i : I) → X i, (i : I) → Y i) :=
.pi fun i ↦ (f i).comp (eval i) | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | piMap | Combine a collection of bundled continuous maps `C(X i, Y i)` into a bundled continuous map
`C(∀ i, X i, ∀ i, Y i)`. |
precomp {ι : Type*} (φ : ι → I) : C((i : I) → X i, (i : ι) → X (φ i)) :=
⟨_, Pi.continuous_precomp' φ⟩ | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | precomp | "Precomposition" as a continuous map between dependent types. |
restrict (f : C(α, β)) : C(s, β) where
toFun := f ∘ ((↑) : s → α)
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | restrict | The restriction of a continuous function `α → β` to a subset `s` of `α`. |
coe_restrict (f : C(α, β)) : ⇑(f.restrict s) = f ∘ ((↑) : s → α) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | coe_restrict | null |
restrict_apply (f : C(α, β)) (s : Set α) (x : s) : f.restrict s x = f x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | restrict_apply | null |
restrict_apply_mk (f : C(α, β)) (s : Set α) (x : α) (hx : x ∈ s) :
f.restrict s ⟨x, hx⟩ = f x :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | restrict_apply_mk | null |
injective_restrict [T2Space β] {s : Set α} (hs : Dense s) :
Injective (restrict s : C(α, β) → C(s, β)) := fun f g h ↦
DFunLike.ext' <| (map_continuous f).ext_on hs (map_continuous g) <|
Set.restrict_eq_restrict_iff.1 <| congr_arg DFunLike.coe h | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | injective_restrict | null |
@[simps]
restrictPreimage (f : C(α, β)) (s : Set β) : C(f ⁻¹' s, s) :=
⟨s.restrictPreimage f, continuous_iff_continuousAt.mpr fun _ ↦
(map_continuousAt f _).restrictPreimage⟩ | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | restrictPreimage | The restriction of a continuous map to the preimage of a set. |
noncomputable mkD (f : α → β) (default : C(α, β)) : C(α, β) :=
open scoped Classical in
if h : Continuous f then ⟨_, h⟩ else default | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD | Interpret `f : α → β` as an element of `C(α, β)`, falling back to the default value
`default : C(α, β)` if `f` is not continuous.
This is mainly intended to be used for `C(α, β)`-valued integration. For example, if a family of
functions `f : ι → α → β` satisfies that `f i` is continuous for almost every `i`, you can write
the `C(α, β)`-valued integral "`∫ i, f i`" as `∫ i, ContinuousMap.mkD (f i) 0`. |
mkD_of_continuous {f : α → β} {g : C(α, β)} (hf : Continuous f) :
mkD f g = ⟨f, hf⟩ := by
simp only [mkD, hf, ↓reduceDIte] | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_of_continuous | null |
mkD_of_not_continuous {f : α → β} {g : C(α, β)} (hf : ¬ Continuous f) :
mkD f g = g := by
simp only [mkD, hf, ↓reduceDIte] | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_of_not_continuous | null |
mkD_apply_of_continuous {f : α → β} {g : C(α, β)} {x : α} (hf : Continuous f) :
mkD f g x = f x := by
rw [mkD_of_continuous hf, coe_mk] | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_apply_of_continuous | null |
mkD_of_continuousOn {s : Set α} {f : α → β} {g : C(s, β)}
(hf : ContinuousOn f s) :
mkD (s.restrict f) g = ⟨s.restrict f, hf.restrict⟩ :=
mkD_of_continuous hf.restrict | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_of_continuousOn | null |
mkD_of_not_continuousOn {s : Set α} {f : α → β} {g : C(s, β)}
(hf : ¬ ContinuousOn f s) :
mkD (s.restrict f) g = g := by
rw [continuousOn_iff_continuous_restrict] at hf
exact mkD_of_not_continuous hf | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_of_not_continuousOn | null |
mkD_apply_of_continuousOn {s : Set α} {f : α → β} {g : C(s, β)} {x : s}
(hf : ContinuousOn f s) :
mkD (s.restrict f) g x = f x := by
rw [mkD_of_continuousOn hf, coe_mk, Set.restrict_apply] | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_apply_of_continuousOn | null |
mkD_eq_self {f g : C(α, β)} : mkD f g = f :=
mkD_of_continuous f.continuous | lemma | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | mkD_eq_self | null |
noncomputable liftCover : C(α, β) :=
haveI H : ⋃ i, S i = Set.univ :=
Set.iUnion_eq_univ_iff.2 fun x ↦ (hS x).imp fun _ ↦ mem_of_mem_nhds
mk (Set.liftCover S (fun i ↦ φ i) hφ H) <| continuous_of_cover_nhds hS fun i ↦ by
rw [continuousOn_iff_continuous_restrict]
simpa +unfoldPartialApp only [Set.restrict, Set.liftCover_coe]
using map_continuous (φ i)
variable {S φ hφ hS}
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftCover | A family `φ i` of continuous maps `C(S i, β)`, where the domains `S i` contain a neighbourhood
of each point in `α` and the functions `φ i` agree pairwise on intersections, can be glued to
construct a continuous map in `C(α, β)`. |
liftCover_coe {i : ι} (x : S i) : liftCover S φ hφ hS x = φ i x := by
rw [liftCover, coe_mk, Set.liftCover_coe _]
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftCover_coe | null |
liftCover_restrict {i : ι} : (liftCover S φ hφ hS).restrict (S i) = φ i := by
ext
simp only [coe_restrict, Function.comp_apply, liftCover_coe]
variable (A : Set (Set α)) (F : ∀ s ∈ A, C(s, β))
(hF : ∀ (s) (hs : s ∈ A) (t) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t),
F s hs ⟨x, hxi⟩ = F t ht ⟨x, hxj⟩)
(hA : ∀ x : α, ∃ i ∈ A, i ∈ 𝓝 x) | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftCover_restrict | null |
noncomputable liftCover' : C(α, β) :=
let F : ∀ i : A, C(i, β) := fun i => F i i.prop
liftCover ((↑) : A → Set α) F (fun i j => hF i i.prop j j.prop)
fun x => let ⟨s, hs, hsx⟩ := hA x; ⟨⟨s, hs⟩, hsx⟩
variable {A F hF hA}
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftCover' | A family `F s` of continuous maps `C(s, β)`, where (1) the domains `s` are taken from a set `A`
of sets in `α` which contain a neighbourhood of each point in `α` and (2) the functions `F s` agree
pairwise on intersections, can be glued to construct a continuous map in `C(α, β)`. |
liftCover_coe' {s : Set α} {hs : s ∈ A} (x : s) : liftCover' A F hF hA x = F s hs x :=
let x' : ((↑) : A → Set α) ⟨s, hs⟩ := x
by delta liftCover'; exact ContinuousMap.liftCover_coe x'
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftCover_coe' | null |
liftCover_restrict' {s : Set α} {hs : s ∈ A} :
(liftCover' A F hF hA).restrict s = F s hs := ext <| liftCover_coe' (hF := hF) (hA := hA) | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftCover_restrict' | null |
inclusion {s t : Set α} (h : s ⊆ t) : C(s, t) where
toFun := Set.inclusion h
continuous_toFun := continuous_inclusion h | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | inclusion | `Set.inclusion` as a bundled continuous map. |
@[simps!]
Function.RightInverse.homeomorph {f' : C(Y, X)} (hf : Function.RightInverse f' f) :
Quotient (Setoid.ker f) ≃ₜ Y where
toEquiv := Setoid.quotientKerEquivOfRightInverse _ _ hf
continuous_toFun := isQuotientMap_quot_mk.continuous_iff.mpr (map_continuous f)
continuous_invFun := continuous_quotient_mk'.comp (map_continuous f') | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | Function.RightInverse.homeomorph | `Setoid.quotientKerEquivOfRightInverse` as a homeomorphism. |
@[simps!]
noncomputable homeomorph (hf : IsQuotientMap f) : Quotient (Setoid.ker f) ≃ₜ Y where
toEquiv := Setoid.quotientKerEquivOfSurjective _ hf.surjective
continuous_toFun := isQuotientMap_quot_mk.continuous_iff.mpr hf.continuous
continuous_invFun := by
rw [hf.continuous_iff]
convert continuous_quotient_mk'
ext
simp only [Equiv.invFun_as_coe, Function.comp_apply,
(Setoid.quotientKerEquivOfSurjective f hf.surjective).symm_apply_eq]
rfl
variable (hf : IsQuotientMap f) (g : C(X, Z)) (h : Function.FactorsThrough g f) | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | homeomorph | The homeomorphism from the quotient of a quotient map to its codomain. This is
`Setoid.quotientKerEquivOfSurjective` as a homeomorphism. |
@[simps]
noncomputable lift : C(Y, Z) where
toFun := ((fun i ↦ Quotient.liftOn' i g (fun _ _ (hab : f _ = f _) ↦ h hab)) :
Quotient (Setoid.ker f) → Z) ∘ hf.homeomorph.symm
continuous_toFun := Continuous.comp (continuous_quot_lift _ g.2) (Homeomorph.continuous _) | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | lift | Descend a continuous map, which is constant on the fibres, along a quotient map. |
@[simp]
lift_comp : (hf.lift g h).comp f = g := by
ext
simpa using h (Function.rightInverse_surjInv _ _) | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | lift_comp | The obvious triangle induced by `IsQuotientMap.lift` commutes:
```
g
X --→ Z
| ↗
f | / hf.lift g h
v /
Y
``` |
@[simps]
noncomputable liftEquiv : { g : C(X, Z) // Function.FactorsThrough g f} ≃ C(Y, Z) where
toFun g := hf.lift g g.prop
invFun g := ⟨g.comp f, fun _ _ h ↦ by simp only [ContinuousMap.comp_apply]; rw [h]⟩
left_inv := by intro; simp
right_inv := by
intro g
ext a
simpa using congrArg g (Function.rightInverse_surjInv hf.surjective a) | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | liftEquiv | `IsQuotientMap.lift` as an equivalence. |
instContinuousMapClass : ContinuousMapClass (α ≃ₜ β) α β where
map_continuous f := f.continuous_toFun
@[simp] | instance | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | instContinuousMapClass | null |
coe_refl : (Homeomorph.refl α : C(α, α)) = ContinuousMap.id α :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | coe_refl | null |
coe_trans : (f.trans g : C(α, γ)) = (g : C(β, γ)).comp f :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | coe_trans | null |
@[simp]
symm_comp_toContinuousMap :
(f.symm : C(β, α)).comp (f : C(α, β)) = ContinuousMap.id α := by
rw [← coe_trans, self_trans_symm, coe_refl] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | symm_comp_toContinuousMap | Left inverse to a continuous map from a homeomorphism, mirroring `Equiv.symm_comp_self`. |
@[simp]
toContinuousMap_comp_symm :
(f : C(α, β)).comp (f.symm : C(β, α)) = ContinuousMap.id β := by
rw [← coe_trans, symm_trans_self, coe_refl] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | toContinuousMap_comp_symm | Right inverse to a continuous map from a homeomorphism, mirroring `Equiv.self_comp_symm`. |
compactlySupported (α γ : Type*) [TopologicalSpace α] [NonUnitalNormedRing γ] :
TwoSidedIdeal (α →ᵇ γ) :=
.mk' {z | HasCompactSupport z} .zero .add .neg .mul_left .mul_right
variable {α γ : Type*} [TopologicalSpace α] [NonUnitalNormedRing γ]
@[inherit_doc]
scoped[BoundedContinuousFunction] notation
"C_cb(" α ", " γ ")" => compactlySupported α γ | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | compactlySupported | The two-sided ideal of compactly supported functions. |
mem_compactlySupported {f : α →ᵇ γ} :
f ∈ C_cb(α, γ) ↔ HasCompactSupport f :=
TwoSidedIdeal.mem_mk' {z : α →ᵇ γ | HasCompactSupport z} .zero .add .neg .mul_left .mul_right f | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | mem_compactlySupported | null |
exist_norm_eq [c : Nonempty α] {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) : ∃ (x : α),
‖f x‖ = ‖f‖ := by
by_cases hs : (tsupport f).Nonempty
· obtain ⟨x, _, hmax⟩ := mem_compactlySupported.mp h |>.exists_isMaxOn hs <|
(map_continuous f).norm.continuousOn
refine ⟨x, le_antisymm (norm_coe_le_norm f x) (norm_le (norm_nonneg _) |>.mpr fun y ↦ ?_)⟩
by_cases hy : y ∈ tsupport f
· exact hmax hy
· simp [image_eq_zero_of_notMem_tsupport hy]
· suffices f = 0 by simp [this]
rwa [not_nonempty_iff_eq_empty, tsupport_eq_empty_iff, ← coe_zero, ← DFunLike.ext'_iff] at hs | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | exist_norm_eq | null |
norm_lt_iff_of_compactlySupported {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) {M : ℝ}
(M0 : 0 < M) : ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M := by
refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩
· obtain (he | he) := isEmpty_or_nonempty α
· simpa
· obtain ⟨x, hx⟩ := exist_norm_eq h
exact fun h ↦ hx ▸ h x | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | norm_lt_iff_of_compactlySupported | null |
norm_lt_iff_of_nonempty_compactlySupported [c : Nonempty α] {f : α →ᵇ γ}
(h : f ∈ C_cb(α, γ)) {M : ℝ} : ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M := by
obtain (hM | hM) := lt_or_ge 0 M
· exact norm_lt_iff_of_compactlySupported h hM
· exact ⟨fun h ↦ False.elim <| (h.trans_le hM).not_ge (by positivity),
fun h ↦ False.elim <| (h (Classical.arbitrary α) |>.trans_le hM).not_ge (by positivity)⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | norm_lt_iff_of_nonempty_compactlySupported | null |
compactlySupported_eq_top_of_isCompact (h : IsCompact (Set.univ : Set α)) :
C_cb(α, γ) = ⊤ :=
eq_top_iff.mpr fun _ _ ↦ h.of_isClosed_subset (isClosed_tsupport _) (subset_univ _)
/- This is intentionally not marked `@[simp]` to prevent Lean looking for a `CompactSpace α` | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | compactlySupported_eq_top_of_isCompact | null |
every time it sees `C_cb(α, γ)`. -/ | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | every | null |
compactlySupported_eq_top [CompactSpace α] : C_cb(α, γ) = ⊤ :=
compactlySupported_eq_top_of_isCompact CompactSpace.isCompact_univ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | compactlySupported_eq_top | null |
compactlySupported_eq_top_iff [Nontrivial γ] :
C_cb(α, γ) = ⊤ ↔ IsCompact (Set.univ : Set α) := by
refine ⟨fun h ↦ ?_, compactlySupported_eq_top_of_isCompact⟩
obtain ⟨x, hx⟩ := exists_ne (0 : γ)
simpa [tsupport, Function.support_const hx]
using (mem_compactlySupported (f := const α x).mp (by simp [h])).isCompact | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | compactlySupported_eq_top_iff | null |
ofCompactSupport (g : α → γ) (hg₁ : Continuous g) (hg₂ : HasCompactSupport g) : α →ᵇ γ where
toFun := g
continuous_toFun := hg₁
map_bounded' := by
obtain (hs | hs) := (tsupport g).eq_empty_or_nonempty
· exact ⟨0, by simp [tsupport_eq_empty_iff.mp hs]⟩
· obtain ⟨z, _, hmax⟩ := hg₂.exists_isMaxOn hs <| hg₁.norm.continuousOn
refine ⟨2 * ‖g z‖, dist_le_two_norm' fun x ↦ ?_⟩
by_cases hx : x ∈ tsupport g
· exact isMaxOn_iff.mp hmax x hx
· simp [image_eq_zero_of_notMem_tsupport hx] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | ofCompactSupport | A compactly supported continuous function is automatically bounded. This constructor gives
an object of `α →ᵇ γ` from `g : α → γ` and these assumptions. |
ofCompactSupport_mem (g : α → γ) (hg₁ : Continuous g) (hg₂ : HasCompactSupport g) :
ofCompactSupport g hg₁ hg₂ ∈ C_cb(α, γ) := mem_compactlySupported.mpr hg₂ | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Normed"
] | Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean | ofCompactSupport_mem | null |
CocompactMap (α : Type u) (β : Type v) [TopologicalSpace α] [TopologicalSpace β] :
Type max u v
extends ContinuousMap α β where
/-- The cocompact filter on `α` tends to the cocompact filter on `β` under the function -/
cocompact_tendsto' : Tendsto toFun (cocompact α) (cocompact β) | structure | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | CocompactMap | A *cocompact continuous map* is a continuous function between topological spaces which
tends to the cocompact filter along the cocompact filter. Functions for which preimages of compact
sets are compact always satisfy this property, and the converse holds for cocompact continuous maps
when the codomain is Hausdorff (see `CocompactMap.tendsto_of_forall_preimage` and
`CocompactMap.isCompact_preimage`).
Cocompact maps thus generalise proper maps, with which they correspond when the codomain is
Hausdorff. |
CocompactMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
[TopologicalSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where
/-- The cocompact filter on `α` tends to the cocompact filter on `β` under the function -/
cocompact_tendsto (f : F) : Tendsto f (cocompact α) (cocompact β) | class | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | CocompactMapClass | `CocompactMapClass F α β` states that `F` is a type of cocompact continuous maps.
You should also extend this typeclass when you extend `CocompactMap`. |
@[coe]
toCocompactMap (f : F) : CocompactMap α β :=
{ (f : C(α, β)) with
cocompact_tendsto' := cocompact_tendsto f } | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | toCocompactMap | Turn an element of a type `F` satisfying `CocompactMapClass F α β` into an actual
`CocompactMap`. This is declared as the default coercion from `F` to `CocompactMap α β`. |
@[simp]
coe_toContinuousMap {f : CocompactMap α β} : (f.toContinuousMap : α → β) = f :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | coe_toContinuousMap | null |
ext {f g : CocompactMap α β} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | ext | null |
protected copy (f : CocompactMap α β) (f' : α → β) (h : f' = f) : CocompactMap α β where
toFun := f'
continuous_toFun := by
rw [h]
exact f.continuous_toFun
cocompact_tendsto' := by
simp_rw [h]
exact f.cocompact_tendsto'
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | copy | Copy of a `CocompactMap` with a new `toFun` equal to the old one. Useful
to fix definitional equalities. |
coe_copy (f : CocompactMap α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | coe_copy | null |
copy_eq (f : CocompactMap α β) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | copy_eq | null |
coe_mk (f : C(α, β)) (h : Tendsto f (cocompact α) (cocompact β)) :
⇑(⟨f, h⟩ : CocompactMap α β) = f :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | coe_mk | null |
protected id : CocompactMap α α :=
⟨ContinuousMap.id _, tendsto_id⟩
@[simp, norm_cast] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | id | The identity as a cocompact continuous map. |
coe_id : ⇑(CocompactMap.id α) = id :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | coe_id | null |
comp (f : CocompactMap β γ) (g : CocompactMap α β) : CocompactMap α γ :=
⟨f.toContinuousMap.comp g, (cocompact_tendsto f).comp (cocompact_tendsto g)⟩
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | comp | The composition of cocompact continuous maps, as a cocompact continuous map. |
coe_comp (f : CocompactMap β γ) (g : CocompactMap α β) : ⇑(comp f g) = f ∘ g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | coe_comp | null |
comp_apply (f : CocompactMap β γ) (g : CocompactMap α β) (a : α) : comp f g a = f (g a) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | comp_apply | null |
comp_assoc (f : CocompactMap γ δ) (g : CocompactMap β γ) (h : CocompactMap α β) :
(f.comp g).comp h = f.comp (g.comp h) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | comp_assoc | null |
id_comp (f : CocompactMap α β) : (CocompactMap.id _).comp f = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | id_comp | null |
comp_id (f : CocompactMap α β) : f.comp (CocompactMap.id _) = f :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | comp_id | null |
tendsto_of_forall_preimage {f : α → β} (h : ∀ s, IsCompact s → IsCompact (f ⁻¹' s)) :
Tendsto f (cocompact α) (cocompact β) := fun s hs =>
match mem_cocompact.mp hs with
| ⟨t, ht, hts⟩ =>
mem_map.mpr (mem_cocompact.mpr ⟨f ⁻¹' t, h t ht, by simpa using preimage_mono hts⟩) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | tendsto_of_forall_preimage | null |
isCompact_preimage_of_isClosed (f : CocompactMap α β)
⦃s : Set β⦄ (hs : IsCompact s) (h's : IsClosed s) :
IsCompact (f ⁻¹' s) := by
obtain ⟨t, ht, hts⟩ :=
mem_cocompact'.mp
(by
simpa only [preimage_image_preimage, preimage_compl] using
mem_map.mp
(cocompact_tendsto f <|
mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩))
exact
ht.of_isClosed_subset (h's.preimage <| map_continuous f) (by simpa using hts) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | isCompact_preimage_of_isClosed | Preimages of compact closed sets are compact under a cocompact continuous map. |
isCompact_preimage [T2Space β] (f : CocompactMap α β) ⦃s : Set β⦄ (hs : IsCompact s) :
IsCompact (f ⁻¹' s) :=
isCompact_preimage_of_isClosed f hs hs.isClosed | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | isCompact_preimage | If the codomain is Hausdorff, preimages of compact sets are compact under a cocompact
continuous map. |
@[simps]
Homeomorph.toCocompactMap {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
(f : α ≃ₜ β) : CocompactMap α β where
toFun := f
continuous_toFun := f.continuous
cocompact_tendsto' := by
refine CocompactMap.tendsto_of_forall_preimage fun K hK => ?_
have := K.preimage_equiv_eq_image_symm f.toEquiv
simp only [coe_toEquiv] at this
rw [this]
exact hK.image f.symm.continuous | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/CocompactMap.lean | Homeomorph.toCocompactMap | A homeomorphism is a cocompact map. |
@[simps -fullyApplied]
equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
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 | equivBoundedOfCompact | When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`. |
isUniformInducing_equivBoundedOfCompact : IsUniformInducing (equivBoundedOfCompact α β) :=
IsUniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ 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 | isUniformInducing_equivBoundedOfCompact | null |
isUniformEmbedding_equivBoundedOfCompact : IsUniformEmbedding (equivBoundedOfCompact α β) :=
{ isUniformInducing_equivBoundedOfCompact α β with
injective := (equivBoundedOfCompact α β).injective } | 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 | isUniformEmbedding_equivBoundedOfCompact | null |
@[simps! -fullyApplied apply symm_apply]
addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm | 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 | addEquivBoundedOfCompact | When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`. |
instPseudoMetricSpace : PseudoMetricSpace C(α, β) :=
(isUniformEmbedding_equivBoundedOfCompact α β).comapPseudoMetricSpace _ | 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 | instPseudoMetricSpace | null |
instMetricSpace {β : Type*} [MetricSpace β] :
MetricSpace C(α, β) :=
(isUniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _ | 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 | instMetricSpace | null |
@[simps! -fullyApplied toEquiv apply symm_apply]
isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β | 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 | isometryEquivBoundedOfCompact | When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`. |
@[simp]
_root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
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 | _root_.BoundedContinuousFunction.dist_mkOfCompact | null |
_root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
open BoundedContinuousFunction | 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.dist_toContinuousMap | null |
dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply] | 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_apply_le_dist | The pointwise distance is controlled by the distance between functions, by definition. |
dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply] | 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_le | The distance between two functions is controlled by the supremum of the pointwise distances. |
dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply] | 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_le_iff_of_nonempty | null |
dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply] | 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_iff_of_nonempty | null |
dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.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_nonempty | null |
dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply] | 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_iff | null |
dist_eq_iSup : dist f g = ⨆ x, dist (f x) (g x) := by
simp [← isometryEquivBoundedOfCompact α β |>.dist_eq f g,
BoundedContinuousFunction.dist_eq_iSup] | 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_eq_iSup | null |
nndist_eq_iSup : nndist f g = ⨆ x, nndist (f x) (g x) := by
simp [← isometryEquivBoundedOfCompact α β |>.nndist_eq f g,
BoundedContinuousFunction.nndist_eq_iSup] | 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 | nndist_eq_iSup | null |
edist_eq_iSup : edist f g = ⨆ (x : α), edist (f x) (g x) := by
simp [← isometryEquivBoundedOfCompact α β |>.edist_eq f g,
BoundedContinuousFunction.edist_eq_iSup] | 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 | edist_eq_iSup | null |
@[simp]
_root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
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 | _root_.BoundedContinuousFunction.norm_mkOfCompact | null |
_root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
open BoundedContinuousFunction | 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.norm_toContinuousMap_eq | null |
norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm 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_coe_le_norm | null |
dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y | 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_le_two_norm | Distance between the images of any two points is at most twice the norm of the function. |
norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0 | 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_le | The norm of a function is controlled by the supremum of the pointwise norms. |
norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _ | 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_le_of_nonempty | null |
norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0 | 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_lt_iff | null |
nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0 | 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 | nnnorm_lt_iff | null |
norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _ | 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_lt_iff_of_nonempty | null |
nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty | 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 | nnnorm_lt_iff_of_nonempty | null |
apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm 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 | apply_le_norm | null |
neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs (f 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 | neg_norm_le_apply | null |
nnnorm_eq_iSup_nnnorm : ‖f‖₊ = ⨆ x : α, ‖f x‖₊ :=
(mkOfCompact f).nnnorm_eq_iSup_nnnorm | 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 | nnnorm_eq_iSup_nnnorm | null |
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