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stringclasses 32
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instCompleteSpace [CompleteSpace β] : CompleteSpace (α →ᵇ β) :=
complete_of_cauchySeq_tendsto fun (f : ℕ → α →ᵇ β) (hf : CauchySeq f) => by
/- We have to show that `f n` converges to a bounded continuous function.
For this, we prove pointwise convergence to define the limit, then check
it is a continuous bounded function, and then check the norm convergence. -/
rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩
have f_bdd := fun x n m N hn hm => le_trans (dist_coe_le_dist x) (b_bound n m N hn hm)
have fx_cau : ∀ x, CauchySeq fun n => f n x :=
fun x => cauchySeq_iff_le_tendsto_0.2 ⟨b, b0, f_bdd x, b_lim⟩
choose F hF using fun x => cauchySeq_tendsto_of_complete (fx_cau x)
/- `F : α → β`, `hF : ∀ (x : α), Tendsto (fun n ↦ ↑(f n) x) atTop (𝓝 (F x))`
`F` is the desired limit function. Check that it is uniformly approximated by `f N`. -/
have fF_bdd : ∀ x N, dist (f N x) (F x) ≤ b N :=
fun x N => le_of_tendsto (tendsto_const_nhds.dist (hF x))
(Filter.eventually_atTop.2 ⟨N, fun n hn => f_bdd x N n N (le_refl N) hn⟩)
refine ⟨⟨⟨F, ?_⟩, ?_⟩, ?_⟩
· -- Check that `F` is continuous, as a uniform limit of continuous functions
have : TendstoUniformly (fun n x => f n x) F atTop := by
refine Metric.tendstoUniformly_iff.2 fun ε ε0 => ?_
refine ((tendsto_order.1 b_lim).2 ε ε0).mono fun n hn x => ?_
rw [dist_comm]
exact lt_of_le_of_lt (fF_bdd x n) hn
exact this.continuous (Eventually.of_forall fun N => (f N).continuous)
· -- Check that `F` is bounded
rcases (f 0).bounded with ⟨C, hC⟩
refine ⟨C + (b 0 + b 0), fun x y => ?_⟩
calc
dist (F x) (F y) ≤ dist (f 0 x) (f 0 y) + (dist (f 0 x) (F x) + dist (f 0 y) (F y)) :=
dist_triangle4_left _ _ _ _
_ ≤ C + (b 0 + b 0) := add_le_add (hC x y) (add_le_add (fF_bdd x 0) (fF_bdd y 0))
· -- Check that `F` is close to `f N` in distance terms
refine tendsto_iff_dist_tendsto_zero.2 (squeeze_zero (fun _ => dist_nonneg) ?_ b_lim)
exact fun N => (dist_le (b0 _)).2 fun x => fF_bdd x N
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instCompleteSpace
|
Bounded continuous functions taking values in a complete space form a complete space.
|
compContinuous {δ : Type*} [TopologicalSpace δ] (f : α →ᵇ β) (g : C(δ, α)) : δ →ᵇ β where
toContinuousMap := f.1.comp g
map_bounded' := f.map_bounded'.imp fun _ hC _ _ => hC _ _
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
compContinuous
|
Composition of a bounded continuous function and a continuous function.
|
coe_compContinuous {δ : Type*} [TopologicalSpace δ] (f : α →ᵇ β) (g : C(δ, α)) :
⇑(f.compContinuous g) = f ∘ g := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_compContinuous
| null |
compContinuous_apply {δ : Type*} [TopologicalSpace δ] (f : α →ᵇ β) (g : C(δ, α)) (x : δ) :
f.compContinuous g x = f (g x) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
compContinuous_apply
| null |
lipschitz_compContinuous {δ : Type*} [TopologicalSpace δ] (g : C(δ, α)) :
LipschitzWith 1 fun f : α →ᵇ β => f.compContinuous g :=
LipschitzWith.mk_one fun _ _ => (dist_le dist_nonneg).2 fun x => dist_coe_le_dist (g x)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
lipschitz_compContinuous
| null |
continuous_compContinuous {δ : Type*} [TopologicalSpace δ] (g : C(δ, α)) :
Continuous fun f : α →ᵇ β => f.compContinuous g :=
(lipschitz_compContinuous g).continuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
continuous_compContinuous
| null |
restrict (f : α →ᵇ β) (s : Set α) : s →ᵇ β :=
f.compContinuous <| (ContinuousMap.id _).restrict s
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
restrict
|
Restrict a bounded continuous function to a set.
|
coe_restrict (f : α →ᵇ β) (s : Set α) : ⇑(f.restrict s) = f ∘ (↑) := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_restrict
| null |
restrict_apply (f : α →ᵇ β) (s : Set α) (x : s) : f.restrict s x = f x := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
restrict_apply
| null |
comp (G : β → γ) {C : ℝ≥0} (H : LipschitzWith C G) (f : α →ᵇ β) : α →ᵇ γ :=
⟨⟨fun x => G (f x), H.continuous.comp f.continuous⟩,
let ⟨D, hD⟩ := f.bounded
⟨max C 0 * D, fun x y =>
calc
dist (G (f x)) (G (f y)) ≤ C * dist (f x) (f y) := H.dist_le_mul _ _
_ ≤ max C 0 * dist (f x) (f y) := by gcongr; apply le_max_left
_ ≤ max C 0 * D := by gcongr; apply hD
⟩⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
comp
|
Composition (in the target) of a bounded continuous function with a Lipschitz map again
gives a bounded continuous function.
|
comp_apply (G : β → γ) {C : ℝ≥0} (H : LipschitzWith C G) (f : α →ᵇ β) (a : α) :
(f.comp G H) a = G (f a) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
comp_apply
| null |
lipschitz_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
LipschitzWith C (comp G H : (α →ᵇ β) → α →ᵇ γ) :=
LipschitzWith.of_dist_le_mul fun f g =>
(dist_le (mul_nonneg C.2 dist_nonneg)).2 fun x =>
calc
dist (G (f x)) (G (g x)) ≤ C * dist (f x) (g x) := H.dist_le_mul _ _
_ ≤ C * dist f g := by gcongr; apply dist_coe_le_dist
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
lipschitz_comp
|
The composition operator (in the target) with a Lipschitz map is Lipschitz.
|
uniformContinuous_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
UniformContinuous (comp G H : (α →ᵇ β) → α →ᵇ γ) :=
(lipschitz_comp H).uniformContinuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
uniformContinuous_comp
|
The composition operator (in the target) with a Lipschitz map is uniformly continuous.
|
continuous_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
Continuous (comp G H : (α →ᵇ β) → α →ᵇ γ) :=
(lipschitz_comp H).continuous
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
continuous_comp
|
The composition operator (in the target) with a Lipschitz map is continuous.
|
codRestrict (s : Set β) (f : α →ᵇ β) (H : ∀ x, f x ∈ s) : α →ᵇ s :=
⟨⟨s.codRestrict f H, f.continuous.subtype_mk _⟩, f.bounded⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
codRestrict
|
Restriction (in the target) of a bounded continuous function taking values in a subset.
|
@[simps]
noncomputable indicator (s : Set α) (hs : IsClopen s) : BoundedContinuousFunction α ℝ where
toFun := s.indicator 1
continuous_toFun := continuous_indicator (by simp [hs]) <| continuous_const.continuousOn
map_bounded' := ⟨1, fun x y ↦ by by_cases hx : x ∈ s <;> by_cases hy : y ∈ s <;> simp [hx, hy]⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
indicator
|
A version of `Function.extend` for bounded continuous maps. We assume that the domain has
discrete topology, so we only need to verify boundedness. -/
nonrec def extend (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : δ →ᵇ β where
toFun := extend f g h
continuous_toFun := continuous_of_discreteTopology
map_bounded' := by
rw [← isBounded_range_iff, range_extend f.injective]
exact g.isBounded_range.union (h.isBounded_image _)
@[simp]
theorem extend_apply (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) (x : α) : extend f g h (f x) = g x :=
f.injective.extend_apply _ _ _
@[simp]
nonrec theorem extend_comp (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : extend f g h ∘ f = g :=
extend_comp f.injective _ _
nonrec theorem extend_apply' {f : α ↪ δ} {x : δ} (hx : x ∉ range f) (g : α →ᵇ β) (h : δ →ᵇ β) :
extend f g h x = h x :=
extend_apply' _ _ _ hx
theorem extend_of_empty [IsEmpty α] (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : extend f g h = h :=
DFunLike.coe_injective <| Function.extend_of_isEmpty f g h
@[simp]
theorem dist_extend_extend (f : α ↪ δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
dist (g₁.extend f h₁) (g₂.extend f h₂) =
max (dist g₁ g₂) (dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ)) := by
refine le_antisymm ((dist_le <| le_max_iff.2 <| Or.inl dist_nonneg).2 fun x => ?_) (max_le ?_ ?_)
· rcases em (∃ y, f y = x) with (⟨x, rfl⟩ | hx)
· simp only [extend_apply]
exact (dist_coe_le_dist x).trans (le_max_left _ _)
· simp only [extend_apply' hx]
lift x to ((range f)ᶜ : Set δ) using hx
calc
dist (h₁ x) (h₂ x) = dist (h₁.restrict (range f)ᶜ x) (h₂.restrict (range f)ᶜ x) := rfl
_ ≤ dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ) := dist_coe_le_dist x
_ ≤ _ := le_max_right _ _
· refine (dist_le dist_nonneg).2 fun x => ?_
rw [← extend_apply f g₁ h₁, ← extend_apply f g₂ h₂]
exact dist_coe_le_dist _
· refine (dist_le dist_nonneg).2 fun x => ?_
calc
dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by
rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]
_ ≤ _ := dist_coe_le_dist _
theorem isometry_extend (f : α ↪ δ) (h : δ →ᵇ β) : Isometry fun g : α →ᵇ β => extend f g h :=
Isometry.of_dist_eq fun g₁ g₂ => by simp
end Extend
/-- The indicator function of a clopen set, as a bounded continuous function.
|
@[to_additive] instOne : One (α →ᵇ β) := ⟨const α 1⟩
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instOne
| null |
coe_one : ((1 : α →ᵇ β) : α → β) = 1 := rfl
@[to_additive (attr := simp)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_one
| null |
mkOfCompact_one [CompactSpace α] : mkOfCompact (1 : C(α, β)) = 1 := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfCompact_one
| null |
forall_coe_one_iff_one (f : α →ᵇ β) : (∀ x, f x = 1) ↔ f = 1 :=
(@DFunLike.ext_iff _ _ _ _ f 1).symm
@[to_additive (attr := simp)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
forall_coe_one_iff_one
| null |
one_compContinuous [TopologicalSpace γ] (f : C(γ, α)) : (1 : α →ᵇ β).compContinuous f = 1 :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
one_compContinuous
| null |
@[to_additive]
instMul [Mul R] [BoundedMul R] [ContinuousMul R] :
Mul (α →ᵇ R) where
mul f g :=
{ toFun := fun x ↦ f x * g x
continuous_toFun := f.continuous.mul g.continuous
map_bounded' := mul_bounded_of_bounded_of_bounded (map_bounded f) (map_bounded g) }
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instMul
| null |
coe_mul [Mul R] [BoundedMul R] [ContinuousMul R] (f g : α →ᵇ R) : ⇑(f * g) = f * g := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_mul
| null |
mul_apply [Mul R] [BoundedMul R] [ContinuousMul R] (f g : α →ᵇ R) (x : α) :
(f * g) x = f x * g x := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mul_apply
| null |
coe_nsmulRec [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β]
(f : α →ᵇ β) : ∀ n, ⇑(nsmulRec n f) = n • ⇑f
| 0 => by rw [nsmulRec, zero_smul, coe_zero]
| n + 1 => by rw [nsmulRec, succ_nsmul, coe_add, coe_nsmulRec _ n]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_nsmulRec
| null |
instSMulNat [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] :
SMul ℕ (α →ᵇ β) where
smul n f :=
{ toContinuousMap := n • f.toContinuousMap
map_bounded' := by simpa [coe_nsmulRec] using (nsmulRec n f).map_bounded' }
@[to_additive existing instSMulNat]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instSMulNat
| null |
instPow [Monoid R] [BoundedMul R] [ContinuousMul R] : Pow (α →ᵇ R) ℕ where
pow f n :=
{ toFun := fun x ↦ (f x) ^ n
continuous_toFun := f.continuous.pow n
map_bounded' := by
obtain ⟨C, hC⟩ := Metric.isBounded_iff.mp <| isBounded_pow (isBounded_range f) n
exact ⟨C, fun x y ↦ hC (by simp) (by simp)⟩ }
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instPow
| null |
coe_pow [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ℕ) (f : α →ᵇ R) :
⇑(f ^ n) = (⇑f) ^ n := rfl
@[to_additive (attr := simp)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_pow
| null |
pow_apply [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ℕ) (f : α →ᵇ R) (x : α) :
(f ^ n) x = f x ^ n := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
pow_apply
| null |
instMonoid [Monoid R] [BoundedMul R] [ContinuousMul R] :
Monoid (α →ᵇ R) :=
Injective.monoid _ DFunLike.coe_injective' rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instMonoid
| null |
instCommMonoid [CommMonoid R] [BoundedMul R] [ContinuousMul R] :
CommMonoid (α →ᵇ R) where
__ := instMonoid
mul_comm f g := by ext x; simp [mul_apply, mul_comm]
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instCommMonoid
| null |
instMulOneClass [MulOneClass R] [BoundedMul R] [ContinuousMul R] : MulOneClass (α →ᵇ R) :=
DFunLike.coe_injective.mulOneClass _ coe_one coe_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instMulOneClass
| null |
@[to_additive (attr := simps)
/-- Composition on the left by a (lipschitz-continuous) homomorphism of topological `AddMonoid`s,
as a `AddMonoidHom`. Similar to `AddMonoidHom.compLeftContinuous`. -/]
protected _root_.MonoidHom.compLeftContinuousBounded (α : Type*)
[TopologicalSpace α] [PseudoMetricSpace β] [Monoid β] [BoundedMul β] [ContinuousMul β]
[PseudoMetricSpace γ] [Monoid γ] [BoundedMul γ] [ContinuousMul γ]
(g : β →* γ) {C : NNReal} (hg : LipschitzWith C g) :
(α →ᵇ β) →* (α →ᵇ γ) where
toFun f := f.comp g hg
map_one' := ext fun _ => g.map_one
map_mul' _ _ := ext fun _ => g.map_mul _ _
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
_root_.MonoidHom.compLeftContinuousBounded
|
Composition on the left by a (lipschitz-continuous) homomorphism of topological monoids, as a
`MonoidHom`. Similar to `MonoidHom.compLeftContinuous`.
|
@[simp]
mkOfCompact_add [CompactSpace α] [Add β] [BoundedAdd β] [ContinuousAdd β] (f g : C(α, β)) :
mkOfCompact (f + g) = mkOfCompact f + mkOfCompact g := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfCompact_add
| null |
add_compContinuous [Add β] [BoundedAdd β] [ContinuousAdd β] [TopologicalSpace γ]
(f g : α →ᵇ β) (h : C(γ, α)) :
(g + f).compContinuous h = g.compContinuous h + f.compContinuous h := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
add_compContinuous
| null |
@[simps]
coeFnAddHom [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] : (α →ᵇ β) →+ α → β where
toFun := (⇑)
map_zero' := coe_zero
map_add' := coe_add
variable (α β)
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coeFnAddHom
|
Coercion of a `NormedAddGroupHom` is an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`.
|
@[simps]
toContinuousMapAddHom [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] : (α →ᵇ β) →+ C(α, β) where
toFun := toContinuousMap
map_zero' := rfl
map_add' := by
intros
ext
simp
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
toContinuousMapAddHom
|
The additive map forgetting that a bounded continuous function is bounded.
|
@[simp]
coe_sum {ι : Type*} (s : Finset ι) (f : ι → α →ᵇ β) :
⇑(∑ i ∈ s, f i) = ∑ i ∈ s, (f i : α → β) :=
map_sum coeFnAddHom f s
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_sum
| null |
sum_apply {ι : Type*} (s : Finset ι) (f : ι → α →ᵇ β) (a : α) :
(∑ i ∈ s, f i) a = ∑ i ∈ s, f i a := by simp
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
sum_apply
| null |
instLipschitzAdd : LipschitzAdd (α →ᵇ β) where
lipschitz_add :=
⟨LipschitzAdd.C β, by
have C_nonneg := (LipschitzAdd.C β).coe_nonneg
rw [lipschitzWith_iff_dist_le_mul]
rintro ⟨f₁, g₁⟩ ⟨f₂, g₂⟩
rw [dist_le (mul_nonneg C_nonneg dist_nonneg)]
intro x
refine le_trans (lipschitz_with_lipschitz_const_add ⟨f₁ x, g₁ x⟩ ⟨f₂ x, g₂ x⟩) ?_
gcongr
apply max_le_max <;> exact dist_coe_le_dist x⟩
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instLipschitzAdd
| null |
instSub : Sub (α →ᵇ R) where
sub f g :=
{ toFun := fun x ↦ (f x - g x),
map_bounded' := sub_bounded_of_bounded_of_bounded f.map_bounded' g.map_bounded' }
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instSub
|
The pointwise difference of two bounded continuous functions is again bounded continuous.
|
sub_apply {x : α} : (f - g) x = f x - g x := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
sub_apply
| null |
coe_sub : ⇑(f - g) = f - g := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_sub
| null |
@[simp]
natCast_apply [NatCast β] (n : ℕ) (x : α) : (n : α →ᵇ β) x = n := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
natCast_apply
| null |
@[simp]
intCast_apply [IntCast β] (m : ℤ) (x : α) : (m : α →ᵇ β) x = m := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
intCast_apply
| null |
instSemiring {R : Type*} [TopologicalSpace α] [PseudoMetricSpace R]
[Semiring R] [BoundedMul R] [ContinuousMul R] [BoundedAdd R] [ContinuousAdd R] :
Semiring (α →ᵇ R) :=
Injective.semiring _ DFunLike.coe_injective'
rfl rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl)
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instSemiring
| null |
instSMul : SMul 𝕜 (α →ᵇ β) where
smul c f :=
{ toContinuousMap := c • f.toContinuousMap
map_bounded' :=
let ⟨b, hb⟩ := f.bounded
⟨dist c 0 * b, fun x y => by
refine (dist_smul_pair c (f x) (f y)).trans ?_
gcongr
apply hb⟩ }
@[simp]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instSMul
| null |
coe_smul (c : 𝕜) (f : α →ᵇ β) : ⇑(c • f) = fun x => c • f x := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_smul
| null |
smul_apply (c : 𝕜) (f : α →ᵇ β) (x : α) : (c • f) x = c • f x := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
smul_apply
| null |
instIsScalarTower {𝕜' : Type*} [PseudoMetricSpace 𝕜'] [Zero 𝕜'] [SMul 𝕜' β]
[IsBoundedSMul 𝕜' β] [SMul 𝕜' 𝕜] [IsScalarTower 𝕜' 𝕜 β] :
IsScalarTower 𝕜' 𝕜 (α →ᵇ β) where
smul_assoc _ _ _ := ext fun _ ↦ smul_assoc ..
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instIsScalarTower
| null |
instSMulCommClass {𝕜' : Type*} [PseudoMetricSpace 𝕜'] [Zero 𝕜'] [SMul 𝕜' β]
[IsBoundedSMul 𝕜' β] [SMulCommClass 𝕜' 𝕜 β] :
SMulCommClass 𝕜' 𝕜 (α →ᵇ β) where
smul_comm _ _ _ := ext fun _ ↦ smul_comm ..
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instSMulCommClass
| null |
instIsCentralScalar [SMul 𝕜ᵐᵒᵖ β] [IsCentralScalar 𝕜 β] : IsCentralScalar 𝕜 (α →ᵇ β) where
op_smul_eq_smul _ _ := ext fun _ => op_smul_eq_smul _ _
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instIsCentralScalar
| null |
instIsBoundedSMul : IsBoundedSMul 𝕜 (α →ᵇ β) where
dist_smul_pair' c f₁ f₂ := by
rw [dist_le (mul_nonneg dist_nonneg dist_nonneg)]
intro x
refine (dist_smul_pair c (f₁ x) (f₂ x)).trans ?_
gcongr
apply dist_coe_le_dist
dist_pair_smul' c₁ c₂ f := by
rw [dist_le (by positivity)]
intro x
refine (dist_pair_smul c₁ c₂ (f x)).trans ?_
gcongr
apply dist_coe_le_dist (g := 0)
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instIsBoundedSMul
| null |
instMulAction : MulAction 𝕜 (α →ᵇ β) :=
DFunLike.coe_injective.mulAction _ coe_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instMulAction
| null |
instDistribMulAction : DistribMulAction 𝕜 (α →ᵇ β) :=
DFunLike.coe_injective.distribMulAction ⟨⟨_, coe_zero⟩, coe_add⟩ coe_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instDistribMulAction
| null |
instModule : Module 𝕜 (α →ᵇ β) :=
DFunLike.coe_injective.module _ ⟨⟨_, coe_zero⟩, coe_add⟩ coe_smul
variable (𝕜)
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instModule
| null |
@[simps]
evalCLM (x : α) : (α →ᵇ β) →L[𝕜] β where
toFun f := f x
map_add' _ _ := add_apply _ _ _
map_smul' _ _ := smul_apply _ _ _
variable (α β)
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
evalCLM
|
The evaluation at a point, as a continuous linear map from `α →ᵇ β` to `β`.
|
@[simps]
toContinuousMapLinearMap : (α →ᵇ β) →ₗ[𝕜] C(α, β) where
toFun := toContinuousMap
map_smul' _ _ := rfl
map_add' _ _ := rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
toContinuousMapLinearMap
|
The linear map forgetting that a bounded continuous function is bounded.
|
NNReal.upper_bound {α : Type*} [TopologicalSpace α] (f : α →ᵇ ℝ≥0) (x : α) :
f x ≤ nndist f 0 := by
have key : nndist (f x) ((0 : α →ᵇ ℝ≥0) x) ≤ nndist f 0 := @dist_coe_le_dist α ℝ≥0 _ _ f 0 x
simp only [coe_zero, Pi.zero_apply] at key
rwa [NNReal.nndist_zero_eq_val' (f x)] at key
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
NNReal.upper_bound
| null |
instNorm : Norm (α →ᵇ β) := ⟨(dist · 0)⟩
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
instNorm
| null |
norm_def : ‖f‖ = dist f 0 := rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_def
| null |
norm_eq (f : α →ᵇ β) : ‖f‖ = sInf { C : ℝ | 0 ≤ C ∧ ∀ x : α, ‖f x‖ ≤ C } := by
simp [norm_def, BoundedContinuousFunction.dist_eq]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_eq
|
The norm of a bounded continuous function is the supremum of `‖f x‖`.
We use `sInf` to ensure that the definition works if `α` has no elements.
|
norm_eq_of_nonempty [h : Nonempty α] : ‖f‖ = sInf { C : ℝ | ∀ x : α, ‖f x‖ ≤ C } := by
obtain ⟨a⟩ := h
rw [norm_eq]
congr
ext
simp only [and_iff_right_iff_imp]
exact fun h' => le_trans (norm_nonneg (f a)) (h' a)
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_eq_of_nonempty
|
When the domain is non-empty, we do not need the `0 ≤ C` condition in the formula for `‖f‖` as a
`sInf`.
|
norm_eq_zero_of_empty [IsEmpty α] : ‖f‖ = 0 :=
dist_zero_of_empty
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_eq_zero_of_empty
| null |
norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
calc
‖f x‖ = dist (f x) ((0 : α →ᵇ β) x) := by simp [dist_zero_right]
_ ≤ ‖f‖ := dist_coe_le_dist _
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_coe_le_norm
| null |
neg_norm_le_apply (f : α →ᵇ ℝ) (x : α) :
-‖f‖ ≤ f x := (abs_le.mp (norm_coe_le_norm f x)).1
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
neg_norm_le_apply
| null |
apply_le_norm (f : α →ᵇ ℝ) (x : α) :
f x ≤ ‖f‖ := (abs_le.mp (norm_coe_le_norm f x)).2
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
apply_le_norm
| null |
dist_le_two_norm' {f : γ → β} {C : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (x y : γ) :
dist (f x) (f y) ≤ 2 * C :=
calc
dist (f x) (f y) ≤ ‖f x‖ + ‖f y‖ := dist_le_norm_add_norm _ _
_ ≤ C + C := add_le_add (hC x) (hC y)
_ = 2 * C := (two_mul _).symm
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
dist_le_two_norm'
| null |
dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
dist_le_two_norm' f.norm_coe_le_norm x y
variable {f}
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
dist_le_two_norm
|
Distance between the images of any two points is at most twice the norm of the function.
|
norm_le (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C := by
simpa using @dist_le _ _ _ _ f 0 _ C0
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_le
|
The norm of a function is controlled by the supremum of the pointwise norms.
|
norm_le_of_nonempty [Nonempty α] {f : α →ᵇ β} {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M := by
simp_rw [norm_def, ← dist_zero_right]
exact dist_le_iff_of_nonempty
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_le_of_nonempty
| null |
norm_lt_iff_of_compact [CompactSpace α] {f : α →ᵇ β} {M : ℝ} (M0 : 0 < M) :
‖f‖ < M ↔ ∀ x, ‖f x‖ < M := by
simp_rw [norm_def, ← dist_zero_right]
exact dist_lt_iff_of_compact M0
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_lt_iff_of_compact
| null |
norm_lt_iff_of_nonempty_compact [Nonempty α] [CompactSpace α] {f : α →ᵇ β} {M : ℝ} :
‖f‖ < M ↔ ∀ x, ‖f x‖ < M := by
simp_rw [norm_def, ← dist_zero_right]
exact dist_lt_iff_of_nonempty_compact
variable (f)
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_lt_iff_of_nonempty_compact
| null |
norm_const_le (b : β) : ‖const α b‖ ≤ ‖b‖ :=
(norm_le (norm_nonneg b)).2 fun _ => le_rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_const_le
|
Norm of `const α b` is less than or equal to `‖b‖`. If `α` is nonempty,
then it is equal to `‖b‖`.
|
norm_const_eq [h : Nonempty α] (b : β) : ‖const α b‖ = ‖b‖ :=
le_antisymm (norm_const_le b) <| h.elim fun x => (const α b).norm_coe_le_norm x
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_const_eq
| null |
ofNormedAddCommGroup {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β]
(f : α → β) (Hf : Continuous f) (C : ℝ) (H : ∀ x, ‖f x‖ ≤ C) : α →ᵇ β :=
⟨⟨fun n => f n, Hf⟩, ⟨_, dist_le_two_norm' H⟩⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
ofNormedAddCommGroup
|
Constructing a bounded continuous function from a uniformly bounded continuous
function taking values in a normed group.
|
coe_ofNormedAddCommGroup {α : Type u} {β : Type v} [TopologicalSpace α]
[SeminormedAddCommGroup β] (f : α → β) (Hf : Continuous f) (C : ℝ) (H : ∀ x, ‖f x‖ ≤ C) :
(ofNormedAddCommGroup f Hf C H : α → β) = f := rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
coe_ofNormedAddCommGroup
| null |
norm_ofNormedAddCommGroup_le {f : α → β} (hfc : Continuous f) {C : ℝ} (hC : 0 ≤ C)
(hfC : ∀ x, ‖f x‖ ≤ C) : ‖ofNormedAddCommGroup f hfc C hfC‖ ≤ C :=
(norm_le hC).2 hfC
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_ofNormedAddCommGroup_le
| null |
ofNormedAddCommGroupDiscrete {α : Type u} {β : Type v} [TopologicalSpace α] [DiscreteTopology α]
[SeminormedAddCommGroup β] (f : α → β) (C : ℝ) (H : ∀ x, norm (f x) ≤ C) : α →ᵇ β :=
ofNormedAddCommGroup f continuous_of_discreteTopology C H
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
ofNormedAddCommGroupDiscrete
|
Constructing a bounded continuous function from a uniformly bounded
function on a discrete space, taking values in a normed group.
|
coe_ofNormedAddCommGroupDiscrete {α : Type u} {β : Type v} [TopologicalSpace α]
[DiscreteTopology α] [SeminormedAddCommGroup β] (f : α → β) (C : ℝ) (H : ∀ x, ‖f x‖ ≤ C) :
(ofNormedAddCommGroupDiscrete f C H : α → β) = f := rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
coe_ofNormedAddCommGroupDiscrete
| null |
normComp : α →ᵇ ℝ :=
f.comp norm lipschitzWith_one_norm
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
normComp
|
Taking the pointwise norm of a bounded continuous function with values in a
`SeminormedAddCommGroup` yields a bounded continuous function with values in ℝ.
|
coe_normComp : (f.normComp : α → ℝ) = norm ∘ f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
coe_normComp
| null |
norm_normComp : ‖f.normComp‖ = ‖f‖ := by
simp only [norm_eq, coe_normComp, norm_norm, Function.comp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_normComp
| null |
bddAbove_range_norm_comp : BddAbove <| Set.range <| norm ∘ f :=
(@isBounded_range _ _ _ _ f.normComp).bddAbove
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
bddAbove_range_norm_comp
| null |
norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ := by
simp_rw [norm_def, dist_eq_iSup, coe_zero, Pi.zero_apply, dist_zero_right]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_eq_iSup_norm
| null |
instNormOneClass [Nonempty α] [One β] [NormOneClass β] : NormOneClass (α →ᵇ β) where
norm_one := by simp only [norm_eq_iSup_norm, coe_one, Pi.one_apply, norm_one, ciSup_const]
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
instNormOneClass
|
If `‖(1 : β)‖ = 1`, then `‖(1 : α →ᵇ β)‖ = 1` if `α` is nonempty.
|
nnnorm_le (C : ℝ≥0) : ‖f‖₊ ≤ C ↔ ∀ x : α, ‖f x‖₊ ≤ C :=
norm_le C.prop
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
nnnorm_le
|
The pointwise opposite of a bounded continuous function is again bounded continuous. -/
instance : Neg (α →ᵇ β) :=
⟨fun f =>
ofNormedAddCommGroup (-f) f.continuous.neg ‖f‖ fun x =>
norm_neg ((⇑f) x) ▸ f.norm_coe_le_norm x⟩
@[simp]
theorem coe_neg : ⇑(-f) = -f := rfl
theorem neg_apply : (-f) x = -f x := rfl
@[simp]
theorem mkOfCompact_neg [CompactSpace α] (f : C(α, β)) : mkOfCompact (-f) = -mkOfCompact f := rfl
@[simp]
theorem mkOfCompact_sub [CompactSpace α] (f g : C(α, β)) :
mkOfCompact (f - g) = mkOfCompact f - mkOfCompact g := rfl
@[simp]
theorem coe_zsmulRec : ∀ z, ⇑(zsmulRec (· • ·) z f) = z • ⇑f
| Int.ofNat n => by rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmul, natCast_zsmul]
| Int.negSucc n => by rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmul]
instance instSMulInt : SMul ℤ (α →ᵇ β) where
smul n f :=
{ toContinuousMap := n • f.toContinuousMap
map_bounded' := by simpa using (zsmulRec (· • ·) n f).map_bounded' }
@[simp]
theorem coe_zsmul (r : ℤ) (f : α →ᵇ β) : ⇑(r • f) = r • ⇑f := rfl
@[simp]
theorem zsmul_apply (r : ℤ) (f : α →ᵇ β) (v : α) : (r • f) v = r • f v := rfl
instance instAddCommGroup : AddCommGroup (α →ᵇ β) := fast_instance%
DFunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => coe_nsmul _ _)
fun _ _ => coe_zsmul _ _
instance instSeminormedAddCommGroup : SeminormedAddCommGroup (α →ᵇ β) where
dist_eq f g := by simp only [norm_eq, dist_eq, dist_eq_norm, sub_apply]
instance instNormedAddCommGroup {α β} [TopologicalSpace α] [NormedAddCommGroup β] :
NormedAddCommGroup (α →ᵇ β) :=
{ instSeminormedAddCommGroup with
eq_of_dist_eq_zero }
theorem nnnorm_def : ‖f‖₊ = nndist f 0 := rfl
theorem nnnorm_coe_le_nnnorm (x : α) : ‖f x‖₊ ≤ ‖f‖₊ :=
norm_coe_le_norm _ _
theorem nndist_le_two_nnnorm (x y : α) : nndist (f x) (f y) ≤ 2 * ‖f‖₊ :=
dist_le_two_norm _ _ _
/-- The `nnnorm` of a function is controlled by the supremum of the pointwise `nnnorm`s.
|
nnnorm_const_le (b : β) : ‖const α b‖₊ ≤ ‖b‖₊ :=
norm_const_le _
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
nnnorm_const_le
| null |
nnnorm_const_eq [Nonempty α] (b : β) : ‖const α b‖₊ = ‖b‖₊ :=
Subtype.ext <| norm_const_eq _
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
nnnorm_const_eq
| null |
nnnorm_eq_iSup_nnnorm : ‖f‖₊ = ⨆ x : α, ‖f x‖₊ :=
Subtype.ext <| (norm_eq_iSup_norm f).trans <| by simp_rw [val_eq_coe, NNReal.coe_iSup, coe_nnnorm]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
nnnorm_eq_iSup_nnnorm
| null |
enorm_eq_iSup_enorm : ‖f‖ₑ = ⨆ x, ‖f x‖ₑ := by
simpa only [← edist_zero_eq_enorm] using edist_eq_iSup
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
enorm_eq_iSup_enorm
| null |
abs_diff_coe_le_dist : ‖f x - g x‖ ≤ dist f g := by
rw [dist_eq_norm]
exact (f - g).norm_coe_le_norm x
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
abs_diff_coe_le_dist
| null |
coe_le_coe_add_dist {f g : α →ᵇ ℝ} : f x ≤ g x + dist f g :=
sub_le_iff_le_add'.1 <| (abs_le.1 <| @dist_coe_le_dist _ _ _ _ f g x).2
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
coe_le_coe_add_dist
| null |
norm_compContinuous_le [TopologicalSpace γ] (f : α →ᵇ β) (g : C(γ, α)) :
‖f.compContinuous g‖ ≤ ‖f‖ :=
((lipschitz_compContinuous g).dist_le_mul f 0).trans <| by
rw [NNReal.coe_one, one_mul, dist_zero_right]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
norm_compContinuous_le
| null |
instNormedSpace [NormedField 𝕜] [NormedSpace 𝕜 β] : NormedSpace 𝕜 (α →ᵇ β) :=
⟨fun c f => by
refine norm_ofNormedAddCommGroup_le _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) ?_
exact fun x =>
norm_smul c (f x) ▸ mul_le_mul_of_nonneg_left (f.norm_coe_le_norm _) (norm_nonneg _)⟩
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 β]
variable [SeminormedAddCommGroup γ] [NormedSpace 𝕜 γ]
variable (α)
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
instNormedSpace
| null |
protected _root_.ContinuousLinearMap.compLeftContinuousBounded (g : β →L[𝕜] γ) :
(α →ᵇ β) →L[𝕜] α →ᵇ γ :=
LinearMap.mkContinuous
{ toFun := fun f =>
ofNormedAddCommGroup (g ∘ f) (g.continuous.comp f.continuous) (‖g‖ * ‖f‖) fun x =>
g.le_opNorm_of_le (f.norm_coe_le_norm x)
map_add' := fun f g => by ext; simp
map_smul' := fun c f => by ext; simp } ‖g‖ fun f =>
norm_ofNormedAddCommGroup_le _ (mul_nonneg (norm_nonneg g) (norm_nonneg f))
(fun x => by exact g.le_opNorm_of_le (f.norm_coe_le_norm x))
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
_root_.ContinuousLinearMap.compLeftContinuousBounded
|
Postcomposition of bounded continuous functions into a normed module by a continuous linear map
is a continuous linear map.
Upgraded version of `ContinuousLinearMap.compLeftContinuous`, similar to `LinearMap.compLeft`.
|
_root_.ContinuousLinearMap.compLeftContinuousBounded_apply (g : β →L[𝕜] γ) (f : α →ᵇ β)
(x : α) : (g.compLeftContinuousBounded α f) x = g (f x) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
_root_.ContinuousLinearMap.compLeftContinuousBounded_apply
| null |
instNonUnitalRing : NonUnitalRing (α →ᵇ R) := fast_instance%
DFunLike.coe_injective.nonUnitalRing _ coe_zero coe_add coe_mul coe_neg coe_sub
(fun _ _ => coe_nsmul _ _) fun _ _ => coe_zsmul _ _
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
instNonUnitalRing
| null |
instNonUnitalSeminormedRing : NonUnitalSeminormedRing (α →ᵇ R) where
__ := instSeminormedAddCommGroup
__ := instNonUnitalRing
norm_mul_le f g := norm_ofNormedAddCommGroup_le _ (by positivity)
(fun x ↦ (norm_mul_le _ _).trans <| mul_le_mul
(norm_coe_le_norm f x) (norm_coe_le_norm g x) (norm_nonneg _) (norm_nonneg _))
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
instNonUnitalSeminormedRing
| null |
instNonUnitalSeminormedCommRing [NonUnitalSeminormedCommRing R] :
NonUnitalSeminormedCommRing (α →ᵇ R) where
mul_comm _ _ := ext fun _ ↦ mul_comm ..
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] |
Mathlib/Topology/ContinuousMap/Bounded/Normed.lean
|
instNonUnitalSeminormedCommRing
| null |
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