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 ⌀ |
|---|---|---|---|---|---|---|
ContinuousMap.induction_on_of_compact {𝕜 : Type*} [RCLike 𝕜] {s : Set 𝕜} [CompactSpace s]
{p : C(s, 𝕜) → Prop} (const : ∀ r, p (.const s r)) (id : p (.restrict s <| .id 𝕜))
(star_id : p (star (.restrict s <| .id 𝕜)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(frequently : ∀ f, (∃ᶠ g in 𝓝 f, p g) → p f) (f : C(s, 𝕜)) :
p f := by
refine f.induction_on const id star_id add mul fun h f ↦ frequently f ?_
have := polynomialFunctions.starClosure_topologicalClosure s ▸ mem_top (x := f)
rw [← SetLike.mem_coe, topologicalClosure_coe, mem_closure_iff_frequently] at this
exact this.mp <| .of_forall h | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMap.induction_on_of_compact | null |
@[ext (iff := false)]
ContinuousMap.algHom_ext_map_X {A : Type*} [Semiring A]
[Algebra ℝ A] [TopologicalSpace A] [T2Space A] {s : Set ℝ} [CompactSpace s]
{φ ψ : C(s, ℝ) →ₐ[ℝ] A} (hφ : Continuous φ) (hψ : Continuous ψ)
(h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by
suffices (⊤ : Subalgebra ℝ C(s, ℝ)) ≤ AlgHom.equalizer φ ψ from
AlgHom.ext fun x => this (by trivial)
rw [← polynomialFunctions.topologicalClosure s]
exact Subalgebra.topologicalClosure_minimal
(polynomialFunctions.le_equalizer s φ ψ h) (isClosed_eq hφ hψ) | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMap.algHom_ext_map_X | Continuous algebra homomorphisms from `C(s, ℝ)` into an `ℝ`-algebra `A` which agree
at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. |
@[ext (iff := false)]
ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type*} [RCLike 𝕜] [Ring A] [StarRing A]
[Algebra 𝕜 A] [TopologicalSpace A] [T2Space A] {s : Set 𝕜} [CompactSpace s]
{φ ψ : C(s, 𝕜) →⋆ₐ[𝕜] A} (hφ : Continuous φ) (hψ : Continuous ψ)
(h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by
suffices (⊤ : StarSubalgebra 𝕜 C(s, 𝕜)) ≤ StarAlgHom.equalizer φ ψ from
StarAlgHom.ext fun x => this mem_top
rw [← polynomialFunctions.starClosure_topologicalClosure s]
exact StarSubalgebra.topologicalClosure_minimal
(polynomialFunctions.starClosure_le_equalizer s φ ψ h) (isClosed_eq hφ hψ) | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMap.starAlgHom_ext_map_X | Continuous star algebra homomorphisms from `C(s, 𝕜)` into a star `𝕜`-algebra `A` which agree
at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. |
adjoin_id_eq_span_one_union (s : Set 𝕜) :
((StarAlgebra.adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))}) : Set C(s, 𝕜)) =
span 𝕜 ({(1 : C(s, 𝕜))} ∪ (adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))})) := by
ext x
rw [SetLike.mem_coe, SetLike.mem_coe, ← StarAlgebra.adjoin_nonUnitalStarSubalgebra,
← StarSubalgebra.mem_toSubalgebra, ← Subalgebra.mem_toSubmodule,
StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, span_union, span_eq_toSubmodule]
open Pointwise in | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | adjoin_id_eq_span_one_union | null |
adjoin_id_eq_span_one_add (s : Set 𝕜) :
((StarAlgebra.adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))}) : Set C(s, 𝕜)) =
(span 𝕜 {(1 : C(s, 𝕜))} : Set C(s, 𝕜)) + (adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))}) := by
ext x
rw [SetLike.mem_coe, ← StarAlgebra.adjoin_nonUnitalStarSubalgebra,
← StarSubalgebra.mem_toSubalgebra, ← Subalgebra.mem_toSubmodule,
StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, mem_sup]
simp [Set.mem_add] | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | adjoin_id_eq_span_one_add | null |
nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom {s : Set 𝕜} (h0 : 0 ∈ s) :
(adjoin 𝕜 {restrict s (.id 𝕜)} : Set C(s, 𝕜)) ⊆
RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) := by
intro f hf
induction hf using adjoin_induction with
| mem f hf =>
obtain rfl := Set.mem_singleton_iff.mp hf
rfl
| add f g _ _ hf hg => exact add_mem hf hg
| zero => exact zero_mem _
| mul f g _ _ _ hg => exact Ideal.mul_mem_left _ f hg
| smul r f _ hf =>
rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
rw [map_smul, hf, smul_zero]
| star f _ hf =>
rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
rw [map_star, hf, star_zero] | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom | null |
ker_evalStarAlgHom_inter_adjoin_id (s : Set 𝕜) (h0 : 0 ∈ s) :
(StarAlgebra.adjoin 𝕜 {restrict s (.id 𝕜)} : Set C(s, 𝕜)) ∩
RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) = adjoin 𝕜 {restrict s (.id 𝕜)} := by
ext f
constructor
· rintro ⟨hf₁, hf₂⟩
rw [SetLike.mem_coe] at hf₂ ⊢
simp_rw [adjoin_id_eq_span_one_add, Set.mem_add, SetLike.mem_coe, mem_span_singleton] at hf₁
obtain ⟨-, ⟨r, rfl⟩, f, hf, rfl⟩ := hf₁
have := nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom h0 hf
simp only [SetLike.mem_coe, RingHom.mem_ker, evalStarAlgHom_apply] at hf₂ this
rw [add_apply, this, add_zero, smul_apply, one_apply, smul_eq_mul, mul_one] at hf₂
rwa [hf₂, zero_smul, zero_add]
· simp only [Set.mem_inter_iff, SetLike.mem_coe]
refine fun hf ↦ ⟨?_, nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom h0 hf⟩
exact adjoin_le_starAlgebra_adjoin _ _ hf
open RingHom Filter Topology in | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ker_evalStarAlgHom_inter_adjoin_id | null |
AlgHom.closure_ker_inter {F S K A : Type*} [CommRing K] [Ring A] [Algebra K A]
[TopologicalSpace K] [T1Space K] [TopologicalSpace A] [ContinuousSub A] [ContinuousSMul K A]
[FunLike F A K] [AlgHomClass F K A K] [SetLike S A] [OneMemClass S A] [AddSubgroupClass S A]
[SMulMemClass S K A] (φ : F) (hφ : Continuous φ) (s : S) :
closure (s ∩ RingHom.ker φ) = closure s ∩ (ker φ : Set A) := by
refine subset_antisymm ?_ ?_
· simpa only [ker_eq, (isClosed_singleton.preimage hφ).closure_eq]
using closure_inter_subset_inter_closure s (ker φ : Set A)
· intro x ⟨hxs, (hxφ : φ x = 0)⟩
rw [mem_closure_iff_clusterPt, ClusterPt] at hxs
have : Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 s) (𝓝 x) := by
conv => congr; rfl; rfl; rw [← sub_zero x, ← zero_smul K 1, ← hxφ]
exact Filter.tendsto_inf_left (Continuous.tendsto (by fun_prop) x)
refine mem_closure_of_tendsto this <| eventually_inf_principal.mpr ?_
filter_upwards [] with g hg using
⟨sub_mem hg (SMulMemClass.smul_mem _ <| one_mem _), by simp [RingHom.mem_ker]⟩ | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | AlgHom.closure_ker_inter | null |
ker_evalStarAlgHom_eq_closure_adjoin_id (s : Set 𝕜) (h0 : 0 ∈ s) [CompactSpace s] :
(RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) : Set C(s, 𝕜)) =
closure (adjoin 𝕜 {(restrict s (.id 𝕜))}) := by
rw [← ker_evalStarAlgHom_inter_adjoin_id s h0,
AlgHom.closure_ker_inter (φ := evalStarAlgHom 𝕜 𝕜 (X := s) ⟨0, h0⟩) (continuous_eval_const _) _]
convert (Set.univ_inter _).symm
rw [← Polynomial.toContinuousMapOn_X_eq_restrict_id, ← Polynomial.toContinuousMapOnAlgHom_apply,
← polynomialFunctions.starClosure_eq_adjoin_X s]
congrm(($(polynomialFunctions.starClosure_topologicalClosure s) : Set C(s, 𝕜))) | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ker_evalStarAlgHom_eq_closure_adjoin_id | null |
ContinuousMapZero.adjoin_id_dense (s : Set 𝕜) [Fact (0 ∈ s)]
[CompactSpace s] : Dense (adjoin 𝕜 {(.id s : C(s, 𝕜)₀)} : Set C(s, 𝕜)₀) := by
have h0' : 0 ∈ s := Fact.out
rw [dense_iff_closure_eq,
← isClosedEmbedding_toContinuousMap.injective.preimage_image (closure _),
← isClosedEmbedding_toContinuousMap.closure_image_eq, ← coe_toContinuousMapHom,
← NonUnitalStarSubalgebra.coe_map, NonUnitalStarAlgHom.map_adjoin_singleton,
toContinuousMapHom_apply, toContinuousMap_id,
← ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id s h0']
apply Set.eq_univ_of_forall fun f ↦ ?_
simp only [Set.mem_preimage, toContinuousMapHom_apply, SetLike.mem_coe, RingHom.mem_ker,
ContinuousMap.evalStarAlgHom_apply, ContinuousMap.coe_coe]
exact map_zero f
open NonUnitalStarAlgebra in | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMapZero.adjoin_id_dense | If `s : Set 𝕜` with `RCLike 𝕜` is compact and contains `0`, then the non-unital star subalgebra
generated by the identity function in `C(s, 𝕜)₀` is dense. This can be seen as a version of the
Weierstrass approximation theorem. |
ContinuousMapZero.elemental_eq_top {𝕜 : Type*} [RCLike 𝕜] (s : Set 𝕜) [Fact (0 ∈ s)]
[CompactSpace s] : elemental 𝕜 (ContinuousMapZero.id s) = ⊤ :=
SetLike.ext'_iff.mpr (adjoin_id_dense s).closure_eq | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMapZero.elemental_eq_top | null |
@[elab_as_elim]
ContinuousMapZero.induction_on {s : Set 𝕜} [Fact (0 ∈ s)]
{p : C(s, 𝕜)₀ → Prop} (zero : p 0) (id : p (.id s)) (star_id : p (star (.id s)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(smul : ∀ (r : 𝕜) f, p f → p (r • f))
(closure : (∀ f ∈ adjoin 𝕜 {(.id s : C(s, 𝕜)₀)}, p f) → ∀ f, p f) (f : C(s, 𝕜)₀) :
p f := by
refine closure (fun f hf => ?_) f
induction hf using NonUnitalAlgebra.adjoin_induction with
| mem f hf =>
simp only [Set.mem_union, Set.mem_singleton_iff, Set.mem_star] at hf
rw [star_eq_iff_star_eq, eq_comm (b := f)] at hf
obtain (rfl | rfl) := hf
all_goals assumption
| zero => exact zero
| add _ _ _ _ hf hg => exact add _ _ hf hg
| mul _ _ _ _ hf hg => exact mul _ _ hf hg
| smul _ _ _ hf => exact smul _ _ hf
open Topology in
@[elab_as_elim] | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMapZero.induction_on | An induction principle for `C(s, 𝕜)₀`. |
ContinuousMapZero.induction_on_of_compact {s : Set 𝕜} [Fact (0 ∈ s)]
[CompactSpace s] {p : C(s, 𝕜)₀ → Prop} (zero : p 0) (id : p (.id s))
(star_id : p (star (.id s))) (add : ∀ f g, p f → p g → p (f + g))
(mul : ∀ f g, p f → p g → p (f * g)) (smul : ∀ (r : 𝕜) f, p f → p (r • f))
(frequently : ∀ f, (∃ᶠ g in 𝓝 f, p g) → p f) (f : C(s, 𝕜)₀) :
p f := by
refine f.induction_on zero id star_id add mul smul fun h f ↦ frequently f ?_
have := (ContinuousMapZero.adjoin_id_dense s).closure_eq ▸ Set.mem_univ (x := f)
exact mem_closure_iff_frequently.mp this |>.mp <| .of_forall h | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMapZero.induction_on_of_compact | null |
ContinuousMapZero.nonUnitalStarAlgHom_apply_mul_eq_zero {𝕜 A : Type*}
[RCLike 𝕜] [NonUnitalSemiring A] [Star A] [TopologicalSpace A] [ContinuousMul A]
[T2Space A] [DistribMulAction 𝕜 A] [IsScalarTower 𝕜 A A] {s : Set 𝕜} [Fact (0 ∈ s)]
[CompactSpace s] (φ : C(s, 𝕜)₀ →⋆ₙₐ[𝕜] A) (a : A) (hmul_id : φ (.id s) * a = 0)
(hmul_star_id : φ (star (.id s)) * a = 0) (hφ : Continuous φ) (f : C(s, 𝕜)₀) :
φ f * a = 0 := by
induction f using ContinuousMapZero.induction_on_of_compact with
| zero => simp [map_zero]
| id => exact hmul_id
| star_id => exact hmul_star_id
| add _ _ h₁ h₂ => simp only [map_add, add_mul, h₁, h₂, zero_add]
| mul _ _ _ h => simp only [map_mul, mul_assoc, h, mul_zero]
| smul _ _ h => rw [map_smul, smul_mul_assoc, h, smul_zero]
| frequently f h => exact h.mem_of_closed <| isClosed_eq (by fun_prop) continuous_zero | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMapZero.nonUnitalStarAlgHom_apply_mul_eq_zero | null |
ContinuousMapZero.mul_nonUnitalStarAlgHom_apply_eq_zero {𝕜 A : Type*}
[RCLike 𝕜] [NonUnitalSemiring A] [Star A] [TopologicalSpace A] [ContinuousMul A]
[T2Space A] [DistribMulAction 𝕜 A] [SMulCommClass 𝕜 A A] {s : Set 𝕜} [Fact (0 ∈ s)]
[CompactSpace s] (φ : C(s, 𝕜)₀ →⋆ₙₐ[𝕜] A) (a : A) (hmul_id : a * φ (.id s) = 0)
(hmul_star_id : a * φ (star (.id s)) = 0) (hφ : Continuous φ) (f : C(s, 𝕜)₀) :
a * φ f = 0 := by
induction f using ContinuousMapZero.induction_on_of_compact with
| zero => simp [map_zero]
| id => exact hmul_id
| star_id => exact hmul_star_id
| add _ _ h₁ h₂ => simp only [map_add, mul_add, h₁, h₂, zero_add]
| mul _ _ h _ => simp only [map_mul, ← mul_assoc, h, zero_mul]
| smul _ _ h => rw [map_smul, mul_smul_comm, h, smul_zero]
| frequently f h => exact h.mem_of_closed <| isClosed_eq (by fun_prop) continuous_zero | lemma | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.Star.Real",
"Mathlib.Topology.Algebra.StarSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalStarAlgebra",
"Mathlib.Topology.ContinuousMap.ContinuousMapZero",
"Mathlib.Topology.ContinuousMap.Lattice",
"... | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | ContinuousMapZero.mul_nonUnitalStarAlgHom_apply_eq_zero | null |
eq_induced_by_maps_to_sierpinski (X : Type*) [t : TopologicalSpace X] :
t = ⨅ u : Opens X, sierpinskiSpace.induced (· ∈ u) := by
apply le_antisymm
· rw [le_iInf_iff]
exact fun u => Continuous.le_induced (isOpen_iff_continuous_mem.mp u.2)
· intro u h
rw [← generateFrom_iUnion_isOpen]
apply isOpen_generateFrom_of_mem
simp only [Set.mem_iUnion, Set.mem_setOf_eq, isOpen_induced_iff]
exact ⟨⟨u, h⟩, {True}, isOpen_singleton_true, by simp [Set.preimage]⟩
variable (X : Type*) [TopologicalSpace X] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/T0Sierpinski.lean | eq_induced_by_maps_to_sierpinski | null |
productOfMemOpens : C(X, Opens X → Prop) where
toFun x u := x ∈ u
continuous_toFun := continuous_pi_iff.2 fun u => continuous_Prop.2 u.isOpen | def | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/T0Sierpinski.lean | productOfMemOpens | The continuous map from `X` to the product of copies of the Sierpinski space, (one copy for each
open subset `u` of `X`). The `u` coordinate of `productOfMemOpens x` is given by `x ∈ u`. |
productOfMemOpens_isInducing : IsInducing (productOfMemOpens X) := by
convert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x ∈ u
apply eq_induced_by_maps_to_sierpinski | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/T0Sierpinski.lean | productOfMemOpens_isInducing | null |
productOfMemOpens_injective [T0Space X] : Function.Injective (productOfMemOpens X) := by
intro x1 x2 h
apply Inseparable.eq
rw [← IsInducing.inseparable_iff (productOfMemOpens_isInducing X), h] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/T0Sierpinski.lean | productOfMemOpens_injective | null |
productOfMemOpens_isEmbedding [T0Space X] : IsEmbedding (productOfMemOpens X) :=
.mk (productOfMemOpens_isInducing X) (productOfMemOpens_injective X) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/ContinuousMap/T0Sierpinski.lean | productOfMemOpens_isEmbedding | null |
@[simps]
noncomputable unitsOfForallIsUnit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) : C(X, Rˣ) where
toFun x := (h x).unit
continuous_toFun := continuous_isUnit_unit h | def | Topology | [
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Units.lean | unitsOfForallIsUnit | Equivalence between continuous maps into the units of a monoid with continuous multiplication
and the units of the monoid of continuous maps. -/
-- `simps` generates some lemmas here with LHS not in simp normal form,
-- so we write them out manually below.
@[to_additive (attr := simps apply_val_apply symm_apply_apply_val)
/-- Equivalence between continuous maps into the additive units of an additive monoid with
continuous addition and the additive units of the additive monoid of continuous maps. -/]
def unitsLift : C(X, Mˣ) ≃ C(X, M)ˣ where
toFun f :=
{ val := ⟨fun x => f x, Units.continuous_val.comp f.continuous⟩
inv := ⟨fun x => ↑(f x)⁻¹, Units.continuous_val.comp (continuous_inv.comp f.continuous)⟩
val_inv := ext fun _ => Units.mul_inv _
inv_val := ext fun _ => Units.inv_mul _ }
invFun f :=
{ toFun := fun x =>
⟨(f : C(X, M)) x, (↑f⁻¹ : C(X, M)) x,
ContinuousMap.congr_fun f.mul_inv x, ContinuousMap.congr_fun f.inv_mul x⟩
continuous_toFun := continuous_induced_rng.2 <|
(f : C(X, M)).continuous.prodMk <|
MulOpposite.continuous_op.comp (↑f⁻¹ : C(X, M)).continuous }
@[to_additive (attr := simp)]
lemma unitsLift_apply_inv_apply (f : C(X, Mˣ)) (x : X) :
(↑(ContinuousMap.unitsLift f)⁻¹ : C(X, M)) x = (f x)⁻¹ :=
rfl
@[to_additive (attr := simp)]
lemma unitsLift_symm_apply_apply_inv' (f : C(X, M)ˣ) (x : X) :
(ContinuousMap.unitsLift.symm f x)⁻¹ = (↑f⁻¹ : C(X, M)) x := by
rfl
end Monoid
section NormedRing
variable [NormedRing R] [CompleteSpace R]
theorem continuous_isUnit_unit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) :
Continuous fun x => (h x).unit := by
refine
continuous_induced_rng.2
(Continuous.prodMk f.continuous
(MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))
have := NormedRing.inverse_continuousAt (h x).unit
simp only
simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢
exact this.comp (f.continuousAt x)
/-- Construct a continuous map into the group of units of a normed ring from a function into the
normed ring and a proof that every element of the range is a unit. |
canLift :
CanLift C(X, R) C(X, Rˣ) (fun f => ⟨fun x => f x, Units.continuous_val.comp f.continuous⟩)
fun f => ∀ x, IsUnit (f x) where
prf f h := ⟨unitsOfForallIsUnit h, by ext; rfl⟩ | instance | Topology | [
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Units.lean | canLift | null |
isUnit_iff_forall_isUnit (f : C(X, R)) : IsUnit f ↔ ∀ x, IsUnit (f x) :=
Iff.intro (fun h => fun x => ⟨unitsLift.symm h.unit x, rfl⟩) fun h =>
⟨ContinuousMap.unitsLift (unitsOfForallIsUnit h), by ext; rfl⟩ | theorem | Topology | [
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Units.lean | isUnit_iff_forall_isUnit | null |
isUnit_iff_forall_ne_zero (f : C(X, R)) : IsUnit f ↔ ∀ x, f x ≠ 0 := by
simp_rw [f.isUnit_iff_forall_isUnit, isUnit_iff_ne_zero] | theorem | Topology | [
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Units.lean | isUnit_iff_forall_ne_zero | null |
spectrum_eq_preimage_range (f : C(X, R)) :
spectrum 𝕜 f = algebraMap _ _ ⁻¹' Set.range f := by
ext x
simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,
Classical.not_not, Set.mem_range,
sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,
smul_apply, one_apply] | theorem | Topology | [
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Units.lean | spectrum_eq_preimage_range | null |
spectrum_eq_range [CompleteSpace 𝕜] (f : C(X, 𝕜)) : spectrum 𝕜 f = Set.range f := by
rw [spectrum_eq_preimage_range, Algebra.algebraMap_self]
exact Set.preimage_id | theorem | Topology | [
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Units.lean | spectrum_eq_range | null |
polynomialFunctions_closure_eq_top' : (polynomialFunctions I).topologicalClosure = ⊤ := by
apply top_unique
rintro f -
refine mem_closure_of_tendsto (bernsteinApproximation_uniform f) <| .of_forall fun n ↦ ?_
apply Subalgebra.sum_mem
rintro i -
rw [← SetLike.mem_coe, polynomialFunctions_coe]
use bernsteinPolynomial ℝ n i * .C (f (bernstein.z i))
ext
simp [bernstein] | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Bernstein",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/Weierstrass.lean | polynomialFunctions_closure_eq_top' | The special case of the Weierstrass approximation theorem for the interval `[0,1]`.
This is just a matter of unravelling definitions and using the Bernstein approximations. |
polynomialFunctions_closure_eq_top (a b : ℝ) :
(polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤ := by
rcases lt_or_ge a b with h | h
· -- We can pullback continuous functions on `[a,b]` to continuous functions on `[0,1]`,
let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) :=
compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm
let W' : C(Set.Icc a b, ℝ) ≃ₜ C(I, ℝ) := (iccHomeoI a b h).arrowCongr (.refl _)
have w : (W : C(Set.Icc a b, ℝ) → C(I, ℝ)) = W' := rfl
have p := polynomialFunctions_closure_eq_top'
apply_fun fun s => s.comap W at p
simp only [Algebra.comap_top] at p
rw [Subalgebra.topologicalClosure_comap_homeomorph _ W W' w] at p
rw [polynomialFunctions.comap_compRightAlgHom_iccHomeoI] at p
exact p
· -- Otherwise, `b ≤ a`, and the interval is a subsingleton,
subsingleton [(Set.subsingleton_Icc_of_ge h).coe_sort] | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Bernstein",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/Weierstrass.lean | polynomialFunctions_closure_eq_top | The **Weierstrass Approximation Theorem**:
polynomials functions on `[a, b] ⊆ ℝ` are dense in `C([a,b],ℝ)`
(While we could deduce this as an application of the Stone-Weierstrass theorem,
our proof of that relies on the fact that `abs` is in the closure of polynomials on `[-M, M]`,
so we may as well get this done first.) |
continuousMap_mem_polynomialFunctions_closure (a b : ℝ) (f : C(Set.Icc a b, ℝ)) :
f ∈ (polynomialFunctions (Set.Icc a b)).topologicalClosure := by
rw [polynomialFunctions_closure_eq_top _ _]
simp
open scoped Polynomial | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Bernstein",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/Weierstrass.lean | continuousMap_mem_polynomialFunctions_closure | An alternative statement of Weierstrass' theorem.
Every real-valued continuous function on `[a,b]` is a uniform limit of polynomials. |
exists_polynomial_near_continuousMap (a b : ℝ) (f : C(Set.Icc a b, ℝ)) (ε : ℝ)
(pos : 0 < ε) : ∃ p : ℝ[X], ‖p.toContinuousMapOn _ - f‖ < ε := by
have w := mem_closure_iff_frequently.mp (continuousMap_mem_polynomialFunctions_closure _ _ f)
rw [Metric.nhds_basis_ball.frequently_iff] at w
obtain ⟨-, H, ⟨m, ⟨-, rfl⟩⟩⟩ := w ε pos
rw [Metric.mem_ball, dist_eq_norm] at H
exact ⟨m, H⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Bernstein",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/Weierstrass.lean | exists_polynomial_near_continuousMap | An alternative statement of Weierstrass' theorem,
for those who like their epsilons.
Every real-valued continuous function on `[a,b]` is within any `ε > 0` of some polynomial. |
exists_polynomial_near_of_continuousOn (a b : ℝ) (f : ℝ → ℝ)
(c : ContinuousOn f (Set.Icc a b)) (ε : ℝ) (pos : 0 < ε) :
∃ p : ℝ[X], ∀ x ∈ Set.Icc a b, |p.eval x - f x| < ε := by
let f' : C(Set.Icc a b, ℝ) := ⟨fun x => f x, continuousOn_iff_continuous_restrict.mp c⟩
obtain ⟨p, b⟩ := exists_polynomial_near_continuousMap a b f' ε pos
use p
rw [norm_lt_iff _ pos] at b
intro x m
exact b ⟨x, m⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Bernstein",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/Weierstrass.lean | exists_polynomial_near_of_continuousOn | Another alternative statement of Weierstrass's theorem,
for those who like epsilons, but not bundled continuous functions.
Every real-valued function `ℝ → ℝ` which is continuous on `[a,b]`
can be approximated to within any `ε > 0` on `[a,b]` by some polynomial. |
ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β]
[TopologicalSpace β] : Type max u v extends ContinuousMap α β where
/-- The function tends to zero along the `cocompact` filter. -/
zero_at_infty' : Tendsto toFun (cocompact α) (𝓝 0)
@[inherit_doc]
scoped[ZeroAtInfty] notation (priority := 2000) "C₀(" α ", " β ")" => ZeroAtInftyContinuousMap α β
@[inherit_doc]
scoped[ZeroAtInfty] notation α " →C₀ " β => ZeroAtInftyContinuousMap α β
open ZeroAtInfty | structure | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | ZeroAtInftyContinuousMap | `C₀(α, β)` is the type of continuous functions `α → β` which vanish at infinity from a
topological space to a metric space with a zero element.
When possible, instead of parametrizing results over `(f : C₀(α, β))`,
you should parametrize over `(F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F)`.
When you extend this structure, make sure to extend `ZeroAtInftyContinuousMapClass`. |
ZeroAtInftyContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
[Zero β] [TopologicalSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where
/-- Each member of the class tends to zero along the `cocompact` filter. -/
zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0) | class | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | ZeroAtInftyContinuousMapClass | `ZeroAtInftyContinuousMapClass F α β` states that `F` is a type of continuous maps which
vanish at infinity.
You should also extend this typeclass when you extend `ZeroAtInftyContinuousMap`. |
instFunLike : FunLike C₀(α, β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instFunLike | null |
instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass C₀(α, β) α β where
map_continuous f := f.continuous_toFun
zero_at_infty f := f.zero_at_infty' | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instZeroAtInftyContinuousMapClass | null |
instCoeTC : CoeTC F C₀(α, β) :=
⟨fun f =>
{ toFun := f
continuous_toFun := map_continuous f
zero_at_infty' := zero_at_infty f }⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instCoeTC | null |
coe_toContinuousMap (f : C₀(α, β)) : (f.toContinuousMap : α → β) = f :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_toContinuousMap | null |
ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | ext | null |
coe_mk {f : α → β} (hf : Continuous f) (hf' : Tendsto f (cocompact α) (𝓝 0)) :
{ toFun := f,
continuous_toFun := hf,
zero_at_infty' := hf' : ZeroAtInftyContinuousMap α β} = f :=
rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_mk | null |
protected copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : C₀(α, β) where
toFun := f'
continuous_toFun := by
rw [h]
exact f.continuous_toFun
zero_at_infty' := by
simp_rw [h]
exact f.zero_at_infty'
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | copy | Copy of a `ZeroAtInftyContinuousMap` with a new `toFun` equal to the old one. Useful
to fix definitional equalities. |
coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_copy | null |
copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | copy_eq | null |
eq_of_empty [IsEmpty α] (f g : C₀(α, β)) : f = g :=
ext <| IsEmpty.elim ‹_› | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | eq_of_empty | null |
@[simps]
ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β) where
toFun f :=
{ toFun := f
continuous_toFun := f.continuous
zero_at_infty' := by simp }
invFun f := f | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | ContinuousMap.liftZeroAtInfty | A continuous function on a compact space is automatically a continuous function vanishing at
infinity. |
zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [FunLike G α β]
[ContinuousMapClass G α β] [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β where
map_continuous := map_continuous
zero_at_infty := by simp | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | zeroAtInftyContinuousMapClass.ofCompact | A continuous function on a compact space is automatically a continuous function vanishing at
infinity. This is not an instance to avoid type class loops. |
instZero [Zero β] : Zero C₀(α, β) :=
⟨⟨0, tendsto_const_nhds⟩⟩ | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instZero | null |
instInhabited [Zero β] : Inhabited C₀(α, β) :=
⟨0⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instInhabited | null |
coe_zero [Zero β] : ⇑(0 : C₀(α, β)) = 0 :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_zero | null |
zero_apply [Zero β] : (0 : C₀(α, β)) x = 0 :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | zero_apply | null |
instMul [MulZeroClass β] [ContinuousMul β] : Mul C₀(α, β) :=
⟨fun f g =>
⟨f * g, by simpa only [mul_zero] using (zero_at_infty f).mul (zero_at_infty g)⟩⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instMul | null |
coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : ⇑(f * g) = f * g :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_mul | null |
mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : (f * g) x = f x * g x :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | mul_apply | null |
instMulZeroClass [MulZeroClass β] [ContinuousMul β] : MulZeroClass C₀(α, β) :=
DFunLike.coe_injective.mulZeroClass _ coe_zero coe_mul | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instMulZeroClass | null |
instSemigroupWithZero [SemigroupWithZero β] [ContinuousMul β] :
SemigroupWithZero C₀(α, β) :=
DFunLike.coe_injective.semigroupWithZero _ coe_zero coe_mul | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instSemigroupWithZero | null |
instAdd [AddZeroClass β] [ContinuousAdd β] : Add C₀(α, β) :=
⟨fun f g => ⟨f + g, by simpa only [add_zero] using (zero_at_infty f).add (zero_at_infty g)⟩⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instAdd | null |
coe_add [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : ⇑(f + g) = f + g :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_add | null |
add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f + g) x = f x + g x :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | add_apply | null |
instAddZeroClass [AddZeroClass β] [ContinuousAdd β] : AddZeroClass C₀(α, β) :=
DFunLike.coe_injective.addZeroClass _ coe_zero coe_add | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instAddZeroClass | null |
instSMul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] :
SMul R C₀(α, β) :=
⟨fun r f => ⟨r • f, by simpa [smul_zero] using (zero_at_infty f).const_smul r⟩⟩
@[simp, norm_cast] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instSMul | null |
coe_smul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
(f : C₀(α, β)) : ⇑(r • f) = r • ⇑f :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_smul | null |
smul_apply [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
(r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | smul_apply | null |
instAddMonoid : AddMonoid C₀(α, β) :=
DFunLike.coe_injective.addMonoid _ coe_zero coe_add fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instAddMonoid | null |
instAddCommMonoid [AddCommMonoid β] [ContinuousAdd β] : AddCommMonoid C₀(α, β) :=
DFunLike.coe_injective.addCommMonoid _ coe_zero coe_add fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instAddCommMonoid | null |
instNeg : Neg C₀(α, β) :=
⟨fun f => ⟨-f, by simpa only [neg_zero] using (zero_at_infty f).neg⟩⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNeg | null |
coe_neg : ⇑(-f) = -f :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_neg | null |
neg_apply : (-f) x = -f x :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | neg_apply | null |
instSub : Sub C₀(α, β) :=
⟨fun f g => ⟨f - g, by simpa only [sub_zero] using (zero_at_infty f).sub (zero_at_infty g)⟩⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instSub | null |
coe_sub : ⇑(f - g) = f - g :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | coe_sub | null |
sub_apply : (f - g) x = f x - g x :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | sub_apply | null |
instAddGroup : AddGroup C₀(α, β) :=
DFunLike.coe_injective.addGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instAddGroup | null |
instAddCommGroup [AddCommGroup β] [IsTopologicalAddGroup β] : AddCommGroup C₀(α, β) :=
DFunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => rfl) fun _ _ =>
rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instAddCommGroup | null |
instIsCentralScalar [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [SMulWithZero Rᵐᵒᵖ β]
[ContinuousConstSMul R β] [IsCentralScalar R β] : IsCentralScalar R C₀(α, β) :=
⟨fun _ _ => ext fun _ => op_smul_eq_smul _ _⟩ | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instIsCentralScalar | null |
instSMulWithZero [Zero β] {R : Type*} [Zero R] [SMulWithZero R β]
[ContinuousConstSMul R β] : SMulWithZero R C₀(α, β) :=
Function.Injective.smulWithZero ⟨_, coe_zero⟩ DFunLike.coe_injective coe_smul | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instSMulWithZero | null |
instMulActionWithZero [Zero β] {R : Type*} [MonoidWithZero R] [MulActionWithZero R β]
[ContinuousConstSMul R β] : MulActionWithZero R C₀(α, β) :=
Function.Injective.mulActionWithZero ⟨_, coe_zero⟩ DFunLike.coe_injective coe_smul | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instMulActionWithZero | null |
instModule [AddCommMonoid β] [ContinuousAdd β] {R : Type*} [Semiring R] [Module R β]
[ContinuousConstSMul R β] : Module R C₀(α, β) :=
Function.Injective.module R ⟨⟨_, coe_zero⟩, coe_add⟩ DFunLike.coe_injective coe_smul | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instModule | null |
instNonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] [IsTopologicalSemiring β] :
NonUnitalNonAssocSemiring C₀(α, β) :=
DFunLike.coe_injective.nonUnitalNonAssocSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalNonAssocSemiring | null |
instNonUnitalSemiring [NonUnitalSemiring β] [IsTopologicalSemiring β] :
NonUnitalSemiring C₀(α, β) :=
DFunLike.coe_injective.nonUnitalSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalSemiring | null |
instNonUnitalCommSemiring [NonUnitalCommSemiring β] [IsTopologicalSemiring β] :
NonUnitalCommSemiring C₀(α, β) :=
DFunLike.coe_injective.nonUnitalCommSemiring _ coe_zero coe_add coe_mul fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalCommSemiring | null |
instNonUnitalNonAssocRing [NonUnitalNonAssocRing β] [IsTopologicalRing β] :
NonUnitalNonAssocRing C₀(α, β) :=
DFunLike.coe_injective.nonUnitalNonAssocRing _ coe_zero coe_add coe_mul coe_neg coe_sub
(fun _ _ => rfl) fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalNonAssocRing | null |
instNonUnitalRing [NonUnitalRing β] [IsTopologicalRing β] : NonUnitalRing C₀(α, β) :=
DFunLike.coe_injective.nonUnitalRing _ coe_zero coe_add coe_mul coe_neg coe_sub (fun _ _ => rfl)
fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalRing | null |
instNonUnitalCommRing [NonUnitalCommRing β] [IsTopologicalRing β] :
NonUnitalCommRing C₀(α, β) :=
DFunLike.coe_injective.nonUnitalCommRing _ coe_zero coe_add coe_mul coe_neg coe_sub
(fun _ _ => rfl) fun _ _ => rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalCommRing | null |
instIsScalarTower {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
[IsTopologicalSemiring β] [Module R β] [ContinuousConstSMul R β] [IsScalarTower R β β] :
IsScalarTower R C₀(α, β) C₀(α, β) where
smul_assoc r f g := by
ext
simp only [smul_eq_mul, coe_mul, coe_smul, Pi.mul_apply, Pi.smul_apply]
rw [← smul_eq_mul, ← smul_eq_mul, smul_assoc] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instIsScalarTower | null |
instSMulCommClass {R : Type*} [Semiring R] [NonUnitalNonAssocSemiring β]
[IsTopologicalSemiring β] [Module R β] [ContinuousConstSMul R β] [SMulCommClass R β β] :
SMulCommClass R C₀(α, β) C₀(α, β) where
smul_comm r f g := by
ext
simp only [smul_eq_mul, coe_smul, coe_mul, Pi.smul_apply, Pi.mul_apply]
rw [← smul_eq_mul, ← smul_eq_mul, smul_comm] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instSMulCommClass | null |
uniformContinuous (f : F) : UniformContinuous (f : β → γ) :=
(map_continuous f).uniformContinuous_of_tendsto_cocompact (zero_at_infty f) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | uniformContinuous | null |
protected bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C := by
obtain ⟨K : Set α, hK₁, hK₂⟩ := mem_cocompact.mp
(tendsto_def.mp (zero_at_infty (f : F)) _ (closedBall_mem_nhds (0 : β) zero_lt_one))
obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).isBounded.subset_closedBall (0 : β)
refine ⟨max C 1 + max C 1, fun x y => ?_⟩
have : ∀ x, f x ∈ closedBall (0 : β) (max C 1) := by
intro x
by_cases hx : x ∈ K
· exact (mem_closedBall.mp <| hC ⟨x, hx, rfl⟩).trans (le_max_left _ _)
· exact (mem_closedBall.mp <| mem_preimage.mp (hK₂ hx)).trans (le_max_right _ _)
exact (dist_triangle (f x) 0 (f y)).trans
(add_le_add (mem_closedBall.mp <| this x) (mem_closedBall'.mp <| this y)) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | bounded | null |
isBounded_range (f : C₀(α, β)) : IsBounded (range f) :=
isBounded_range_iff.2 (ZeroAtInftyContinuousMap.bounded f) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | isBounded_range | null |
isBounded_image (f : C₀(α, β)) (s : Set α) : IsBounded (f '' s) :=
f.isBounded_range.subset <| image_subset_range _ _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | isBounded_image | null |
@[simps!]
toBCF (f : C₀(α, β)) : α →ᵇ β :=
⟨f, map_bounded f⟩ | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | toBCF | Construct a bounded continuous function from a continuous function vanishing at infinity. |
toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) := fun f g h => by
ext x
simpa only using DFunLike.congr_fun h x | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | toBCF_injective | null |
noncomputable instPseudoMetricSpace : PseudoMetricSpace C₀(α, β) := fast_instance%
PseudoMetricSpace.induced toBCF inferInstance | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instPseudoMetricSpace | The type of continuous functions vanishing at infinity, with the uniform distance induced by the
inclusion `ZeroAtInftyContinuousMap.toBCF`, is a pseudo-metric space. |
noncomputable instMetricSpace {β : Type*} [MetricSpace β] [Zero β] :
MetricSpace C₀(α, β) := fast_instance%
MetricSpace.induced _ (toBCF_injective α β) inferInstance
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instMetricSpace | The type of continuous functions vanishing at infinity, with the uniform distance induced by the
inclusion `ZeroAtInftyContinuousMap.toBCF`, is a metric space. |
dist_toBCF_eq_dist {f g : C₀(α, β)} : dist f.toBCF g.toBCF = dist f g :=
rfl
open BoundedContinuousFunction | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | dist_toBCF_eq_dist | null |
tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l := by
simpa only [Metric.tendsto_nhds] using
@BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBCF)
f.toBCF l | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | tendsto_iff_tendstoUniformly | Convergence in the metric on `C₀(α, β)` is uniform convergence. |
isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) := by tauto | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | isometry_toBCF | null |
isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) := by
refine isClosed_iff_clusterPt.mpr fun f hf => ?_
rw [clusterPt_principal_iff] at hf
have : Tendsto f (cocompact α) (𝓝 0) := by
refine Metric.tendsto_nhds.mpr fun ε hε => ?_
obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f <| half_pos hε)
refine (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
(Eventually.of_forall fun x hx => ?_)
calc
dist (f x) 0 ≤ dist (g.toBCF x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
_ < dist g.toBCF f + ε / 2 := add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx
_ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2)
exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | isClosed_range_toBCF | null |
instCompleteSpace [CompleteSpace β] : CompleteSpace C₀(α, β) :=
(completeSpace_iff_isComplete_range isometry_toBCF.isUniformInducing).mpr
isClosed_range_toBCF.isComplete | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instCompleteSpace | Continuous functions vanishing at infinity taking values in a complete space form a
complete space. |
noncomputable instSeminormedAddCommGroup [SeminormedAddCommGroup β] :
SeminormedAddCommGroup C₀(α, β) := fast_instance%
SeminormedAddCommGroup.induced _ _ (⟨⟨toBCF, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instSeminormedAddCommGroup | null |
noncomputable instNormedAddCommGroup [NormedAddCommGroup β] :
NormedAddCommGroup C₀(α, β) := fast_instance%
NormedAddCommGroup.induced _ _ (⟨⟨toBCF, rfl⟩, fun _ _ => rfl⟩ : C₀(α, β) →+ α →ᵇ β)
(toBCF_injective α β)
variable [SeminormedAddCommGroup β] {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β]
@[simp] | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNormedAddCommGroup | null |
norm_toBCF_eq_norm {f : C₀(α, β)} : ‖f.toBCF‖ = ‖f‖ :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | norm_toBCF_eq_norm | null |
noncomputable instNonUnitalSeminormedRing [NonUnitalSeminormedRing β] :
NonUnitalSeminormedRing C₀(α, β) :=
{ instNonUnitalRing, instSeminormedAddCommGroup with
norm_mul_le f g := norm_mul_le f.toBCF g.toBCF } | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Star",
"Mathlib.Topology.ContinuousMap.CocompactMap"
] | Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean | instNonUnitalSeminormedRing | null |
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