Loogle!

Result

Found 163 declarations mentioning DenseRange.

• DenseRange 📋 Mathlib.Topology.Defs.Basic
  {X : Type u} [TopologicalSpace X] {α : Type u_1} (f : α → X) : Prop
• DenseRange.of_comp 📋 Mathlib.Topology.Closure
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {β : Type u_2} {f : α → X} {g : β → α} (h : DenseRange (f ∘ g)) : DenseRange f
• denseRange_id 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] : DenseRange id
• DenseRange.some 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (hf : DenseRange f) (x : X) : α
• DenseRange.nonempty 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} [h : Nonempty X] (hf : DenseRange f) : Nonempty α
• Function.Surjective.denseRange 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (hf : Function.Surjective f) : DenseRange f
• DenseRange.nonempty_iff 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (hf : DenseRange f) : Nonempty α ↔ Nonempty X
• denseRange_subtype_val 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {p : X → Prop} : DenseRange Subtype.val ↔ Dense {x | p x}
• DenseRange.closure_range 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} (h : DenseRange f) : closure (Set.range f) = Set.univ
• denseRange_iff_closure_range 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} : DenseRange f ↔ closure (Set.range f) = Set.univ
• DenseRange.exists_mem_open 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} {s : Set X} (hf : DenseRange f) (ho : IsOpen s) (hs : s.Nonempty) : ∃ a, f a ∈ s
• DenseRange.dense_image 📋 Mathlib.Topology.Continuous
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) (hs : Dense s) : Dense (f '' s)
• Dense.denseRange_val 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : Dense s) : DenseRange Subtype.val
• DenseRange.subset_closure_image_preimage_of_isOpen 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {f : α → X} {s : Set X} (hf : DenseRange f) (hs : IsOpen s) : s ⊆ closure (f '' f ⁻¹' s)
• DenseRange.comp 📋 Mathlib.Topology.Continuous
  {Y : Type u_2} {Z : Type u_3} [TopologicalSpace Y] [TopologicalSpace Z] {α : Type u_4} {g : Y → Z} {f : α → Y} (hg : DenseRange g) (hf : DenseRange f) (cg : Continuous g) : DenseRange (g ∘ f)
• DenseRange.dense_of_mapsTo 📋 Mathlib.Topology.Continuous
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) (hs : Dense s) {t : Set Y} (ht : Set.MapsTo f s t) : Dense t
• DenseRange.mem_nhds 📋 Mathlib.Topology.Continuous
  {X : Type u_1} [TopologicalSpace X] {x : X} {α : Type u_4} {f : α → X} {s : Set X} (h : DenseRange f) (hs : s ∈ nhds x) : ∃ a, f a ∈ s
• denseRange_discrete 📋 Mathlib.Topology.Order
  {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α] {ι : Type u_3} {f : ι → α} : DenseRange f ↔ Function.Surjective f
• DenseRange.prodMap 📋 Mathlib.Topology.Constructions.SumProd
  {Y : Type v} {Z : Type u_2} [TopologicalSpace Y] [TopologicalSpace Z] {ι : Type u_5} {κ : Type u_6} {f : ι → Y} {g : κ → Z} (hf : DenseRange f) (hg : DenseRange g) : DenseRange (Prod.map f g)
• DenseRange.quotient 📋 Mathlib.Topology.Constructions
  {X : Type u} {Y : Type v} [Setoid X] [TopologicalSpace X] {f : Y → X} (hf : DenseRange f) : DenseRange (Quotient.mk' ∘ f)
• denseRange_inclusion_iff 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] {s t : Set X} (hst : s ⊆ t) : DenseRange (Set.inclusion hst) ↔ t ⊆ closure s
• DenseRange.nhdsWithin_neBot 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_1} [TopologicalSpace α] {ι : Type u_3} {f : ι → α} (h : DenseRange f) (x : α) : (nhdsWithin x (Set.range f)).NeBot
• DenseRange.piMap 📋 Mathlib.Topology.NhdsWithin
  {ι : Type u_3} {X : ι → Type u_4} {Y : ι → Type u_5} [(i : ι) → TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (hf : ∀ (i : ι), DenseRange (f i)) : DenseRange (Pi.map f)
• DenseRange.separableSpace' 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] {ι : Type u_2} [Countable ι] (u : ι → α) (hu : DenseRange u) : TopologicalSpace.SeparableSpace α
• TopologicalSpace.SeparableSpace.of_denseRange 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] {ι : Type u_2} [Countable ι] (u : ι → α) (hu : DenseRange u) : TopologicalSpace.SeparableSpace α
• TopologicalSpace.denseRange_denseSeq 📋 Mathlib.Topology.Bases
  (α : Type u) [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [Nonempty α] : DenseRange (TopologicalSpace.denseSeq α)
• TopologicalSpace.exists_dense_seq 📋 Mathlib.Topology.Bases
  (α : Type u) [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [Nonempty α] : ∃ u, DenseRange u
• DenseRange.separableSpace 📋 Mathlib.Topology.Bases
  {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [TopologicalSpace β] {f : α → β} (h : DenseRange f) (h' : Continuous f) : TopologicalSpace.SeparableSpace β
• IsOpenMap.denseRange_of_isPreirreducibleSpace 📋 Mathlib.Topology.Irreducible
  {U : Type u_3} {X : Type u_4} [TopologicalSpace U] [Nonempty U] [TopologicalSpace X] (f : U → X) (hf : IsOpenMap f) [PreirreducibleSpace X] : DenseRange f
• DenseRange.preconnectedSpace 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [PreconnectedSpace α] {f : α → β} (hf : DenseRange f) (hc : Continuous f) : PreconnectedSpace β
• IsDenseInducing.dense 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (self : IsDenseInducing i) : DenseRange i
• IsDenseInducing.mk 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (toIsInducing : Topology.IsInducing i) (dense : DenseRange i) : IsDenseInducing i
• isClosed_property 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → Prop} (he : DenseRange e) (hp : IsClosed {x | p x}) (h : ∀ (a : α), p (e a)) (b : β) : p b
• DenseRange.induction_on 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} (he : DenseRange e) {p : β → Prop} (b₀ : β) (hp : IsClosed {b | p b}) (ih : ∀ (a : α), p (e a)) : p b₀
• isClosed_property2 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2}) (h : ∀ (a₁ a₂ : α), p (e a₁) (e a₂)) (b₁ b₂ : β) : p b₁ b₂
• DenseRange.induction_on₂ 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2}) (h : ∀ (a₁ a₂ : α), p (e a₁) (e a₂)) (b₁ b₂ : β) : p b₁ b₂
• DenseRange.equalizer 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β] [TopologicalSpace γ] [T2Space γ] {f : α → β} (hfd : DenseRange f) {g h : β → γ} (hg : Continuous g) (hh : Continuous h) (H : g ∘ f = h ∘ f) : g = h
• IsDenseEmbedding.mk' 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α → β) (c : Continuous e) (dense : DenseRange e) (injective : Function.Injective e) (H : ∀ (a : α), ∀ s ∈ nhds a, ∃ t ∈ nhds (e a), ∀ (b : α), e b ∈ t → b ∈ s) : IsDenseEmbedding e
• isClosed_property3 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2.1 q.2.2}) (h : ∀ (a₁ a₂ a₃ : α), p (e a₁) (e a₂) (e a₃)) (b₁ b₂ b₃ : β) : p b₁ b₂ b₃
• DenseRange.induction_on₃ 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2.1 q.2.2}) (h : ∀ (a₁ a₂ a₃ : α), p (e a₁) (e a₂) (e a₃)) (b₁ b₂ b₃ : β) : p b₁ b₂ b₃
• DenseRange.iUnion_uniformity_ball 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} [UniformSpace α] {ι : Type u_2} {xs : ι → α} (xs_dense : DenseRange xs) {U : SetRel α α} (hU : U ∈ uniformity α) : ⋃ i, UniformSpace.ball (xs i) U = Set.univ
• IsUniformEmbedding.isDenseEmbedding 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : IsUniformEmbedding f) (hd : DenseRange f) : IsDenseEmbedding f
• IsUniformInducing.isDenseInducing 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (h : IsUniformInducing f) (hd : DenseRange f) : IsDenseInducing f
• completeSpace_extension 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {m : β → α} (hm : IsUniformInducing m) (dense : DenseRange m) (h : ∀ (f : Filter β), Cauchy f → ∃ x, Filter.map m f ≤ nhds x) : CompleteSpace α
• uniformly_extend_exists 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] (a : α) : ∃ c, Filter.Tendsto f (Filter.comap e (nhds a)) (nhds c)
• uniformContinuous_uniformly_extend 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] : UniformContinuous (⋯.extend f)
• uniformly_extend_of_ind 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [T0Space γ] (b : β) : ⋯.extend f (e b) = f b
• uniformly_extend_unique 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} [T0Space γ] {g : α → γ} (hg : ∀ (b : β), g (e b) = f b) (hc : Continuous g) : ⋯.extend f = g
• uniformly_extend_spec 📋 Mathlib.Topology.UniformSpace.UniformEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] (a : α) : Filter.Tendsto f (Filter.comap e (nhds a)) (nhds (⋯.extend f a))
• DenseRange.topologicalClosure_map_addSubgroup 📋 Mathlib.Topology.Algebra.Group.Subgroup
  {G : Type u_1} {H : Type u_2} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] [AddGroup H] [TopologicalSpace H] [IsTopologicalAddGroup H] {f : G →+ H} (hf : Continuous ⇑f) (hf' : DenseRange ⇑f) {s : AddSubgroup G} (hs : s.topologicalClosure = ⊤) : (AddSubgroup.map f s).topologicalClosure = ⊤
• DenseRange.topologicalClosure_map_subgroup 📋 Mathlib.Topology.Algebra.Group.Subgroup
  {G : Type u_1} {H : Type u_2} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {f : G →* H} (hf : Continuous ⇑f) (hf' : DenseRange ⇑f) {s : Subgroup G} (hs : s.topologicalClosure = ⊤) : (Subgroup.map f s).topologicalClosure = ⊤
• not_denseRange_zpow 📋 Mathlib.Topology.Algebra.Order.Group
  {G : Type u_1} [TopologicalSpace G] [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [OrderTopology G] [Nontrivial G] [DenselyOrdered G] {a : G} : ¬DenseRange fun x => a ^ x
• not_denseRange_zsmul 📋 Mathlib.Topology.Algebra.Order.Group
  {G : Type u_1} [TopologicalSpace G] [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [OrderTopology G] [Nontrivial G] [DenselyOrdered G] {a : G} : ¬DenseRange fun x => x • a
• denseRange_zpow_iff_surjective 📋 Mathlib.Topology.Algebra.Order.Group
  {G : Type u_1} [TopologicalSpace G] [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [OrderTopology G] {a : G} : (DenseRange fun x => a ^ x) ↔ Function.Surjective fun x => a ^ x
• denseRange_zsmul_iff_surjective 📋 Mathlib.Topology.Algebra.Order.Group
  {G : Type u_1} [TopologicalSpace G] [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [OrderTopology G] {a : G} : (DenseRange fun x => x • a) ↔ Function.Surjective fun x => x • a
• TopologicalSpace.IsOpenCover.denseRange_iff_restrictPreimage 📋 Mathlib.Topology.LocalAtTarget
  {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : TopologicalSpace.IsOpenCover U) : DenseRange f ↔ ∀ (i : ι), DenseRange ((U i).carrier.restrictPreimage f)
• PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) : DenseRange (PrimeSpectrum.comap f) ↔ RingHom.ker f ≤ nilradical R
• PrimeSpectrum.denseRange_comap_iff_minimalPrimes 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) : DenseRange (PrimeSpectrum.comap f) ↔ ∀ (I : Ideal R) (h : I ∈ minimalPrimes R), { asIdeal := I, isPrime := ⋯ } ∈ Set.range (PrimeSpectrum.comap f)
• DenseRange.topologicalClosure_map_submodule 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : DenseRange ⇑f) {s : Submodule R₁ M₁} (hs : s.topologicalClosure = ⊤) : (Submodule.map (↑f) s).topologicalClosure = ⊤
• EReal.denseRange_ratCast 📋 Mathlib.Topology.Order.Real
  : DenseRange fun r => ↑↑r
• DenseRange.exists_dist_lt 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε
• Metric.denseRange_iff 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} : DenseRange f ↔ ∀ (x : α), ∀ r > 0, ∃ y, dist x (f y) < r
• DenseRange.exists_seq_strictAnti_tendsto 📋 Mathlib.Topology.Order.IsLUB
  {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {β : Type u_3} [LinearOrder β] [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f) (x : α) : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), f (u n) ∈ Set.Ioi x) ∧ Filter.Tendsto (f ∘ u) Filter.atTop (nhds x)
• DenseRange.exists_seq_strictMono_tendsto 📋 Mathlib.Topology.Order.IsLUB
  {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {β : Type u_3} [LinearOrder β] [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f) (x : α) : ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Set.Iio x) ∧ Filter.Tendsto (f ∘ u) Filter.atTop (nhds x)
• DenseRange.exists_seq_strictAnti_tendsto_of_lt 📋 Mathlib.Topology.Order.IsLUB
  {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {β : Type u_3} [LinearOrder β] [DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f) (hmono : Monotone f) (hlt : x < y) : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), f (u n) ∈ Set.Ioo x y) ∧ Filter.Tendsto (f ∘ u) Filter.atTop (nhds x)
• DenseRange.exists_seq_strictMono_tendsto_of_lt 📋 Mathlib.Topology.Order.IsLUB
  {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {β : Type u_3} [LinearOrder β] [DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f) (hmono : Monotone f) (hlt : y < x) : ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Set.Ioo y x) ∧ Filter.Tendsto (f ∘ u) Filter.atTop (nhds x)
• Rat.denseRange_cast 📋 Mathlib.Topology.Algebra.Order.Archimedean
  {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] [Archimedean 𝕜] : DenseRange Rat.cast
• Monotone.continuous_of_denseRange 📋 Mathlib.Topology.Order.MonotoneContinuity
  {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} (h_mono : Monotone f) (h_dense : DenseRange f) : Continuous f
• AbstractCompletion.dense 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type u} [UniformSpace α] (self : AbstractCompletion.{v, u} α) : DenseRange self.coe
• AbstractCompletion.mk 📋 Mathlib.Topology.UniformSpace.AbstractCompletion
  {α : Type u} [UniformSpace α] (space : Type v) (coe : α → space) (uniformStruct : UniformSpace space) (complete : CompleteSpace space) (separation : T0Space space) (isUniformInducing : IsUniformInducing coe) (dense : DenseRange coe) : AbstractCompletion.{v, u} α
• CauchyFilter.denseRange_pureCauchy 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u} [UniformSpace α] : DenseRange CauchyFilter.pureCauchy
• UniformSpace.Completion.denseRange_coe 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] : DenseRange UniformSpace.Completion.coe'
• UniformSpace.Completion.denseRange_coe₂ 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] : DenseRange fun x => (↑x.1, ↑x.2)
• UniformSpace.Completion.denseRange_coe₃ 📋 Mathlib.Topology.UniformSpace.Completion
  {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] : DenseRange fun x => (↑x.1, ↑x.2.1, ↑x.2.2)
• NormedField.denseRange_nnnorm 📋 Mathlib.Analysis.Normed.Field.Lemmas
  (α : Type u_1) [DenselyNormedField α] : DenseRange nnnorm
• DenseRange.topologicalClosure_map_subalgebra 📋 Mathlib.Topology.Algebra.Algebra
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [IsSemitopologicalSemiring A] [IsSemitopologicalSemiring B] {f : A →A[R] B} (hf' : DenseRange ⇑f) {s : Subalgebra R A} (hs : s.topologicalClosure = ⊤) : (Subalgebra.map (↑f) s).topologicalClosure = ⊤
• denseRange_smul 📋 Mathlib.Dynamics.Minimal
  (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] (x : α) : DenseRange fun c => c • x
• denseRange_vadd 📋 Mathlib.Dynamics.Minimal
  (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] (x : α) : DenseRange fun c => c +ᵥ x
• IsDenseInducing.extendRingHom 📋 Mathlib.Topology.Algebra.UniformRing
  {α : Type u_1} [UniformSpace α] [Semiring α] {β : Type u_2} [UniformSpace β] [Semiring β] [IsTopologicalSemiring β] {γ : Type u_3} [UniformSpace γ] [Semiring γ] [IsTopologicalSemiring γ] [T2Space γ] [CompleteSpace γ] {i : α →+* β} {f : α →+* γ} (ue : IsUniformInducing ⇑i) (dr : DenseRange ⇑i) (hf : UniformContinuous ⇑f) : β →+* γ
• ContinuousLinearMap.extend_unique 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] {e : E →L[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_e : IsUniformInducing ⇑e) (g : Eₗ →SL[σ₁₂] F) (H : g ∘SL e = f) : f.extend e = g
• ContinuousLinearMap.extend_zero 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [CompleteSpace F] {e : E →L[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_e : IsUniformInducing ⇑e) : ContinuousLinearMap.extend 0 e = 0
• ContinuousLinearMap.extend_eq 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {Eₗ : Type u_5} [AddCommGroup E] [UniformSpace E] [IsUniformAddGroup E] [AddCommGroup F] [UniformSpace F] [IsUniformAddGroup F] [T0Space F] [AddCommMonoid Eₗ] [UniformSpace Eₗ] [ContinuousAdd Eₗ] [Semiring 𝕜] [Semiring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] [Module 𝕜 Eₗ] [ContinuousConstSMul 𝕜 Eₗ] [ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [CompleteSpace F] {e : E →L[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_e : IsUniformInducing ⇑e) (x : E) : (f.extend e) (e x) = f x
• LinearMap.extendOfIsometry 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup Eₗ] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [NormedAddCommGroup F] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [CompleteSpace F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_norm : ∀ (x : E), ‖f x‖ = ‖e x‖) : Eₗ →ₛₗᵢ[σ₁₂] F
• LinearMap.opNorm_extendOfNorm_le 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedAddCommGroup F] [SeminormedAddCommGroup Eₗ] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜 Eₗ] [AddCommGroup E] [Module 𝕜 E] [CompleteSpace F] {f : E →ₛₗ[σ₁₂] F} {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) {C : ℝ} (hC : 0 ≤ C) (h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖) : ‖f.extendOfNorm e‖ ≤ C
• LinearMap.toContinuousLinearMap_extendOfIsometry 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup Eₗ] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [NormedAddCommGroup F] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [CompleteSpace F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_norm : ∀ (x : E), ‖f x‖ = ‖e x‖) : (f.extendOfIsometry h_dense h_norm).toContinuousLinearMap = f.extendOfNorm e
• LinearMap.norm_extendOfNorm_apply_le 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [AddCommGroup E] [SeminormedAddCommGroup Eₗ] [NormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [CompleteSpace F] {f : E →ₛₗ[σ₁₂] F} {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (C : ℝ) (h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖) (x : Eₗ) : ‖(f.extendOfNorm e) x‖ ≤ C * ‖x‖
• LinearMap.extendOfIsometry_apply 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup Eₗ] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [NormedAddCommGroup F] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [CompleteSpace F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_norm : ∀ (x : E), ‖f x‖ = ‖e x‖) (x : Eₗ) : (f.extendOfIsometry h_dense h_norm) x = (f.extendOfNorm e) x
• LinearMap.extendOfIsometry_unique 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup Eₗ] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [NormedAddCommGroup F] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [CompleteSpace F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_norm : ∀ (x : E), ‖f x‖ = ‖e x‖) (g : Eₗ →ₛₗᵢ[σ₁₂] F) (H : g.toLinearMap ∘ₛₗ e = f) : f.extendOfIsometry h_dense h_norm = g
• LinearMap.extendOfIsometry_eq 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup Eₗ] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [NormedAddCommGroup F] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [CompleteSpace F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_norm : ∀ (x : E), ‖f x‖ = ‖e x‖) (x : E) : (f.extendOfIsometry h_dense h_norm) (e x) = f x
• LinearMap.extendOfNorm_unique 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [AddCommGroup E] [SeminormedAddCommGroup Eₗ] [NormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [CompleteSpace F] {f : E →ₛₗ[σ₁₂] F} {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (C : ℝ) (h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖) (g : Eₗ →SL[σ₁₂] F) (H : ↑g ∘ₛₗ e = f) : f.extendOfNorm e = g
• LinearMap.extendOfNorm_eq 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [AddCommGroup E] [SeminormedAddCommGroup Eₗ] [NormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] [IsBoundedSMul 𝕜₂ F] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [CompleteSpace F] {f : E →ₛₗ[σ₁₂] F} {e : E →ₗ[𝕜] Eₗ} (h_dense : DenseRange ⇑e) (h_norm : ∃ C, ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖) (x : E) : (f.extendOfNorm e) (e x) = f x
• LinearEquiv.extendOfIsometry 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedField 𝕜] [NormedField 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜₂ F] [NormedAddCommGroup Eₗ] [NormedSpace 𝕜 Eₗ] [CompleteSpace Eₗ] [NormedAddCommGroup Fₗ] [NormedSpace 𝕜₂ Fₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_dense₂ : DenseRange ⇑e₂) (h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖) : Eₗ ≃ₛₗᵢ[σ₁₂] Fₗ
• LinearEquiv.extend 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) : Eₗ ≃SL[σ₁₂] Fₗ
• LinearEquiv.extendOfIsometry_apply 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedField 𝕜] [NormedField 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜₂ F] [NormedAddCommGroup Eₗ] [NormedSpace 𝕜 Eₗ] [CompleteSpace Eₗ] [NormedAddCommGroup Fₗ] [NormedSpace 𝕜₂ Fₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_dense₂ : DenseRange ⇑e₂) (h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖) (x : Eₗ) : (f.extendOfIsometry e₁ e₂ h_dense₁ h_dense₂ h_norm) x = ((e₂ ∘ₛₗ ↑f).extendOfNorm e₁) x
• LinearEquiv.extendOfIsometry_eq 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedField 𝕜] [NormedField 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜₂ F] [NormedAddCommGroup Eₗ] [NormedSpace 𝕜 Eₗ] [CompleteSpace Eₗ] [NormedAddCommGroup Fₗ] [NormedSpace 𝕜₂ Fₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_dense₂ : DenseRange ⇑e₂) (h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖) (x : E) : (f.extendOfIsometry e₁ e₂ h_dense₁ h_dense₂ h_norm) (e₁ x) = e₂ (f x)
• LinearEquiv.extendOfIsometry_symm_apply 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedField 𝕜] [NormedField 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜₂ F] [NormedAddCommGroup Eₗ] [NormedSpace 𝕜 Eₗ] [CompleteSpace Eₗ] [NormedAddCommGroup Fₗ] [NormedSpace 𝕜₂ Fₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_dense₂ : DenseRange ⇑e₂) (h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖) (x : Fₗ) : (f.extendOfIsometry e₁ e₂ h_dense₁ h_dense₂ h_norm).symm x = ((e₁ ∘ₛₗ ↑f.symm).extendOfNorm e₂) x
• LinearEquiv.extendOfIsometry_symm_eq 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedField 𝕜] [NormedField 𝕜₂] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜₂ F] [NormedAddCommGroup Eₗ] [NormedSpace 𝕜 Eₗ] [CompleteSpace Eₗ] [NormedAddCommGroup Fₗ] [NormedSpace 𝕜₂ Fₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_dense₂ : DenseRange ⇑e₂) (h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖) (x : F) : (f.extendOfIsometry e₁ e₂ h_dense₁ h_dense₂ h_norm).symm (e₂ x) = e₁ (f.symm x)
• ContinuousLinearMap.opNorm_extend_le 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedAddCommGroup E] [NormedAddCommGroup Eₗ] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜₂ F] [CompleteSpace F] (f : E →SL[σ₁₂] F) {e : E →L[𝕜] Eₗ} {N : NNReal} [RingHomIsometric σ₁₂] (h_dense : DenseRange ⇑e) (h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖) : ‖f.extend e‖ ≤ ↑N * ‖f‖
• LinearEquiv.extend_apply 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) (x : Eₗ) : (f.extend e₁ e₂ h_dense₁ h_norm₁ h_dense₂ h_norm₂) x = ((e₂ ∘ₛₗ ↑f).extendOfNorm e₁) x
• LinearEquiv.extend_eq 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) (x : E) : (f.extend e₁ e₂ h_dense₁ h_norm₁ h_dense₂ h_norm₂) (e₁ x) = e₂ (f x)
• LinearEquiv.norm_extend_le 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (C : ℝ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) (x : Eₗ) : ‖(f.extend e₁ e₂ h_dense₁ ⋯ h_dense₂ h_norm₂) x‖ ≤ C * ‖x‖
• LinearEquiv.extend_symm_apply 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) (x : Fₗ) : (f.extend e₁ e₂ h_dense₁ h_norm₁ h_dense₂ h_norm₂).symm x = ((e₁ ∘ₛₗ ↑f.symm).extendOfNorm e₂) x
• LinearEquiv.extend_symm_eq 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) (x : F) : (f.extend e₁ e₂ h_dense₁ h_norm₁ h_dense₂ h_norm₂).symm (e₂ x) = e₁ (f.symm x)
• LinearEquiv.norm_extend_symm_le 📋 Mathlib.Analysis.Normed.Operator.Extend
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} {Fₗ : Type u_6} [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [NormedAddCommGroup Eₗ] [AddCommGroup F] [NormedAddCommGroup Fₗ] [Module 𝕜 E] [Module 𝕜 Eₗ] [IsBoundedSMul 𝕜 Eₗ] [Module 𝕜₂ F] [Module 𝕜₂ Fₗ] [IsBoundedSMul 𝕜₂ Fₗ] [CompleteSpace Eₗ] [CompleteSpace Fₗ] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : E ≃ₛₗ[σ₁₂] F) (e₁ : E →ₗ[𝕜] Eₗ) (e₂ : F →ₗ[𝕜₂] Fₗ) (C : ℝ) (h_dense₁ : DenseRange ⇑e₁) (h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖) (h_dense₂ : DenseRange ⇑e₂) (h_norm₂ : ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖) (x : Fₗ) : ‖(f.extend e₁ e₂ h_dense₁ h_norm₁ h_dense₂ ⋯).symm x‖ ≤ C * ‖x‖
• MeasureTheory.Lp.simpleFunc.denseRange 📋 Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
  {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ENNReal} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤) : DenseRange Subtype.val
• MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg 📋 Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
  {α : Type u_1} [MeasurableSpace α] (p : ENNReal) (μ : MeasureTheory.Measure α) (G : Type u_7) [NormedAddCommGroup G] [PartialOrder G] [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤) : DenseRange (MeasureTheory.Lp.simpleFunc.coeSimpleFuncNonnegToLpNonneg p μ G)
• DenseRange.eq_of_inner_left 📋 Mathlib.Analysis.InnerProductSpace.Continuous
  {E : Type u_3} {ι : Type u_4} (𝕜 : Type u_5) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} {f : ι → E} (hf : DenseRange f) (h : ∀ (i : ι), inner 𝕜 x (f i) = inner 𝕜 y (f i)) : x = y
• DenseRange.eq_of_inner_right 📋 Mathlib.Analysis.InnerProductSpace.Continuous
  {E : Type u_3} {ι : Type u_4} (𝕜 : Type u_5) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} {f : ι → E} (hf : DenseRange f) (h : ∀ (i : ι), inner 𝕜 (f i) x = inner 𝕜 (f i) y) : x = y
• DenseRange.eq_zero_of_inner_left 📋 Mathlib.Analysis.InnerProductSpace.Continuous
  {E : Type u_3} {ι : Type u_4} (𝕜 : Type u_5) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E} {f : ι → E} (hf : DenseRange f) (h : ∀ (i : ι), inner 𝕜 x (f i) = 0) : x = 0
• DenseRange.eq_zero_of_inner_right 📋 Mathlib.Analysis.InnerProductSpace.Continuous
  {E : Type u_3} {ι : Type u_4} (𝕜 : Type u_5) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E} {f : ι → E} (hf : DenseRange f) (h : ∀ (i : ι), inner 𝕜 (f i) x = 0) : x = 0
• AlgebraicGeometry.dominant_eq_topologically 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  : @AlgebraicGeometry.IsDominant = AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => DenseRange
• AlgebraicGeometry.IsDominant.denseRange 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.IsDominant f] : DenseRange ⇑f
• AlgebraicGeometry.IsDominant.mk 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (denseRange : DenseRange ⇑f) : AlgebraicGeometry.IsDominant f
• AlgebraicGeometry.Scheme.Hom.denseRange 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsDominant f] : DenseRange ⇑f
• AlgebraicGeometry.isDominant_iff 📋 Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsDominant f ↔ DenseRange ⇑f
• IsDedekindDomain.HeightOneSpectrum.denseRange_algebraMap 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
  {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) : DenseRange ⇑(algebraMap K (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v))
• MvPowerSeries.WithPiTopology.denseRange_toMvPowerSeries 📋 Mathlib.RingTheory.MvPowerSeries.PiTopology
  {σ : Type u_1} {R : Type u_2} [TopologicalSpace R] [CommSemiring R] : DenseRange MvPolynomial.toMvPowerSeries
• PowerSeries.WithPiTopology.denseRange_toPowerSeries 📋 Mathlib.RingTheory.PowerSeries.PiTopology
  (R : Type u_1) [TopologicalSpace R] [CommSemiring R] : DenseRange Polynomial.toPowerSeries
• HasFDerivWithinAt.uniqueDiffWithinAt 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivWithinAt f f' s x) (hs : UniqueDiffWithinAt 𝕜 s x) (h' : DenseRange ⇑f') : UniqueDiffWithinAt 𝕜 (f '' s) (f x)
• UniqueDiffOn.image 📋 Mathlib.Analysis.Calculus.FDeriv.Equiv
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {f' : E → E →L[𝕜] F} (hs : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hd : ∀ x ∈ s, DenseRange ⇑(f' x)) : UniqueDiffOn 𝕜 (f '' s)
• Padic.denseRange_ratCast 📋 Mathlib.NumberTheory.Padics.PadicNumbers
  (p : ℕ) [Fact (Nat.Prime p)] : DenseRange Rat.cast
• AbsoluteValue.denseRange_algebraMap_pi 📋 Mathlib.Analysis.AbsoluteValue.Equivalence
  {F : Type u_1} [Field F] {ι : Type u_2} [Finite ι] {v : ι → AbsoluteValue F ℝ} (h : ∀ (i : ι), (v i).IsNontrivial) (hv : Pairwise fun i j => ¬(v i).IsEquiv (v j)) : DenseRange ⇑(algebraMap F ((i : ι) → WithAbs (v i)))
• UniqueMDiffWithinAt.image_denseRange 📋 Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {s : Set M} {x : M} (hs : UniqueMDiffAt[s] x) {f : M → M'} {f' : E →L[𝕜] E'} (hf : HasMFDerivAt[s] f x f') (hd : DenseRange ⇑f') : UniqueMDiffAt[f '' s] (f x)
• UniqueMDiffOn.image_denseRange' 📋 Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {s : Set M} (hs : UniqueMDiff[s]) {f : M → M'} {f' : M → E →L[𝕜] E'} (hf : ∀ x ∈ s, HasMFDerivAt[s] f x (f' x)) (hd : ∀ x ∈ s, DenseRange ⇑(f' x)) : UniqueMDiff[f '' s]
• UniqueMDiffOn.image_denseRange 📋 Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {s : Set M} (hs : UniqueMDiff[s]) {f : M → M'} (hf : MDiff[s] f) (hd : ∀ x ∈ s, DenseRange ⇑(mfderiv[s] f x)) : UniqueMDiff[f '' s]
• denseRange_pure 📋 Mathlib.Topology.Compactification.StoneCech
  {α : Type u} : DenseRange pure
• denseRange_preStoneCechUnit 📋 Mathlib.Topology.Compactification.StoneCech
  {α : Type u} [TopologicalSpace α] : DenseRange preStoneCechUnit
• denseRange_stoneCechUnit 📋 Mathlib.Topology.Compactification.StoneCech
  {α : Type u} [TopologicalSpace α] : DenseRange stoneCechUnit
• OnePoint.denseRange_coe 📋 Mathlib.Topology.Compactification.OnePoint.Basic
  {X : Type u_1} [TopologicalSpace X] [NoncompactSpace X] : DenseRange OnePoint.some
• BoundedContinuousFunction.toLp_denseRange 📋 Mathlib.MeasureTheory.Function.ContinuousMapDense
  {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] (E : Type u_2) [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) {p : ENNReal} [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (𝕜 : Type u_3) [NormedRing 𝕜] [Module 𝕜 E] [IsBoundedSMul 𝕜 E] [NormedSpace ℝ E] [μ.WeaklyRegular] [MeasureTheory.IsFiniteMeasure μ] (hp : p ≠ ⊤) : DenseRange ⇑(BoundedContinuousFunction.toLp p μ 𝕜)
• ContinuousMap.toLp_denseRange 📋 Mathlib.MeasureTheory.Function.ContinuousMapDense
  {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] (E : Type u_2) [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) {p : ENNReal} [SecondCountableTopologyEither α E] [_i : Fact (1 ≤ p)] (𝕜 : Type u_3) [NormedRing 𝕜] [Module 𝕜 E] [IsBoundedSMul 𝕜 E] [NormedSpace ℝ E] [CompactSpace α] [μ.WeaklyRegular] [MeasureTheory.IsFiniteMeasure μ] (hp : p ≠ ⊤) : DenseRange ⇑(ContinuousMap.toLp p μ 𝕜)
• SchwartzMap.denseRange_toLpCLM 📋 Mathlib.Analysis.Distribution.SchwartzSpace.Basic
  {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopologyEither E F] [FiniteDimensional ℝ E] [BorelSpace E] {p : ENNReal} (hp : p ≠ ⊤) [hp' : Fact (1 ≤ p)] {μ : MeasureTheory.Measure E} [hμ : μ.HasTemperateGrowth] [MeasureTheory.IsFiniteMeasureOnCompacts μ] : DenseRange ⇑(SchwartzMap.toLpCLM ℝ F p μ)
• Polynomial.exists_monic_and_natDegree_eq_and_norm_map_algebraMap_coeff_sub_lt 📋 Mathlib.Analysis.Normed.Field.Approximation
  {K : Type u_1} {L : Type u_2} [Field K] [NormedField L] [Algebra K L] (hd : DenseRange ⇑(algebraMap K L)) {f : Polynomial L} (hf : f.Monic) {ε : ℝ} (hε : ε > 0) : ∃ g, g.Monic ∧ f.natDegree = g.natDegree ∧ ∀ (n : ℕ), ‖(Polynomial.map (algebraMap K L) g).coeff n - f.coeff n‖ < ε
• IsAlgClosed.of_denseRange 📋 Mathlib.Analysis.Normed.Field.Dense
  {K : Type u_1} {L : Type u_2} [Field K] [NontriviallyNormedField L] [CompleteSpace L] [CharZero L] [IsUltrametricDist L] [Algebra K L] (hi : DenseRange ⇑(algebraMap K L)) [IsAlgClosed K] : IsAlgClosed L
• NormedAddCommGroup.denseRange_toCompl 📋 Mathlib.Analysis.Normed.Group.HomCompletion
  {G : Type u_1} [SeminormedAddCommGroup G] : DenseRange ⇑NormedAddCommGroup.toCompl
• DenseRange.zpow_of_ergodic_mul_left 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [MeasurableSpace G] [OpensMeasurableSpace G] {μ : MeasureTheory.Measure G} [μ.IsOpenPosMeasure] {g : G} (hg : Ergodic (fun x => g * x) μ) : DenseRange fun x => g ^ x
• DenseRange.zsmul_of_ergodic_add_left 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [MeasurableSpace G] [OpensMeasurableSpace G] {μ : MeasureTheory.Measure G} [μ.IsOpenPosMeasure] {g : G} (hg : Ergodic (fun x => g + x) μ) : DenseRange fun x => x • g
• ergodic_add_left_of_denseRange_zsmul 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [MeasurableSpace G] [SecondCountableTopology G] [BorelSpace G] {g : G} (hg : DenseRange fun x => x • g) (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsAddLeftInvariant] : Ergodic (fun x => g + x) μ
• ergodic_mul_left_of_denseRange_zpow 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [MeasurableSpace G] [SecondCountableTopology G] [BorelSpace G] {g : G} (hg : DenseRange fun x => g ^ x) (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant] : Ergodic (fun x => g * x) μ
• ergodic_add_left_of_denseRange_nsmul 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [MeasurableSpace G] [SecondCountableTopology G] [BorelSpace G] {g : G} (hg : DenseRange fun x => x • g) (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsAddLeftInvariant] : Ergodic (fun x => g + x) μ
• ergodic_mul_left_of_denseRange_pow 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [MeasurableSpace G] [SecondCountableTopology G] [BorelSpace G] {g : G} (hg : DenseRange fun x => g ^ x) (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant] : Ergodic (fun x => g * x) μ
• ergodic_add_left_iff_denseRange_zsmul 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [MeasurableSpace G] [SecondCountableTopology G] [BorelSpace G] {g : G} (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsAddLeftInvariant] [NeZero μ] : Ergodic (fun x => g + x) μ ↔ DenseRange fun x => x • g
• ergodic_mul_left_iff_denseRange_zpow 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [MeasurableSpace G] [SecondCountableTopology G] [BorelSpace G] {g : G} (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant] [NeZero μ] : Ergodic (fun x => g * x) μ ↔ DenseRange fun x => g ^ x
• ergodic_smul_of_denseRange_pow 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {X : Type u_2} [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X] {M : Type u_3} [Monoid M] [TopologicalSpace M] [MulAction M X] [ContinuousSMul M X] {g : M} (hg : DenseRange fun x => g ^ x) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul M X μ] : Ergodic (fun x => g • x) μ
• ergodic_vadd_of_denseRange_nsmul 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {X : Type u_2} [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X] {M : Type u_3} [AddMonoid M] [TopologicalSpace M] [AddAction M X] [ContinuousVAdd M X] {g : M} (hg : DenseRange fun x => x • g) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicVAdd M X μ] : Ergodic (fun x => g +ᵥ x) μ
• ergodic_smul_of_denseRange_zpow 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousInv G] {X : Type u_2} [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X] [MulAction G X] [ContinuousSMul G X] {g : G} (hg : DenseRange fun x => g ^ x) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul G X μ] : Ergodic (fun x => g • x) μ
• ergodic_vadd_of_denseRange_zsmul 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousNeg G] {X : Type u_2} [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X] [AddAction G X] [ContinuousVAdd G X] {g : G} (hg : DenseRange fun x => x • g) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicVAdd G X μ] : Ergodic (fun x => g +ᵥ x) μ
• AddCircle.denseRange_zsmul_iff 📋 Mathlib.Topology.Instances.AddCircle.DenseSubgroup
  {p : ℝ} [Fact (0 < p)] {a : AddCircle p} : (DenseRange fun x => x • a) ↔ addOrderOf a = 0
• AddCircle.denseRange_zsmul_coe_iff 📋 Mathlib.Topology.Instances.AddCircle.DenseSubgroup
  {a p : ℝ} : (DenseRange fun x => x • ↑a) ↔ Irrational (a / p)
• PadicInt.denseRange_intCast 📋 Mathlib.NumberTheory.Padics.RingHoms
  {p : ℕ} [hp_prime : Fact (Nat.Prime p)] : DenseRange Int.cast
• PadicInt.denseRange_natCast 📋 Mathlib.NumberTheory.Padics.RingHoms
  {p : ℕ} [hp_prime : Fact (Nat.Prime p)] : DenseRange Nat.cast
• NumberField.InfinitePlace.denseRange_algebraMap_pi 📋 Mathlib.NumberTheory.NumberField.InfinitePlace.Basic
  (K : Type u_1) [Field K] [NumberField K] : DenseRange ⇑(algebraMap K ((v : NumberField.InfinitePlace K) → WithAbs ↑v))
• NumberField.InfinitePlace.Completion.denseRange_coe 📋 Mathlib.NumberTheory.NumberField.Completion.InfinitePlace
  {K : Type u_1} [Field K] (v : NumberField.InfinitePlace K) : DenseRange fun x => { toCompletion := ↑x }
• NumberField.InfiniteAdeleRing.denseRange_algebraMap 📋 Mathlib.NumberTheory.NumberField.InfiniteAdeleRing
  (K : Type u_1) [Field K] [NumberField K] : DenseRange ⇑(algebraMap K (NumberField.InfiniteAdeleRing K))
• DenseRange.addChar_eq_of_eval_one_eq 📋 Mathlib.Topology.Algebra.Monoid.AddChar
  {A : Type u_1} {M : Type u_2} [TopologicalSpace A] [AddMonoidWithOne A] [Monoid M] [TopologicalSpace M] [T2Space M] (hdr : DenseRange Nat.cast) {κ₁ κ₂ : AddChar A M} (hκ₁ : Continuous ⇑κ₁) (hκ₂ : Continuous ⇑κ₂) (h : κ₁ 1 = κ₂ 1) : κ₁ = κ₂
• ContinuousMap.denseRange_tensorHom 📋 Mathlib.Topology.UniformSpace.ProdApproximation
  {X : Type u_5} {Y : Type u_6} {R : Type u_7} [TopologicalSpace X] [TopologicalSpace Y] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [CompactSpace X] [T2Space X] [CompactSpace Y] [TotallyDisconnectedSpace X] : DenseRange ⇑ContinuousMap.tensorHom
• ProfiniteGrp.denseRange_toLimit 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits
  (P : ProfiniteGrp.{u}) : DenseRange ⇑(ProfiniteGrp.Hom.hom P.toLimit)
• LaurentSeries.coe_range_dense 📋 Mathlib.RingTheory.LaurentSeries
  {K : Type u_2} [Field K] : DenseRange ⇑(algebraMap (RatFunc K) (LaurentSeries K))
• ProfiniteAddGrp.ProfiniteCompletion.denseRange 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
  (G : AddGrpCat) : DenseRange (ProfiniteAddGrp.ProfiniteCompletion.etaFn G)
• ProfiniteGrp.ProfiniteCompletion.denseRange 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
  (G : GrpCat) : DenseRange (ProfiniteGrp.ProfiniteCompletion.etaFn G)
• denseRange_zpow_iff_pow 📋 Mathlib.Topology.Algebra.Group.SubmonoidClosure
  {G : Type u_1} [Group G] [TopologicalSpace G] [CompactSpace G] [IsTopologicalGroup G] {x : G} : (DenseRange fun x_1 => x ^ x_1) ↔ DenseRange fun x_1 => x ^ x_1
• denseRange_zsmul_iff_nsmul 📋 Mathlib.Topology.Algebra.Group.SubmonoidClosure
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [CompactSpace G] [IsTopologicalAddGroup G] {x : G} : (DenseRange fun x_1 => x_1 • x) ↔ DenseRange fun x_1 => x_1 • x
• KuratowskiEmbedding.embeddingOfSubset_isometry 📋 Mathlib.Topology.MetricSpace.Kuratowski
  {α : Type u} [MetricSpace α] (x : ℕ → α) (H : DenseRange x) : Isometry (KuratowskiEmbedding.embeddingOfSubset x)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c