File: Mathlib/Topology/Algebra/Module/LocallyConvex.lean

full_name: LocallyConvexSpace.convex_basis
code: /-- A `LocallyConvexSpace` is a topological semimodule over an ordered semiring in which convex
neighborhoods of a point form a neighborhood basis at that point. -/
class LocallyConvexSpace (𝕜 E : Type*) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
    [TopologicalSpace E] : Prop where
  convex_basis : ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id
goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ Convex 𝕜 s) id

full_name: LocallyConvexSpace
code: /-- A `LocallyConvexSpace` is a topological semimodule over an ordered semiring in which convex
neighborhoods of a point form a neighborhood basis at that point. -/
class LocallyConvexSpace (𝕜 E : Type*) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
    [TopologicalSpace E] : Prop where
  convex_basis : ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id
goal: Prop

full_name: locallyConvexSpace_iff
code: theorem locallyConvexSpace_iff :
    LocallyConvexSpace 𝕜 E ↔ ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id 
goal: LocallyConvexSpace 𝕜 E ↔ ∀ (x : E), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ Convex 𝕜 s) id

full_name: LocallyConvexSpace.ofBases
code: theorem LocallyConvexSpace.ofBases {ι : Type*} (b : E → ι → Set E) (p : E → ι → Prop)
    (hbasis : ∀ x : E, (𝓝 x).HasBasis (p x) (b x)) (hconvex : ∀ x i, p x i → Convex 𝕜 (b x i)) :
    LocallyConvexSpace 𝕜 E 
goal: LocallyConvexSpace 𝕜 E

full_name: LocallyConvexSpace.convex_basis_zero
code: theorem LocallyConvexSpace.convex_basis_zero [LocallyConvexSpace 𝕜 E] :
    (𝓝 0 : Filter E).HasBasis (fun s => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id 
goal: (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Convex 𝕜 s) id

full_name: locallyConvexSpace_iff_exists_convex_subset
code: theorem locallyConvexSpace_iff_exists_convex_subset :
    LocallyConvexSpace 𝕜 E ↔ ∀ x : E, ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, Convex 𝕜 S ∧ S ⊆ U 
goal: LocallyConvexSpace 𝕜 E ↔ ∀ (x : E), ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, Convex 𝕜 S ∧ S ⊆ U

full_name: LocallyConvexSpace.ofBasisZero
code: theorem LocallyConvexSpace.ofBasisZero {ι : Type*} (b : ι → Set E) (p : ι → Prop)
    (hbasis : (𝓝 0).HasBasis p b) (hconvex : ∀ i, p i → Convex 𝕜 (b i)) :
    LocallyConvexSpace 𝕜 E 
goal: LocallyConvexSpace 𝕜 E

full_name: locallyConvexSpace_iff_zero
code: theorem locallyConvexSpace_iff_zero : LocallyConvexSpace 𝕜 E ↔
    (𝓝 0 : Filter E).HasBasis (fun s : Set E => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id 
goal: LocallyConvexSpace 𝕜 E ↔ (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Convex 𝕜 s) id

full_name: locallyConvexSpace_iff_exists_convex_subset_zero
code: theorem locallyConvexSpace_iff_exists_convex_subset_zero :
    LocallyConvexSpace 𝕜 E ↔ ∀ U ∈ (𝓝 0 : Filter E), ∃ S ∈ (𝓝 0 : Filter E), Convex 𝕜 S ∧ S ⊆ U 
goal: LocallyConvexSpace 𝕜 E ↔ ∀ U ∈ 𝓝 0, ∃ S ∈ 𝓝 0, Convex 𝕜 S ∧ S ⊆ U

full_name: LocallyConvexSpace.toLocallyConnectedSpace
code: instance (priority := 100) LocallyConvexSpace.toLocallyConnectedSpace [Module ℝ E]
    [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] : LocallyConnectedSpace E 
goal: LocallyConnectedSpace E

full_name: LocallyConvexSpace.convex_open_basis_zero
code: theorem LocallyConvexSpace.convex_open_basis_zero [LocallyConvexSpace 𝕜 E] :
    (𝓝 0 : Filter E).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) id 
goal: (𝓝 0).HasBasis (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) id

full_name: Disjoint.exists_open_convexes
code: /-- In a locally convex space, if `s`, `t` are disjoint convex sets, `s` is compact and `t` is
closed, then we can find open disjoint convex sets containing them. -/
theorem Disjoint.exists_open_convexes [LocallyConvexSpace 𝕜 E] {s t : Set E} (disj : Disjoint s t)
    (hs₁ : Convex 𝕜 s) (hs₂ : IsCompact s) (ht₁ : Convex 𝕜 t) (ht₂ : IsClosed t) :
    ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v 
goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v

full_name: locallyConvexSpace_sInf
code: theorem locallyConvexSpace_sInf {ts : Set (TopologicalSpace E)}
    (h : ∀ t ∈ ts, @LocallyConvexSpace 𝕜 E _ _ _ t) : @LocallyConvexSpace 𝕜 E _ _ _ (sInf ts) 
goal: LocallyConvexSpace 𝕜 E

full_name: locallyConvexSpace_iInf
code: theorem locallyConvexSpace_iInf {ts' : ι → TopologicalSpace E}
    (h' : ∀ i, @LocallyConvexSpace 𝕜 E _ _ _ (ts' i)) :
    @LocallyConvexSpace 𝕜 E _ _ _ (⨅ i, ts' i) 
goal: LocallyConvexSpace 𝕜 E

full_name: locallyConvexSpace_inf
code: theorem locallyConvexSpace_inf {t₁ t₂ : TopologicalSpace E} (h₁ : @LocallyConvexSpace 𝕜 E _ _ _ t₁)
    (h₂ : @LocallyConvexSpace 𝕜 E _ _ _ t₂) : @LocallyConvexSpace 𝕜 E _ _ _ (t₁ ⊓ t₂) 
goal: LocallyConvexSpace 𝕜 E

full_name: locallyConvexSpace_induced
code: theorem locallyConvexSpace_induced {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F]
    (f : E →ₗ[𝕜] F) : @LocallyConvexSpace 𝕜 E _ _ _ (t.induced f) 
goal: LocallyConvexSpace 𝕜 E

full_name: Pi.locallyConvexSpace
code: instance Pi.locallyConvexSpace {ι : Type*} {X : ι → Type*} [∀ i, AddCommMonoid (X i)]
    [∀ i, TopologicalSpace (X i)] [∀ i, Module 𝕜 (X i)] [∀ i, LocallyConvexSpace 𝕜 (X i)] :
    LocallyConvexSpace 𝕜 (∀ i, X i) 
goal: LocallyConvexSpace 𝕜 ((i : ι) → X i)

full_name: Prod.locallyConvexSpace
code: instance Prod.locallyConvexSpace [TopologicalSpace E] [TopologicalSpace F] [LocallyConvexSpace 𝕜 E]
    [LocallyConvexSpace 𝕜 F] : LocallyConvexSpace 𝕜 (E × F) 
goal: LocallyConvexSpace 𝕜 (E × F)