File: Mathlib/Analysis/LocallyConvex/Bounded.lean

full_name: Bornology.IsVonNBounded
code: /-- A set `s` is von Neumann bounded if every neighborhood of 0 absorbs `s`. -/
def IsVonNBounded (s : Set E) : Prop 
goal: Prop

full_name: Bornology.isVonNBounded_empty
code: @[simp]
theorem isVonNBounded_empty : IsVonNBounded 𝕜 (∅ : Set E) 
goal: Bornology.IsVonNBounded 𝕜 ∅

full_name: Bornology.isVonNBounded_iff
code: theorem isVonNBounded_iff (s : Set E) : IsVonNBounded 𝕜 s ↔ ∀ V ∈ 𝓝 (0 : E), Absorbs 𝕜 V s 
goal: Bornology.IsVonNBounded 𝕜 s ↔ ∀ V ∈ 𝓝 0, Absorbs 𝕜 V s

full_name: Filter.HasBasis.isVonNBounded_iff
code: theorem _root_.Filter.HasBasis.isVonNBounded_iff {q : ι → Prop} {s : ι → Set E} {A : Set E}
    (h : (𝓝 (0 : E)).HasBasis q s) : IsVonNBounded 𝕜 A ↔ ∀ i, q i → Absorbs 𝕜 (s i) A 
goal: Bornology.IsVonNBounded 𝕜 A ↔ ∀ (i : ι), q i → Absorbs 𝕜 (s i) A

full_name: Bornology.IsVonNBounded.subset
code: /-- Subsets of bounded sets are bounded. -/
theorem IsVonNBounded.subset {s₁ s₂ : Set E} (h : s₁ ⊆ s₂) (hs₂ : IsVonNBounded 𝕜 s₂) :
    IsVonNBounded 𝕜 s₁ 
goal: Bornology.IsVonNBounded 𝕜 s₁

full_name: Bornology.isVonNBounded_union
code: @[simp]
theorem isVonNBounded_union {s t : Set E} :
    IsVonNBounded 𝕜 (s ∪ t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 (s ∪ t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.IsVonNBounded.union
code: /-- The union of two bounded sets is bounded. -/
theorem IsVonNBounded.union {s₁ s₂ : Set E} (hs₁ : IsVonNBounded 𝕜 s₁) (hs₂ : IsVonNBounded 𝕜 s₂) :
    IsVonNBounded 𝕜 (s₁ ∪ s₂) 
goal: Bornology.IsVonNBounded 𝕜 (s₁ ∪ s₂)

full_name: Bornology.IsVonNBounded.of_boundedSpace
code: theorem IsVonNBounded.of_boundedSpace [BoundedSpace 𝕜] {s : Set E} : IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.isVonNBounded_iUnion
code: @[simp]
theorem isVonNBounded_iUnion {ι : Sort*} [Finite ι] {s : ι → Set E} :
    IsVonNBounded 𝕜 (⋃ i, s i) ↔ ∀ i, IsVonNBounded 𝕜 (s i) 
goal: Bornology.IsVonNBounded 𝕜 (⋃ i, s i) ↔ ∀ (i : ι), Bornology.IsVonNBounded 𝕜 (s i)

full_name: Bornology.isVonNBounded_biUnion
code: theorem isVonNBounded_biUnion {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set E} :
    IsVonNBounded 𝕜 (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, IsVonNBounded 𝕜 (s i) 
goal: Bornology.IsVonNBounded 𝕜 (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bornology.IsVonNBounded 𝕜 (s i)

full_name: Bornology.isVonNBounded_sUnion
code: theorem isVonNBounded_sUnion {S : Set (Set E)} (hS : S.Finite) :
    IsVonNBounded 𝕜 (⋃₀ S) ↔ ∀ s ∈ S, IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 (⋃₀ S) ↔ ∀ s ∈ S, Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.IsVonNBounded.add
code: protected theorem IsVonNBounded.add (hs : IsVonNBounded 𝕜 s) (ht : IsVonNBounded 𝕜 t) :
    IsVonNBounded 𝕜 (s + t) 
goal: Bornology.IsVonNBounded 𝕜 (s + t)

full_name: Bornology.IsVonNBounded.neg
code: protected theorem IsVonNBounded.neg (hs : IsVonNBounded 𝕜 s) : IsVonNBounded 𝕜 (-s) 
goal: Bornology.IsVonNBounded 𝕜 (-s)

full_name: Bornology.isVonNBounded_neg
code: @[simp]
theorem isVonNBounded_neg : IsVonNBounded 𝕜 (-s) ↔ IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 (-s) ↔ Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.IsVonNBounded.sub
code: protected theorem IsVonNBounded.sub (hs : IsVonNBounded 𝕜 s) (ht : IsVonNBounded 𝕜 t) :
    IsVonNBounded 𝕜 (s - t) 
goal: Bornology.IsVonNBounded 𝕜 (s - t)

full_name: Bornology.IsVonNBounded.of_topologicalSpace_le
code: /-- If a topology `t'` is coarser than `t`, then any set `s` that is bounded with respect to
`t` is bounded with respect to `t'`. -/
theorem IsVonNBounded.of_topologicalSpace_le {t t' : TopologicalSpace E} (h : t ≤ t') {s : Set E}
    (hs : @IsVonNBounded 𝕜 E _ _ _ t s) : @IsVonNBounded 𝕜 E _ _ _ t' s 
goal: Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.isVonNBounded_iff_tendsto_smallSets_nhds
code: lemma isVonNBounded_iff_tendsto_smallSets_nhds {𝕜 E : Type*} [NormedDivisionRing 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] {S : Set E} :
    IsVonNBounded 𝕜 S ↔ Tendsto (· • S : 𝕜 → Set E) (𝓝 0) (𝓝 0).smallSets 
goal: Bornology.IsVonNBounded 𝕜 S ↔ Filter.Tendsto (fun x => x • S) (𝓝 0) (𝓝 0).smallSets

full_name: Bornology.isVonNBounded_pi_iff
code: lemma isVonNBounded_pi_iff {𝕜 ι : Type*} {E : ι → Type*} [NormedDivisionRing 𝕜]
    [∀ i, AddCommGroup (E i)] [∀ i, Module 𝕜 (E i)] [∀ i, TopologicalSpace (E i)]
    {S : Set (∀ i, E i)} : IsVonNBounded 𝕜 S ↔ ∀ i, IsVonNBounded 𝕜 (eval i '' S) 
goal: Bornology.IsVonNBounded 𝕜 S ↔ ∀ (i : ι), Bornology.IsVonNBounded 𝕜 (Function.eval i '' S)

full_name: Bornology.IsVonNBounded.image
code: /-- A continuous linear image of a bounded set is bounded. -/
theorem IsVonNBounded.image {σ : 𝕜₁ →+* 𝕜₂} [RingHomSurjective σ] [RingHomIsometric σ] {s : Set E}
    (hs : IsVonNBounded 𝕜₁ s) (f : E →SL[σ] F) : IsVonNBounded 𝕜₂ (f '' s) 
goal: Bornology.IsVonNBounded 𝕜₂ (⇑f '' s)

full_name: Bornology.IsVonNBounded.smul_tendsto_zero
code: theorem IsVonNBounded.smul_tendsto_zero [NormedField 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
    {S : Set E} {ε : ι → 𝕜} {x : ι → E} {l : Filter ι}
    (hS : IsVonNBounded 𝕜 S) (hxS : ∀ᶠ n in l, x n ∈ S) (hε : Tendsto ε l (𝓝 0)) :
    Tendsto (ε • x) l (𝓝 0) 
goal: Filter.Tendsto (ε • x) l (𝓝 0)

full_name: Bornology.isVonNBounded_of_smul_tendsto_zero
code: theorem isVonNBounded_of_smul_tendsto_zero {ε : ι → 𝕜} {l : Filter ι} [l.NeBot]
    (hε : ∀ᶠ n in l, ε n ≠ 0) {S : Set E}
    (H : ∀ x : ι → E, (∀ n, x n ∈ S) → Tendsto (ε • x) l (𝓝 0)) : IsVonNBounded 𝕜 S 
goal: Bornology.IsVonNBounded 𝕜 S

full_name: Bornology.isVonNBounded_iff_smul_tendsto_zero
code: /-- Given any sequence `ε` of scalars which tends to `𝓝[≠] 0`, we have that a set `S` is bounded
  if and only if for any sequence `x : ℕ → S`, `ε • x` tends to 0. This actually works for any
  indexing type `ι`, but in the special case `ι = ℕ` we get the important fact that convergent
  sequences fully characterize bounded sets. -/
theorem isVonNBounded_iff_smul_tendsto_zero {ε : ι → 𝕜} {l : Filter ι} [l.NeBot]
    (hε : Tendsto ε l (𝓝[≠] 0)) {S : Set E} :
    IsVonNBounded 𝕜 S ↔ ∀ x : ι → E, (∀ n, x n ∈ S) → Tendsto (ε • x) l (𝓝 0) 
goal: Bornology.IsVonNBounded 𝕜 S ↔ ∀ (x : ι → E), (∀ (n : ι), x n ∈ S) → Filter.Tendsto (ε • x) l (𝓝 0)

full_name: Bornology.isVonNBounded_singleton
code: /-- Singletons are bounded. -/
theorem isVonNBounded_singleton (x : E) : IsVonNBounded 𝕜 ({x} : Set E) 
goal: Bornology.IsVonNBounded 𝕜 {x}

full_name: Bornology.isVonNBounded_insert
code: @[simp]
theorem isVonNBounded_insert (x : E) {s : Set E} :
    IsVonNBounded 𝕜 (insert x s) ↔ IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 (insert x s) ↔ Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.IsVonNBounded.vadd
code: protected theorem IsVonNBounded.vadd (hs : IsVonNBounded 𝕜 s) (x : E) :
    IsVonNBounded 𝕜 (x +ᵥ s) 
goal: Bornology.IsVonNBounded 𝕜 (x +ᵥ s)

full_name: Bornology.isVonNBounded_vadd
code: @[simp]
theorem isVonNBounded_vadd (x : E) : IsVonNBounded 𝕜 (x +ᵥ s) ↔ IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 (x +ᵥ s) ↔ Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.IsVonNBounded.of_add_right
code: theorem IsVonNBounded.of_add_right (hst : IsVonNBounded 𝕜 (s + t)) (hs : s.Nonempty) :
    IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.IsVonNBounded.of_add_left
code: theorem IsVonNBounded.of_add_left (hst : IsVonNBounded 𝕜 (s + t)) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.isVonNBounded_add_of_nonempty
code: theorem isVonNBounded_add_of_nonempty (hs : s.Nonempty) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 (s + t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 (s + t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.isVonNBounded_add
code: theorem isVonNBounded_add :
    IsVonNBounded 𝕜 (s + t) ↔ s = ∅ ∨ t = ∅ ∨ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 (s + t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.isVonNBounded_add_self
code: @[simp]
theorem isVonNBounded_add_self : IsVonNBounded 𝕜 (s + s) ↔ IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 (s + s) ↔ Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.IsVonNBounded.of_sub_left
code: theorem IsVonNBounded.of_sub_left (hst : IsVonNBounded 𝕜 (s - t)) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 s

full_name: Bornology.IsVonNBounded.of_sub_right
code: theorem IsVonNBounded.of_sub_right (hst : IsVonNBounded 𝕜 (s - t)) (hs : s.Nonempty) :
    IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.isVonNBounded_sub_of_nonempty
code: theorem isVonNBounded_sub_of_nonempty (hs : s.Nonempty) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 (s - t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 (s - t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.isVonNBounded_sub
code: theorem isVonNBounded_sub :
    IsVonNBounded 𝕜 (s - t) ↔ s = ∅ ∨ t = ∅ ∨ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
goal: Bornology.IsVonNBounded 𝕜 (s - t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t

full_name: Bornology.isVonNBounded_covers
code: /-- The union of all bounded set is the whole space. -/
theorem isVonNBounded_covers : ⋃₀ setOf (IsVonNBounded 𝕜) = (Set.univ : Set E) 
goal: ⋃₀ setOf (Bornology.IsVonNBounded 𝕜) = Set.univ

full_name: Bornology.vonNBornology
code: /-- The von Neumann bornology defined by the von Neumann bounded sets.

Note that this is not registered as an instance, in order to avoid diamonds with the
metric bornology. -/
abbrev vonNBornology : Bornology E 
goal: Bornology E

full_name: Bornology.isBounded_iff_isVonNBounded
code: @[simp]
theorem isBounded_iff_isVonNBounded {s : Set E} :
    @IsBounded _ (vonNBornology 𝕜 E) s ↔ IsVonNBounded 𝕜 s 
goal: Bornology.IsBounded s ↔ Bornology.IsVonNBounded 𝕜 s

full_name: TotallyBounded.isVonNBounded
code: theorem TotallyBounded.isVonNBounded {s : Set E} (hs : TotallyBounded s) :
    Bornology.IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 s

full_name: Filter.Tendsto.isVonNBounded_range
code: theorem Filter.Tendsto.isVonNBounded_range [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E]
    {f : ℕ → E} {x : E} (hf : Tendsto f atTop (𝓝 x)) : Bornology.IsVonNBounded 𝕜 (range f) 
goal: Bornology.IsVonNBounded 𝕜 (Set.range f)

full_name: NormedSpace.isVonNBounded_of_isBounded
code: theorem isVonNBounded_of_isBounded {s : Set E} (h : Bornology.IsBounded s) :
    Bornology.IsVonNBounded 𝕜 s 
goal: Bornology.IsVonNBounded 𝕜 s

full_name: NormedSpace.isVonNBounded_ball
code: theorem isVonNBounded_ball (r : ℝ) : Bornology.IsVonNBounded 𝕜 (Metric.ball (0 : E) r) 
goal: Bornology.IsVonNBounded 𝕜 (Metric.ball 0 r)

full_name: NormedSpace.isVonNBounded_closedBall
code: theorem isVonNBounded_closedBall (r : ℝ) :
    Bornology.IsVonNBounded 𝕜 (Metric.closedBall (0 : E) r) 
goal: Bornology.IsVonNBounded 𝕜 (Metric.closedBall 0 r)

full_name: NormedSpace.isVonNBounded_iff
code: theorem isVonNBounded_iff {s : Set E} : Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s 
goal: Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s

full_name: NormedSpace.isVonNBounded_iff'
code: theorem isVonNBounded_iff' {s : Set E} :
    Bornology.IsVonNBounded 𝕜 s ↔ ∃ r : ℝ, ∀ x ∈ s, ‖x‖ ≤ r 
goal: Bornology.IsVonNBounded 𝕜 s ↔ ∃ r, ∀ x ∈ s, ‖x‖ ≤ r

full_name: NormedSpace.image_isVonNBounded_iff
code: theorem image_isVonNBounded_iff {α : Type*} {f : α → E} {s : Set α} :
    Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∃ r : ℝ, ∀ x ∈ s, ‖f x‖ ≤ r 
goal: Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∃ r, ∀ x ∈ s, ‖f x‖ ≤ r

full_name: NormedSpace.vonNBornology_eq
code: /-- In a normed space, the von Neumann bornology (`Bornology.vonNBornology`) is equal to the
metric bornology. -/
theorem vonNBornology_eq : Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology 
goal: Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology

full_name: NormedSpace.isBounded_iff_subset_smul_ball
code: theorem isBounded_iff_subset_smul_ball {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.ball (0 : E) 1 
goal: Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.ball 0 1

full_name: NormedSpace.isBounded_iff_subset_smul_closedBall
code: theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 
goal: Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1