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/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
module
public import Mathlib.GroupTheory.GroupAction.Pointwise
public import Mathlib.Analysis.LocallyConvex.Basic
public import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
public import Mathlib.Analysis.Seminorm
public import Mathlib.Topology.Bornology.Basic
public import Mathlib.Topology.Algebra.IsUniformGroup.Basic
public import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Von Neumann Boundedness
This file defines natural or von Neumann bounded sets and proves elementary properties.
## Main declarations
* `Bornology.IsVonNBounded`: A set `s` is von Neumann-bounded if every neighborhood of zero
absorbs `s`.
* `Bornology.vonNBornology`: The bornology made of the von Neumann-bounded sets.
## Main results
* `Bornology.IsVonNBounded.of_topologicalSpace_le`: A coarser topology admits more
von Neumann-bounded sets.
* `Bornology.IsVonNBounded.image`: A continuous linear image of a bounded set is bounded.
* `Bornology.isVonNBounded_iff_smul_tendsto_zero`: Given any sequence `Ξ΅` of scalars which tends
to `π[β ] 0`, we have that a set `S` is bounded if and only if for any sequence `x : β β S`,
`Ξ΅ β’ x` tends to 0. This shows that bounded sets are completely determined by sequences, which is
the key fact for proving that sequential continuity implies continuity for linear maps defined on
a bornological space
## References
* [Bourbaki, *Topological Vector Spaces*][bourbaki1987]
-/
@[expose] public section
variable {π π' E F ΞΉ : Type*}
open Set Filter Function
open scoped Topology Pointwise
namespace Bornology
section SeminormedRing
section Zero
variable (π)
variable [SeminormedRing π] [SMul π E] [Zero E]
variable [TopologicalSpace E]
/-- A set `s` is von Neumann bounded if every neighborhood of 0 absorbs `s`. -/
def IsVonNBounded (s : Set E) : Prop :=
β β¦Vβ¦, V β π (0 : E) β Absorbs π V s
variable (E)
@[simp]
theorem isVonNBounded_empty : IsVonNBounded π (β
: Set E) := fun _ _ => Absorbs.empty
variable {π E}
theorem isVonNBounded_iff (s : Set E) : IsVonNBounded π s β β V β π (0 : E), Absorbs π V s :=
Iff.rfl
theorem _root_.Filter.HasBasis.isVonNBounded_iff {q : ΞΉ β Prop} {s : ΞΉ β Set E} {A : Set E}
(h : (π (0 : E)).HasBasis q s) : IsVonNBounded π A β β i, q i β Absorbs π (s i) A := by
refine β¨fun hA i hi => hA (h.mem_of_mem hi), fun hA V hV => ?_β©
rcases h.mem_iff.mp hV with β¨i, hi, hVβ©
exact (hA i hi).mono_left hV
/-- Subsets of bounded sets are bounded. -/
theorem IsVonNBounded.subset {sβ sβ : Set E} (h : sβ β sβ) (hsβ : IsVonNBounded π sβ) :
IsVonNBounded π sβ := fun _ hV => (hsβ hV).mono_right h
@[simp]
theorem isVonNBounded_union {s t : Set E} :
IsVonNBounded π (s βͺ t) β IsVonNBounded π s β§ IsVonNBounded π t := by
simp only [IsVonNBounded, absorbs_union, forall_and]
/-- The union of two bounded sets is bounded. -/
theorem IsVonNBounded.union {sβ sβ : Set E} (hsβ : IsVonNBounded π sβ) (hsβ : IsVonNBounded π sβ) :
IsVonNBounded π (sβ βͺ sβ) := isVonNBounded_union.2 β¨hsβ, hsββ©
@[nontriviality]
theorem IsVonNBounded.of_boundedSpace [BoundedSpace π] {s : Set E} : IsVonNBounded π s := fun _ _ β¦
.of_boundedSpace
@[nontriviality]
theorem IsVonNBounded.of_subsingleton [Subsingleton E] {s : Set E} : IsVonNBounded π s :=
fun U hU β¦ .of_forall fun c β¦ calc
s β univ := subset_univ s
_ = c β’ U := .symm <| Subsingleton.eq_univ_of_nonempty <| (Filter.nonempty_of_mem hU).image _
@[simp]
theorem isVonNBounded_iUnion {ΞΉ : Sort*} [Finite ΞΉ] {s : ΞΉ β Set E} :
IsVonNBounded π (β i, s i) β β i, IsVonNBounded π (s i) := by
simp only [IsVonNBounded, absorbs_iUnion, @forall_comm ΞΉ]
theorem isVonNBounded_biUnion {ΞΉ : Type*} {I : Set ΞΉ} (hI : I.Finite) {s : ΞΉ β Set E} :
IsVonNBounded π (β i β I, s i) β β i β I, IsVonNBounded π (s i) := by
have _ := hI.to_subtype
rw [biUnion_eq_iUnion, isVonNBounded_iUnion, Subtype.forall]
theorem isVonNBounded_sUnion {S : Set (Set E)} (hS : S.Finite) :
IsVonNBounded π (ββ S) β β s β S, IsVonNBounded π s := by
rw [sUnion_eq_biUnion, isVonNBounded_biUnion hS]
end Zero
section ContinuousAdd
variable [SeminormedRing π] [AddZeroClass E] [TopologicalSpace E] [ContinuousAdd E]
[DistribSMul π E] {s t : Set E}
protected theorem IsVonNBounded.add (hs : IsVonNBounded π s) (ht : IsVonNBounded π t) :
IsVonNBounded π (s + t) := fun U hU β¦ by
rcases exists_open_nhds_zero_add_subset hU with β¨V, hVo, hV, hVUβ©
exact ((hs <| hVo.mem_nhds hV).add (ht <| hVo.mem_nhds hV)).mono_left hVU
end ContinuousAdd
section IsTopologicalAddGroup
variable [SeminormedRing π] [AddGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E]
[DistribMulAction π E] {s t : Set E}
protected theorem IsVonNBounded.neg (hs : IsVonNBounded π s) : IsVonNBounded π (-s) := fun U hU β¦ by
rw [β neg_neg U]
exact (hs <| neg_mem_nhds_zero _ hU).neg_neg
@[simp]
theorem isVonNBounded_neg : IsVonNBounded π (-s) β IsVonNBounded π s :=
β¨fun h β¦ neg_neg s βΈ h.neg, fun h β¦ h.negβ©
alias β¨IsVonNBounded.of_neg, _β© := isVonNBounded_neg
protected theorem IsVonNBounded.sub (hs : IsVonNBounded π s) (ht : IsVonNBounded π t) :
IsVonNBounded π (s - t) := by
rw [sub_eq_add_neg]
exact hs.add ht.neg
end IsTopologicalAddGroup
end SeminormedRing
section MultipleTopologies
variable [SeminormedRing π] [AddCommGroup E] [Module π E]
/-- 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 β€ t') {s : Set E}
(hs : @IsVonNBounded π E _ _ _ t s) : @IsVonNBounded π E _ _ _ t' s := fun _ hV =>
hs <| (le_iff_nhds t t').mp h 0 hV
end MultipleTopologies
lemma isVonNBounded_iff_tendsto_smallSets_nhds {π E : Type*} [NormedDivisionRing π]
[AddCommGroup E] [Module π E] [TopologicalSpace E] {S : Set E} :
IsVonNBounded π S β Tendsto (Β· β’ S : π β Set E) (π 0) (π 0).smallSets := by
rw [tendsto_smallSets_iff]
refine forallβ_congr fun V hV β¦ ?_
simp only [absorbs_iff_eventually_nhds_zero (mem_of_mem_nhds hV), mapsTo_iff_image_subset,
image_smul]
alias β¨IsVonNBounded.tendsto_smallSets_nhds, _β© := isVonNBounded_iff_tendsto_smallSets_nhds
lemma isVonNBounded_iff_absorbing_le {π E : Type*} [NormedDivisionRing π]
[AddCommGroup E] [Module π E] [TopologicalSpace E] {S : Set E} :
IsVonNBounded π S β Filter.absorbing π S β€ π 0 :=
.rfl
lemma isVonNBounded_pi_iff {π ΞΉ : Type*} {E : ΞΉ β Type*} [NormedDivisionRing π]
[β i, AddCommGroup (E i)] [β i, Module π (E i)] [β i, TopologicalSpace (E i)]
{S : Set (β i, E i)} : IsVonNBounded π S β β i, IsVonNBounded π (eval i '' S) := by
simp_rw [isVonNBounded_iff_tendsto_smallSets_nhds, nhds_pi, Filter.pi, smallSets_iInf,
smallSets_comap_eq_comap_image, tendsto_iInf, tendsto_comap_iff, Function.comp_def,
β image_smul, image_image, eval, Pi.smul_apply, Pi.zero_apply]
section Image
variable {πβ πβ : Type*} [NormedDivisionRing πβ] [NormedDivisionRing πβ] [AddCommGroup E]
[Module πβ E] [AddCommGroup F] [Module πβ F] [TopologicalSpace E] [TopologicalSpace F]
/-- A continuous linear image of a bounded set is bounded. -/
protected theorem IsVonNBounded.image {Ο : πβ β+* πβ} [RingHomSurjective Ο] [RingHomIsometric Ο]
{s : Set E} (hs : IsVonNBounded πβ s) (f : E βSL[Ο] F) : IsVonNBounded πβ (f '' s) := by
have : map Ο (π 0) = π 0 := by
rw [Ο.isometry.isEmbedding.map_nhds_eq, Ο.surjective.range_eq, nhdsWithin_univ, map_zero]
have hfβ : Tendsto f (π 0) (π 0) := f.continuous.tendsto' 0 0 (map_zero f)
simp only [isVonNBounded_iff_tendsto_smallSets_nhds, β this, tendsto_map'_iff] at hs β’
simpa only [comp_def, image_smul_setββ] using hfβ.image_smallSets.comp hs
end Image
section sequence
theorem IsVonNBounded.smul_tendsto_zero [NormedField π]
[AddCommGroup E] [Module π E] [TopologicalSpace E]
{S : Set E} {Ξ΅ : ΞΉ β π} {x : ΞΉ β E} {l : Filter ΞΉ}
(hS : IsVonNBounded π S) (hxS : βαΆ n in l, x n β S) (hΞ΅ : Tendsto Ξ΅ l (π 0)) :
Tendsto (Ξ΅ β’ x) l (π 0) :=
(hS.tendsto_smallSets_nhds.comp hΞ΅).of_smallSets <| hxS.mono fun _ β¦ smul_mem_smul_set
variable [NontriviallyNormedField π]
[AddCommGroup E] [Module π E] [TopologicalSpace E] [ContinuousSMul π E]
theorem isVonNBounded_of_smul_tendsto_zero {Ξ΅ : ΞΉ β π} {l : Filter ΞΉ} [l.NeBot]
(hΞ΅ : βαΆ n in l, Ξ΅ n β 0) {S : Set E}
(H : β x : ΞΉ β E, (β n, x n β S) β Tendsto (Ξ΅ β’ x) l (π 0)) : IsVonNBounded π S := by
rw [(nhds_basis_balanced π E).isVonNBounded_iff]
by_contra! β¨V, β¨hV, hVbβ©, hVSβ©
have : βαΆ n in l, β x : S, Ξ΅ n β’ (x : E) β V := by
filter_upwards [hΞ΅] with n hn
rw [absorbs_iff_norm] at hVS
push Not at hVS
rcases hVS β(Ξ΅ n)β»ΒΉβ with β¨a, haΞ΅, haSβ©
rcases Set.not_subset.mp haS with β¨x, hxS, hxβ©
refine β¨β¨x, hxSβ©, fun hnx => ?_β©
rw [β Set.mem_inv_smul_set_iffβ hn] at hnx
exact hx (hVb.smul_mono haΞ΅ hnx)
rcases this.choice with β¨x, hxβ©
refine Filter.frequently_false l (Filter.Eventually.frequently ?_)
filter_upwards [hx,
(H (_ β x) fun n => (x n).2).eventually (eventually_mem_set.mpr hV)] using fun n => id
/-- Given any sequence `Ξ΅` of scalars which tends to `π[β ] 0`, we have that a set `S` is bounded
if and only if for any sequence `x : β β S`, `Ξ΅ β’ x` tends to 0. This actually works for any
indexing type `ΞΉ`, but in the special case `ΞΉ = β` we get the important fact that convergent
sequences fully characterize bounded sets. -/
theorem isVonNBounded_iff_smul_tendsto_zero {Ξ΅ : ΞΉ β π} {l : Filter ΞΉ} [l.NeBot]
(hΞ΅ : Tendsto Ξ΅ l (π[β ] 0)) {S : Set E} :
IsVonNBounded π S β β x : ΞΉ β E, (β n, x n β S) β Tendsto (Ξ΅ β’ x) l (π 0) :=
β¨fun hS _ hxS => hS.smul_tendsto_zero (Eventually.of_forall hxS) (le_trans hΞ΅ nhdsWithin_le_nhds),
isVonNBounded_of_smul_tendsto_zero (by exact hΞ΅ self_mem_nhdsWithin)β©
end sequence
/-- If a set is von Neumann bounded with respect to a smaller field,
then it is also von Neumann bounded with respect to a larger field.
See also `Bornology.IsVonNBounded.restrict_scalars` below. -/
theorem IsVonNBounded.extend_scalars [NontriviallyNormedField π]
{E : Type*} [AddCommGroup E] [Module π E]
(π : Type*) [NontriviallyNormedField π] [NormedAlgebra π π]
[Module π E] [TopologicalSpace E] [ContinuousSMul π E] [IsScalarTower π π E]
{s : Set E} (h : IsVonNBounded π s) : IsVonNBounded π s := by
obtain β¨Ξ΅, hΞ΅, hΞ΅ββ© : β Ξ΅ : β β π, Tendsto Ξ΅ atTop (π 0) β§ βαΆ n in atTop, Ξ΅ n β 0 := by
simpa only [tendsto_nhdsWithin_iff] using exists_seq_tendsto (π[β ] (0 : π))
refine isVonNBounded_of_smul_tendsto_zero (Ξ΅ := (Ξ΅ Β· β’ 1)) (by simpa) fun x hx β¦ ?_
have := h.smul_tendsto_zero (.of_forall hx) hΞ΅
simpa only [Pi.smul_def', smul_one_smul]
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E]
variable [TopologicalSpace E]
/-- The closure of a bounded set is bounded. -/
theorem IsVonNBounded.closure [T1Space E] [RegularSpace E] [ContinuousConstSMul π E]
{a : Set E} (ha : IsVonNBounded π a) : IsVonNBounded π (closure a) := by
intro V hV
rcases exists_mem_nhds_isClosed_subset hV with β¨W, hWβ, hWβ, hWββ©
specialize ha hWβ
filter_upwards [ha] with b ha'
grw [ha', closure_smulβ b, closure_subset_iff_isClosed.mpr hWβ, hWβ]
variable [ContinuousSMul π E]
/-- Singletons are bounded. -/
theorem isVonNBounded_singleton (x : E) : IsVonNBounded π ({x} : Set E) := fun _ hV =>
(absorbent_nhds_zero hV).absorbs
@[simp]
theorem isVonNBounded_insert (x : E) {s : Set E} :
IsVonNBounded π (insert x s) β IsVonNBounded π s := by
simp only [β singleton_union, isVonNBounded_union, isVonNBounded_singleton, true_and]
protected alias β¨_, IsVonNBounded.insertβ© := isVonNBounded_insert
/-- Finite sets are bounded. -/
theorem _root_.Set.Finite.isVonNBounded {s : Set E} (hs : s.Finite) :
IsVonNBounded π s := fun _ hV β¦
(absorbent_nhds_zero hV).absorbs_finite hs
section ContinuousAdd
variable [ContinuousAdd E] {s t : Set E}
protected theorem IsVonNBounded.vadd (hs : IsVonNBounded π s) (x : E) :
IsVonNBounded π (x +α΅₯ s) := by
rw [β singleton_vadd]
-- TODO: dot notation timeouts in the next line
exact IsVonNBounded.add (isVonNBounded_singleton x) hs
@[simp]
theorem isVonNBounded_vadd (x : E) : IsVonNBounded π (x +α΅₯ s) β IsVonNBounded π s :=
β¨fun h β¦ by simpa using h.vadd (-x), fun h β¦ h.vadd xβ©
theorem IsVonNBounded.of_add_right (hst : IsVonNBounded π (s + t)) (hs : s.Nonempty) :
IsVonNBounded π t :=
let β¨x, hxβ© := hs
(isVonNBounded_vadd x).mp <| hst.subset <| image_subset_image2_right hx
theorem IsVonNBounded.of_add_left (hst : IsVonNBounded π (s + t)) (ht : t.Nonempty) :
IsVonNBounded π s :=
((add_comm s t).subst hst).of_add_right ht
theorem isVonNBounded_add_of_nonempty (hs : s.Nonempty) (ht : t.Nonempty) :
IsVonNBounded π (s + t) β IsVonNBounded π s β§ IsVonNBounded π t :=
β¨fun h β¦ β¨h.of_add_left ht, h.of_add_right hsβ©, and_imp.2 IsVonNBounded.addβ©
theorem isVonNBounded_add :
IsVonNBounded π (s + t) β s = β
β¨ t = β
β¨ IsVonNBounded π s β§ IsVonNBounded π t := by
rcases s.eq_empty_or_nonempty with rfl | hs; Β· simp
rcases t.eq_empty_or_nonempty with rfl | ht; Β· simp
simp [hs.ne_empty, ht.ne_empty, isVonNBounded_add_of_nonempty hs ht]
@[simp]
theorem isVonNBounded_add_self : IsVonNBounded π (s + s) β IsVonNBounded π s := by
rcases s.eq_empty_or_nonempty with rfl | hs <;> simp [isVonNBounded_add_of_nonempty, *]
theorem IsVonNBounded.of_sub_left (hst : IsVonNBounded π (s - t)) (ht : t.Nonempty) :
IsVonNBounded π s :=
((sub_eq_add_neg s t).subst hst).of_add_left ht.neg
end ContinuousAdd
section IsTopologicalAddGroup
variable [IsTopologicalAddGroup E] {s t : Set E}
theorem IsVonNBounded.of_sub_right (hst : IsVonNBounded π (s - t)) (hs : s.Nonempty) :
IsVonNBounded π t :=
(((sub_eq_add_neg s t).subst hst).of_add_right hs).of_neg
theorem isVonNBounded_sub_of_nonempty (hs : s.Nonempty) (ht : t.Nonempty) :
IsVonNBounded π (s - t) β IsVonNBounded π s β§ IsVonNBounded π t := by
simp [sub_eq_add_neg, isVonNBounded_add_of_nonempty, hs, ht]
theorem isVonNBounded_sub :
IsVonNBounded π (s - t) β s = β
β¨ t = β
β¨ IsVonNBounded π s β§ IsVonNBounded π t := by
simp [sub_eq_add_neg, isVonNBounded_add]
end IsTopologicalAddGroup
/-- The union of all bounded set is the whole space. -/
theorem sUnion_isVonNBounded_eq_univ : ββ setOf (IsVonNBounded π) = (Set.univ : Set E) :=
Set.eq_univ_iff_forall.mpr fun x =>
Set.mem_sUnion.mpr β¨{x}, isVonNBounded_singleton _, Set.mem_singleton _β©
@[deprecated (since := "2025-11-14")]
alias isVonNBounded_covers := sUnion_isVonNBounded_eq_univ
variable (π E)
-- See note [reducible non-instances]
/-- 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 :=
Bornology.ofBounded (setOf (IsVonNBounded π)) (isVonNBounded_empty π E)
(fun _ hs _ ht => hs.subset ht) (fun _ hs _ => hs.union) isVonNBounded_singleton
variable {E}
@[simp]
theorem isBounded_iff_isVonNBounded {s : Set E} :
@IsBounded _ (vonNBornology π E) s β IsVonNBounded π s :=
isBounded_ofBounded_iff _
end NormedField
end Bornology
section IsUniformAddGroup
variable (π) [NormedField π] [AddCommGroup E] [Module π E]
variable [UniformSpace E] [IsUniformAddGroup E] [ContinuousSMul π E]
theorem TotallyBounded.isVonNBounded {s : Set E} (hs : TotallyBounded s) :
Bornology.IsVonNBounded π s := by
if h : β x : π, 1 < βxβ then
letI : NontriviallyNormedField π := β¨hβ©
rw [totallyBounded_iff_subset_finite_iUnion_nhds_zero] at hs
intro U hU
have h : Filter.Tendsto (fun x : E Γ E => x.fst + x.snd) (π 0) (π 0) :=
continuous_add.tendsto' _ _ (zero_add _)
have h' := (nhds_basis_balanced π E).prod (nhds_basis_balanced π E)
simp_rw [β nhds_prod_eq, id] at h'
rcases h.basis_left h' U hU with β¨x, hx, h''β©
rcases hs x.snd hx.2.1 with β¨t, ht, hsβ©
refine Absorbs.mono_right ?_ hs
rw [ht.absorbs_biUnion]
have hx_fstsnd : x.fst + x.snd β U := add_subset_iff.mpr fun z1 hz1 z2 hz2 β¦
h'' <| mk_mem_prod hz1 hz2
refine fun y _ => Absorbs.mono_left ?_ hx_fstsnd
-- TODO: with dot notation, Lean timeouts on the next line. Why?
exact Absorbent.vadd_absorbs (absorbent_nhds_zero hx.1.1) hx.2.2.absorbs_self
else
haveI : BoundedSpace π := β¨Metric.isBounded_iff.2 β¨1, by simp_all [dist_eq_norm]β©β©
exact Bornology.IsVonNBounded.of_boundedSpace
end IsUniformAddGroup
variable (π) in
theorem Filter.Tendsto.isVonNBounded_range [NormedField π] [AddCommGroup E] [Module π E]
[TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousSMul π E]
{f : β β E} {x : E} (hf : Tendsto f atTop (π x)) : Bornology.IsVonNBounded π (range f) :=
letI := IsTopologicalAddGroup.rightUniformSpace E
haveI := isUniformAddGroup_of_addCommGroup (G := E)
hf.cauchySeq.totallyBounded_range.isVonNBounded π
variable (π) in
protected theorem Bornology.IsVonNBounded.restrict_scalars_of_nontrivial
[NormedField π] [NormedRing π'] [NormedAlgebra π π'] [Nontrivial π']
[Zero E] [TopologicalSpace E]
[SMul π E] [MulAction π' E] [IsScalarTower π π' E] {s : Set E}
(h : IsVonNBounded π' s) : IsVonNBounded π s := by
intro V hV
refine (h hV).restrict_scalars <| AntilipschitzWith.tendsto_cobounded (K := β(1 : π')βββ»ΒΉ) ?_
refine AntilipschitzWith.of_le_mul_nndist fun x y β¦ ?_
rw [nndist_eq_nnnorm, nndist_eq_nnnorm, β sub_smul, nnnorm_smul, β div_eq_inv_mul,
mul_div_cancel_rightβ _ (nnnorm_ne_zero_iff.2 one_ne_zero)]
variable (π) in
protected theorem Bornology.IsVonNBounded.restrict_scalars
[NormedField π] [NormedRing π'] [NormedAlgebra π π']
[Zero E] [TopologicalSpace E]
[SMul π E] [MulActionWithZero π' E] [IsScalarTower π π' E] {s : Set E}
(h : IsVonNBounded π' s) : IsVonNBounded π s :=
match subsingleton_or_nontrivial π' with
| .inl _ =>
have : Subsingleton E := MulActionWithZero.subsingleton π' E
IsVonNBounded.of_subsingleton
| .inr _ =>
h.restrict_scalars_of_nontrivial _
section VonNBornologyEqMetric
namespace NormedSpace
section NormedField
variable (π)
variable [NormedField π] [SeminormedAddCommGroup E] [NormedSpace π E]
theorem isVonNBounded_of_isBounded {s : Set E} (h : Bornology.IsBounded s) :
Bornology.IsVonNBounded π s := by
rcases h.subset_ball 0 with β¨r, hrβ©
rw [Metric.nhds_basis_ball.isVonNBounded_iff]
rw [β ball_normSeminorm π E] at hr β’
exact fun Ξ΅ hΞ΅ β¦ ((normSeminorm π E).ball_zero_absorbs_ball_zero hΞ΅).mono_right hr
variable (E)
theorem isVonNBounded_ball (r : β) : Bornology.IsVonNBounded π (Metric.ball (0 : E) r) :=
isVonNBounded_of_isBounded _ Metric.isBounded_ball
theorem isVonNBounded_closedBall (r : β) :
Bornology.IsVonNBounded π (Metric.closedBall (0 : E) r) :=
isVonNBounded_of_isBounded _ Metric.isBounded_closedBall
end NormedField
variable (π)
variable [NontriviallyNormedField π] [SeminormedAddCommGroup E] [NormedSpace π E]
theorem isVonNBounded_iff {s : Set E} : Bornology.IsVonNBounded π s β Bornology.IsBounded s := by
refine β¨fun h β¦ ?_, isVonNBounded_of_isBounded _β©
rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with β¨Ο, hΟ, hΟballβ©
rcases NormedField.exists_lt_norm π Ο with β¨a, haβ©
specialize hΟball a ha.le
rw [β ball_normSeminorm π E, Seminorm.smul_ball_zero (norm_pos_iff.1 <| hΟ.trans ha),
ball_normSeminorm] at hΟball
exact Metric.isBounded_ball.subset hΟball
theorem isVonNBounded_iff' {s : Set E} :
Bornology.IsVonNBounded π s β β r : β, β x β s, βxβ β€ r := by
rw [NormedSpace.isVonNBounded_iff, isBounded_iff_forall_norm_le]
theorem image_isVonNBounded_iff {Ξ± : Type*} {f : Ξ± β E} {s : Set Ξ±} :
Bornology.IsVonNBounded π (f '' s) β β r : β, β x β s, βf xβ β€ r := by
simp_rw [isVonNBounded_iff', Set.forall_mem_image]
/-- In a normed space, the von Neumann bornology (`Bornology.vonNBornology`) is equal to the
metric bornology. -/
theorem vonNBornology_eq : Bornology.vonNBornology π E = PseudoMetricSpace.toBornology := by
rw [Bornology.ext_iff_isBounded]
intro s
rw [Bornology.isBounded_iff_isVonNBounded]
exact isVonNBounded_iff _
theorem isBounded_iff_subset_smul_ball {s : Set E} :
Bornology.IsBounded s β β a : π, s β a β’ Metric.ball (0 : E) 1 := by
rw [β isVonNBounded_iff π]
constructor
Β· intro h
rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with β¨Ο, _, hΟballβ©
rcases NormedField.exists_lt_norm π Ο with β¨a, haβ©
exact β¨a, hΟball a ha.leβ©
Β· rintro β¨a, haβ©
exact ((isVonNBounded_ball π E 1).image (a β’ (1 : E βL[π] E))).subset ha
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
Bornology.IsBounded s β β a : π, s β a β’ Metric.closedBall (0 : E) 1 := by
constructor
Β· rw [isBounded_iff_subset_smul_ball π]
exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall
Β· rw [β isVonNBounded_iff π]
rintro β¨a, haβ©
exact ((isVonNBounded_closedBall π E 1).image (a β’ (1 : E βL[π] E))).subset ha
end NormedSpace
end VonNBornologyEqMetric
section QuasiCompleteSpace
/-- A locally convex space is quasi-complete if every closed and von Neumann bounded set is
complete. -/
class QuasiCompleteSpace (π : Type*) (E : Type*) [Zero E] [UniformSpace E] [SeminormedRing π]
[SMul π E] : Prop where
/-- A locally convex space is quasi-complete if every closed and von Neumann bounded set is
complete. -/
quasiComplete : β β¦s : Set Eβ¦, Bornology.IsVonNBounded π s β IsClosed s β IsComplete s
variable {π : Type*} {E : Type*} [Zero E] [UniformSpace E] [SeminormedRing π] [SMul π E]
/-- A complete space is quasi-complete with respect to any scalar ring. -/
instance [CompleteSpace E] : QuasiCompleteSpace π E where
quasiComplete _ _ := IsClosed.isComplete
/-- [Bourbaki, *Topological Vector Spaces*, III Β§1.6][bourbaki1987] -/
theorem isCompact_closure_of_totallyBounded_quasiComplete {E : Type*} {π : Type*} [NormedField π]
[AddCommGroup E] [Module π E] [UniformSpace E] [IsUniformAddGroup E] [ContinuousSMul π E]
[QuasiCompleteSpace π E] {s : Set E} (hs : TotallyBounded s) : IsCompact (closure s) :=
hs.closure.isCompact_of_isComplete
(QuasiCompleteSpace.quasiComplete (TotallyBounded.isVonNBounded π (TotallyBounded.closure hs))
isClosed_closure)
end QuasiCompleteSpace