fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
le_def {p q : Seminorm 𝕜 E} : p ≤ q ↔ ∀ x, p x ≤ q x :=
Iff.rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | le_def | null |
lt_def {p q : Seminorm 𝕜 E} : p < q ↔ p ≤ q ∧ ∃ x, p x < q x :=
@Pi.lt_def _ _ _ p q | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | lt_def | null |
instSemilatticeSup : SemilatticeSup (Seminorm 𝕜 E) :=
Function.Injective.semilatticeSup _ DFunLike.coe_injective coe_sup | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instSemilatticeSup | null |
comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : Seminorm 𝕜 E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
smul' _ _ := by simp only [map_smulₛₗ _, map_smul_eq_mul, RingHomIsometric.norm_map] } | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp | Composition of a seminorm with a linear map is a seminorm. |
coe_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : ⇑(p.comp f) = p ∘ f :=
rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_comp | null |
comp_apply (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) : (p.comp f) x = p (f x) :=
rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_apply | null |
comp_id (p : Seminorm 𝕜 E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_id | null |
comp_zero (p : Seminorm 𝕜₂ E₂) : p.comp (0 : E →ₛₗ[σ₁₂] E₂) = 0 :=
ext fun _ => map_zero p
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_zero | null |
zero_comp (f : E →ₛₗ[σ₁₂] E₂) : (0 : Seminorm 𝕜₂ E₂).comp f = 0 :=
ext fun _ => rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | zero_comp | null |
comp_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (p : Seminorm 𝕜₃ E₃) (g : E₂ →ₛₗ[σ₂₃] E₃)
(f : E →ₛₗ[σ₁₂] E₂) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_comp | null |
add_comp (p q : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | add_comp | null |
comp_add_le (p : Seminorm 𝕜₂ E₂) (f g : E →ₛₗ[σ₁₂] E₂) :
p.comp (f + g) ≤ p.comp f + p.comp g := fun _ => map_add_le_add p _ _ | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_add_le | null |
smul_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : R) :
(c • p).comp f = c • p.comp f :=
ext fun _ => rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | smul_comp | null |
comp_mono {p q : Seminorm 𝕜₂ E₂} (f : E →ₛₗ[σ₁₂] E₂) (hp : p ≤ q) : p.comp f ≤ q.comp f :=
fun _ => hp _ | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_mono | null |
@[simps]
pullback (f : E →ₛₗ[σ₁₂] E₂) : Seminorm 𝕜₂ E₂ →+ Seminorm 𝕜 E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | pullback | The composition as an `AddMonoidHom`. |
instOrderBot : OrderBot (Seminorm 𝕜 E) where
bot := 0
bot_le := apply_nonneg
@[simp] | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instOrderBot | null |
coe_bot : ⇑(⊥ : Seminorm 𝕜 E) = 0 :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_bot | null |
bot_eq_zero : (⊥ : Seminorm 𝕜 E) = 0 :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | bot_eq_zero | null |
smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (apply_nonneg p x) (NNReal.coe_nonneg b) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | smul_le_smul | null |
finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, apply_nonneg (p i) x⟩ : ℝ≥0) := by
induction s using Finset.cons_induction_on with
| empty =>
rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
| cons a s ha ih =>
rw [Finset.sup_cons, Finset.sup_cons, coe_sup, Pi.sup_apply, NNReal.coe_max, NNReal.coe_mk, ih] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | finset_sup_apply | null |
exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i ↦ (⟨p i x, apply_nonneg _ _⟩ : ℝ≥0)) with ⟨i, hi, hix⟩
rw [finset_sup_apply]
exact ⟨i, hi, congr_arg _ hix⟩ | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | exists_apply_eq_finset_sup | null |
zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl | hs)
· left; rfl
· right; exact exists_apply_eq_finset_sup p hs x | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | zero_or_exists_apply_eq_finset_sup | null |
finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((↑) : ℝ≥0 → ℝ) (NNReal.mul_finset_sup C s (fun i ↦ ⟨p i x, apply_nonneg _ _⟩)) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | finset_sup_smul | null |
finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i ∈ s, p i := by
classical
refine Finset.sup_le_iff.mpr ?_
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | finset_sup_le_sum | null |
finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a := by
lift a to ℝ≥0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | finset_sup_apply_le | null |
le_finset_sup_apply {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {i : ι}
(hi : i ∈ s) : p i x ≤ s.sup p x :=
(Finset.le_sup hi : p i ≤ s.sup p) x | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | le_finset_sup_apply | null |
finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a := by
lift a to ℝ≥0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
· exact h
· exact NNReal.coe_pos.mpr ha | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | finset_sup_apply_lt | null |
norm_sub_map_le_sub (p : Seminorm 𝕜 E) (x y : E) : ‖p x - p y‖ ≤ p (x - y) :=
abs_sub_map_le_sub p x y | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | norm_sub_map_le_sub | null |
comp_smul (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : 𝕜₂) :
p.comp (c • f) = ‖c‖₊ • p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_smul | null |
comp_smul_apply (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : 𝕜₂) (x : E) :
p.comp (c • f) x = ‖c‖ * p (f x) :=
map_smul_eq_mul p _ _ | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | comp_smul_apply | null |
bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
⟨0, by
rintro _ ⟨x, rfl⟩
dsimp; positivity⟩ | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | bddBelow_range_add | Auxiliary lemma to show that the infimum of seminorms is well-defined. |
noncomputable instInf : Min (Seminorm 𝕜 E) where
min p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· rw [norm_zero, zero_mul, zero_smul]
refine
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
(fun i => by positivity)
fun x hx => ⟨0, by rwa [map_zero, sub_zero, map_zero, add_zero]⟩
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, ← map_smul_eq_mul p, ←
map_smul_eq_mul q, smul_sub]
refine
Function.Surjective.iInf_congr ((a⁻¹ • ·) : E → E)
(fun u => ⟨a • u, inv_smul_smul₀ ha u⟩) fun u => ?_
rw [smul_inv_smul₀ ha] }
@[simp] | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instInf | null |
inf_apply (p q : Seminorm 𝕜 E) (x : E) : (p ⊓ q) x = ⨅ u : E, p u + q (x - u) :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | inf_apply | null |
noncomputable instLattice : Lattice (Seminorm 𝕜 E) :=
{ Seminorm.instSemilatticeSup with
inf := (· ⊓ ·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [map_zero, zero_add, sub_zero]; rfl
le_inf := fun a _ _ hab hac _ =>
le_ciInf fun _ => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) } | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instLattice | null |
smul_inf [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, ← smul_one_smul ℝ≥0 r (_ : ℝ), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | smul_inf | null |
noncomputable instSupSet : SupSet (Seminorm 𝕜 E) where
sSup s :=
if h : BddAbove ((↑) '' s : Set (E → ℝ)) then
{ toFun := ⨆ p : s, ((p : Seminorm 𝕜 E) : E → ℝ)
map_zero' := by
rw [iSup_apply, ← @Real.iSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with ⟨q, hq⟩
obtain rfl | h := s.eq_empty_or_nonempty
· simp [Real.iSup_of_isEmpty]
haveI : Nonempty ↑s := h.coe_sort
simp only [iSup_apply]
refine ciSup_le fun i =>
((i : Seminorm 𝕜 E).add_le' x y).trans <| add_le_add
(le_ciSup (f := fun i => (Subtype.val i : Seminorm 𝕜 E).toFun x) ⟨q x, ?_⟩ i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm 𝕜 E).toFun y) ⟨q y, ?_⟩ i)
<;> rw [mem_upperBounds, forall_mem_range]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [← smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm 𝕜 E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else ⊥ | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instSupSet | We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `⊥`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((↑) '' s : Set (E → ℝ))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity). |
protected coe_sSup_eq' {s : Set <| Seminorm 𝕜 E}
(hs : BddAbove ((↑) '' s : Set (E → ℝ))) : ↑(sSup s) = ⨆ p : s, ((p : Seminorm 𝕜 E) : E → ℝ) :=
congr_arg _ (dif_pos hs) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_sSup_eq' | null |
protected bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) :=
⟨fun ⟨q, hq⟩ => ⟨q, forall_mem_image.2 fun _ hp => hq hp⟩, fun H =>
⟨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with ⟨q, hq⟩
exact
le_ciSup ⟨q x, forall_mem_range.mpr fun i : s => hq (mem_image_of_mem _ i.2) x⟩ ⟨p, hp⟩⟩⟩ | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | bddAbove_iff | null |
protected bddAbove_range_iff {ι : Sort*} {p : ι → Seminorm 𝕜 E} :
BddAbove (range p) ↔ ∀ x, BddAbove (range fun i ↦ p i x) := by
rw [Seminorm.bddAbove_iff, ← range_comp, bddAbove_range_pi]; rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | bddAbove_range_iff | null |
protected coe_sSup_eq {s : Set <| Seminorm 𝕜 E} (hs : BddAbove s) :
↑(sSup s) = ⨆ p : s, ((p : Seminorm 𝕜 E) : E → ℝ) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_sSup_eq | null |
protected coe_iSup_eq {ι : Sort*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) :
↑(⨆ i, p i) = ⨆ i, ((p i : Seminorm 𝕜 E) : E → ℝ) := by
rw [← sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm 𝕜 E => (p : E → ℝ)) p | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_iSup_eq | null |
protected sSup_apply {s : Set (Seminorm 𝕜 E)} (hp : BddAbove s) {x : E} :
(sSup s) x = ⨆ p : s, (p : E → ℝ) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | sSup_apply | null |
protected iSup_apply {ι : Sort*} {p : ι → Seminorm 𝕜 E}
(hp : BddAbove (range p)) {x : E} : (⨆ i, p i) x = ⨆ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | iSup_apply | null |
protected sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.iSup_of_isEmpty]
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | sSup_empty | null |
private isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine ⟨fun p hp x => ?_, fun p hp x => ?_⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hs₁, iSup_apply]
· rcases hs₁ with ⟨q, hq⟩
exact le_ciSup ⟨q x, forall_mem_range.mpr fun i : s => hq i.2 x⟩ ⟨p, hp⟩
· exact ciSup_le fun q => hp q.2 x | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | isLUB_sSup | null |
noncomputable instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm 𝕜 E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm 𝕜 E) Seminorm.isLUB_sSup | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instConditionallyCompleteLattice | `Seminorm 𝕜 E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. |
ball (x : E) (r : ℝ) :=
{ y : E | p (y - x) < r } | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball | The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. |
closedBall (x : E) (r : ℝ) :=
{ y : E | p (y - x) ≤ r }
variable {x y : E} {r : ℝ}
@[simp] | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall | The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) ≤ r`. |
mem_ball : y ∈ ball p x r ↔ p (y - x) < r :=
Iff.rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | mem_ball | null |
mem_closedBall : y ∈ closedBall p x r ↔ p (y - x) ≤ r :=
Iff.rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | mem_closedBall | null |
mem_ball_self (hr : 0 < r) : x ∈ ball p x r := by simp [hr] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | mem_ball_self | null |
mem_closedBall_self (hr : 0 ≤ r) : x ∈ closedBall p x r := by simp [hr] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | mem_closedBall_self | null |
mem_ball_zero : y ∈ ball p 0 r ↔ p y < r := by rw [mem_ball, sub_zero] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | mem_ball_zero | null |
mem_closedBall_zero : y ∈ closedBall p 0 r ↔ p y ≤ r := by rw [mem_closedBall, sub_zero] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | mem_closedBall_zero | null |
ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_zero_eq | null |
closedBall_zero_eq : closedBall p 0 r = { y : E | p y ≤ r } :=
Set.ext fun _ => p.mem_closedBall_zero | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_zero_eq | null |
ball_subset_closedBall (x r) : ball p x r ⊆ closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_subset_closedBall | null |
closedBall_eq_biInter_ball (x r) : closedBall p x r = ⋂ ρ > r, ball p x ρ := by
ext y; simp_rw [mem_closedBall, mem_iInter₂, mem_ball, ← forall_gt_iff_le]
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_eq_biInter_ball | null |
ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm 𝕜 E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_zero' | null |
closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm 𝕜 E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_zero' | null |
ball_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff₀ (NNReal.coe_pos.mpr hc)] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_smul | null |
closedBall_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).closedBall x r = p.closedBall x (r / c) := by
ext
rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff₀ (NNReal.coe_pos.mpr hc)] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_smul | null |
ball_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
ball (p ⊔ q) e r = ball p e r ∩ ball q e r := by
simp_rw [ball, ← Set.setOf_and, coe_sup, Pi.sup_apply, sup_lt_iff] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_sup | null |
closedBall_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
closedBall (p ⊔ q) e r = closedBall p e r ∩ closedBall q e r := by
simp_rw [closedBall, ← Set.setOf_and, coe_sup, Pi.sup_apply, sup_le_iff] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_sup | null |
ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction H using Finset.Nonempty.cons_induction with
| singleton => simp
| cons _ _ _ hs ih =>
simp only [Finset.sup'_cons hs, Finset.inf'_cons hs, ball_sup, inf_eq_inter, ih] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_finset_sup' | null |
closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r := by
induction H using Finset.Nonempty.cons_induction with
| singleton => simp
| cons _ _ _ hs ih =>
simp only [Finset.sup'_cons hs, Finset.inf'_cons hs, closedBall_sup, inf_eq_inter, ih] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_finset_sup' | null |
ball_mono {p : Seminorm 𝕜 E} {r₁ r₂ : ℝ} (h : r₁ ≤ r₂) : p.ball x r₁ ⊆ p.ball x r₂ :=
fun _ (hx : _ < _) => hx.trans_le h | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_mono | null |
closedBall_mono {p : Seminorm 𝕜 E} {r₁ r₂ : ℝ} (h : r₁ ≤ r₂) :
p.closedBall x r₁ ⊆ p.closedBall x r₂ := fun _ (hx : _ ≤ _) => hx.trans h | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_mono | null |
ball_antitone {p q : Seminorm 𝕜 E} (h : q ≤ p) : p.ball x r ⊆ q.ball x r := fun _ =>
(h _).trans_lt | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_antitone | null |
closedBall_antitone {p q : Seminorm 𝕜 E} (h : q ≤ p) :
p.closedBall x r ⊆ q.closedBall x r := fun _ => (h _).trans | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_antitone | null |
ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, hy₁, y₂, hy₂, rfl⟩
rw [mem_ball, add_sub_add_comm]
exact (map_add_le_add p _ _).trans_lt (add_lt_add hy₁ hy₂) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_add_ball_subset | null |
closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, hy₁, y₂, hy₂, rfl⟩
rw [mem_closedBall, add_sub_add_comm]
exact (map_add_le_add p _ _).trans (add_le_add hy₁ hy₂) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_add_closedBall_subset | null |
sub_mem_ball (p : Seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.ball y r ↔ x₁ ∈ p.ball (x₂ + y) r := by simp_rw [mem_ball, sub_sub] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | sub_mem_ball | null |
sub_mem_closedBall (p : Seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.closedBall y r ↔ x₁ ∈ p.closedBall (x₂ + y) r := by
simp_rw [mem_closedBall, sub_sub] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | sub_mem_closedBall | null |
vadd_ball (p : Seminorm 𝕜 E) : x +ᵥ p.ball y r = p.ball (x +ᵥ y) r :=
letI := AddGroupSeminorm.toSeminormedAddCommGroup p.toAddGroupSeminorm
Metric.vadd_ball x y r | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | vadd_ball | The image of a ball under addition with a singleton is another ball. |
vadd_closedBall (p : Seminorm 𝕜 E) : x +ᵥ p.closedBall y r = p.closedBall (x +ᵥ y) r :=
letI := AddGroupSeminorm.toSeminormedAddCommGroup p.toAddGroupSeminorm
Metric.vadd_closedBall x y r | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | vadd_closedBall | The image of a closed ball under addition with a singleton is another closed ball. |
ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r := by
ext
simp_rw [ball, mem_preimage, comp_apply, Set.mem_setOf_eq, map_sub] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_comp | null |
closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r := by
ext
simp_rw [closedBall, mem_preimage, comp_apply, Set.mem_setOf_eq, map_sub]
variable (p : Seminorm 𝕜 E) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_comp | null |
preimage_metric_ball {r : ℝ} : p ⁻¹' Metric.ball 0 r = { x | p x < r } := by
ext x
simp only [mem_setOf, mem_preimage, mem_ball_zero_iff, Real.norm_of_nonneg (apply_nonneg p _)] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | preimage_metric_ball | null |
preimage_metric_closedBall {r : ℝ} : p ⁻¹' Metric.closedBall 0 r = { x | p x ≤ r } := by
ext x
simp only [mem_setOf, mem_preimage, mem_closedBall_zero_iff,
Real.norm_of_nonneg (apply_nonneg p _)] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | preimage_metric_closedBall | null |
ball_zero_eq_preimage_ball {r : ℝ} : p.ball 0 r = p ⁻¹' Metric.ball 0 r := by
rw [ball_zero_eq, preimage_metric_ball] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_zero_eq_preimage_ball | null |
closedBall_zero_eq_preimage_closedBall {r : ℝ} :
p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r := by
rw [closedBall_zero_eq, preimage_metric_closedBall]
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_zero_eq_preimage_closedBall | null |
ball_bot {r : ℝ} (x : E) (hr : 0 < r) : ball (⊥ : Seminorm 𝕜 E) x r = Set.univ :=
ball_zero' x hr
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_bot | null |
closedBall_bot {r : ℝ} (x : E) (hr : 0 < r) :
closedBall (⊥ : Seminorm 𝕜 E) x r = Set.univ :=
closedBall_zero' x hr | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_bot | null |
balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
rw [mem_ball_zero, ← hx, map_smul_eq_mul]
calc
_ ≤ p y := mul_le_of_le_one_left (apply_nonneg p _) ha
_ < r := by rwa [mem_ball_zero] at hy | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | balanced_ball_zero | Seminorm-balls at the origin are balanced. |
balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
rw [mem_closedBall_zero, ← hx, map_smul_eq_mul]
calc
_ ≤ p y := mul_le_of_le_one_left (apply_nonneg p _) ha
_ ≤ r := by rwa [mem_closedBall_zero] at hy | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | balanced_closedBall_zero | Closed seminorm-balls at the origin are balanced. |
ball_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 < r) : ball (s.sup p) x r = ⋂ i ∈ s, ball (p i) x r := by
lift r to NNReal using hr.le
simp_rw [ball, iInter_setOf, finset_sup_apply, NNReal.coe_lt_coe,
Finset.sup_lt_iff (show ⊥ < r from hr), ← NNReal.coe_lt_coe, NNReal.coe_mk] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_finset_sup_eq_iInter | null |
closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r := by
lift r to NNReal using hr
simp_rw [closedBall, iInter_setOf, finset_sup_apply, NNReal.coe_le_coe, Finset.sup_le_iff, ←
NNReal.coe_le_coe, NNReal.coe_mk] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_finset_sup_eq_iInter | null |
ball_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf fun i => ball (p i) x r := by
rw [Finset.inf_eq_iInf]
exact ball_finset_sup_eq_iInter _ _ _ hr | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_finset_sup | null |
closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
closedBall (s.sup p) x r = s.inf fun i => closedBall (p i) x r := by
rw [Finset.inf_eq_iInf]
exact closedBall_finset_sup_eq_iInter _ _ _ hr
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_finset_sup | null |
ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ := by
ext
rw [Seminorm.mem_ball, Set.mem_empty_iff_false, iff_false, not_lt]
exact hr.trans (apply_nonneg p _)
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_eq_emptyset | null |
closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ := by
ext
rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false, not_le]
exact hr.trans_le (apply_nonneg _ _) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_eq_emptyset | null |
closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
refine fun a ha b hb ↦ mul_lt_mul' ha hb (apply_nonneg _ _) ?_
exact hr₁.lt_or_gt.resolve_left <| ((norm_nonneg a).trans ha).not_gt | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_smul_ball | null |
ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero, mem_ball_zero_iff,
map_smul_eq_mul]
intro a ha b hb
rw [mul_comm, mul_comm r₁]
refine mul_lt_mul' hb ha (norm_nonneg _) (hr₂.lt_or_gt.resolve_left ?_)
exact ((apply_nonneg p b).trans hb).not_gt | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_smul_closedBall | null |
ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
rcases eq_or_ne r₂ 0 with rfl | hr₂
· simp
· exact (smul_subset_smul_left (ball_subset_closedBall _ _ _)).trans
(ball_smul_closedBall _ _ hr₂) | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ball_smul_ball | null |
closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_closedBall_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
intro a ha b hb
gcongr
exact (norm_nonneg _).trans ha | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | closedBall_smul_closedBall | null |
neg_mem_ball_zero {r : ℝ} {x : E} : -x ∈ ball p 0 r ↔ x ∈ ball p 0 r := by
simp only [mem_ball_zero, map_neg_eq_map] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | neg_mem_ball_zero | null |
neg_mem_closedBall_zero {r : ℝ} {x : E} : -x ∈ closedBall p 0 r ↔ x ∈ closedBall p 0 r := by
simp only [mem_closedBall_zero, map_neg_eq_map]
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | neg_mem_closedBall_zero | null |
neg_ball (p : Seminorm 𝕜 E) (r : ℝ) (x : E) : -ball p x r = ball p (-x) r := by
ext
rw [Set.mem_neg, mem_ball, mem_ball, ← neg_add', sub_neg_eq_add, map_neg_eq_map]
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | neg_ball | null |
neg_closedBall (p : Seminorm 𝕜 E) (r : ℝ) (x : E) :
-closedBall p x r = closedBall p (-x) r := by
ext
rw [Set.mem_neg, mem_closedBall, mem_closedBall, ← neg_add', sub_neg_eq_add, map_neg_eq_map] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | neg_closedBall | null |
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