statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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absorbs.union (hu : absorbs 𝕜 s u) (hv : absorbs 𝕜 s v) : absorbs 𝕜 s (u ∪ v) | begin
obtain ⟨a, ha, hu⟩ := hu,
obtain ⟨b, hb, hv⟩ := hv,
exact ⟨max a b, lt_max_of_lt_left ha,
λ c hc, union_subset (hu _ $ le_of_max_le_left hc) (hv _ $ le_of_max_le_right hc)⟩,
end | lemma | absorbs.union | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"le_of_max_le_left",
"le_of_max_le_right",
"lt_max_of_lt_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs_union : absorbs 𝕜 s (u ∪ v) ↔ absorbs 𝕜 s u ∧ absorbs 𝕜 s v | ⟨λ h, ⟨h.mono_right $ subset_union_left _ _, h.mono_right $ subset_union_right _ _⟩,
λ h, h.1.union h.2⟩ | lemma | absorbs_union | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs_Union_finset {ι : Type*} {t : finset ι} {f : ι → set E} :
absorbs 𝕜 s (⋃ i ∈ t, f i) ↔ ∀ i ∈ t, absorbs 𝕜 s (f i) | begin
classical,
induction t using finset.induction_on with i t ht hi,
{ simp only [finset.not_mem_empty, set.Union_false, set.Union_empty, absorbs_empty,
is_empty.forall_iff, implies_true_iff] },
rw [finset.set_bUnion_insert, absorbs_union, hi],
split; intro h,
{ refine λ _ hi', (finset.mem_insert.mp... | lemma | absorbs_Union_finset | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"absorbs_empty",
"absorbs_union",
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.not_mem_empty",
"finset.set_bUnion_insert",
"is_empty.forall_iff",
"set.Union_empty",
"set.Union_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.absorbs_Union {ι : Type*} {s : set E} {t : set ι} {f : ι → set E} (hi : t.finite) :
absorbs 𝕜 s (⋃ i ∈ t, f i) ↔ ∀ i ∈ t, absorbs 𝕜 s (f i) | begin
lift t to finset ι using hi,
simp only [finset.mem_coe],
exact absorbs_Union_finset,
end | lemma | set.finite.absorbs_Union | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"absorbs_Union_finset",
"finset",
"finset.mem_coe",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent (A : set E) | ∀ x, ∃ r, 0 < r ∧ ∀ a : 𝕜, r ≤ ‖a‖ → x ∈ a • A | def | absorbent | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [] | A set is absorbent if it absorbs every singleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absorbent.subset (hA : absorbent 𝕜 A) (hAB : A ⊆ B) : absorbent 𝕜 B | begin
refine forall_imp (λ x, _) hA,
exact Exists.imp (λ r, and.imp_right $ forall₂_imp $ λ a ha hx, set.smul_set_mono hAB hx),
end | lemma | absorbent.subset | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"Exists.imp",
"absorbent",
"and.imp_right",
"forall_imp",
"forall₂_imp",
"set.smul_set_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent_iff_forall_absorbs_singleton : absorbent 𝕜 A ↔ ∀ x, absorbs 𝕜 A {x} | by simp_rw [absorbs, absorbent, singleton_subset_iff] | lemma | absorbent_iff_forall_absorbs_singleton | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent",
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent.absorbs (hs : absorbent 𝕜 s) {x : E} : absorbs 𝕜 s {x} | absorbent_iff_forall_absorbs_singleton.1 hs _ | lemma | absorbent.absorbs | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent",
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent_iff_nonneg_lt : absorbent 𝕜 A ↔ ∀ x, ∃ r, 0 ≤ r ∧ ∀ ⦃a : 𝕜⦄, r < ‖a‖ → x ∈ a • A | forall_congr $ λ x, ⟨λ ⟨r, hr, hx⟩, ⟨r, hr.le, λ a ha, hx a ha.le⟩, λ ⟨r, hr, hx⟩,
⟨r + 1, add_pos_of_nonneg_of_pos hr zero_lt_one,
λ a ha, hx ((lt_add_of_pos_right r zero_lt_one).trans_le ha)⟩⟩ | lemma | absorbent_iff_nonneg_lt | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent.absorbs_finite {s : set E} (hs : absorbent 𝕜 s) {v : set E} (hv : v.finite) :
absorbs 𝕜 s v | begin
rw ←set.bUnion_of_singleton v,
exact hv.absorbs_Union.mpr (λ _ _, hs.absorbs),
end | lemma | absorbent.absorbs_finite | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent",
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced (A : set E) | ∀ a : 𝕜, ‖a‖ ≤ 1 → a • A ⊆ A | def | balanced | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [] | A set `A` is balanced if `a • A` is contained in `A` whenever `a` has norm at most `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_iff_smul_mem : balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ‖a‖ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s | forall₂_congr $ λ a ha, smul_set_subset_iff | lemma | balanced_iff_smul_mem | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_empty : balanced 𝕜 (∅ : set E) | λ _ _, by { rw smul_set_empty } | lemma | balanced_empty | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_univ : balanced 𝕜 (univ : set E) | λ a ha, subset_univ _ | lemma | balanced_univ | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.union (hA : balanced 𝕜 A) (hB : balanced 𝕜 B) : balanced 𝕜 (A ∪ B) | λ a ha, smul_set_union.subset.trans $ union_subset_union (hA _ ha) $ hB _ ha | lemma | balanced.union | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.inter (hA : balanced 𝕜 A) (hB : balanced 𝕜 B) : balanced 𝕜 (A ∩ B) | λ a ha, smul_set_inter_subset.trans $ inter_subset_inter (hA _ ha) $ hB _ ha | lemma | balanced.inter | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_Union {f : ι → set E} (h : ∀ i, balanced 𝕜 (f i)) : balanced 𝕜 (⋃ i, f i) | λ a ha, (smul_set_Union _ _).subset.trans $ Union_mono $ λ _, h _ _ ha | lemma | balanced_Union | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_Union₂ {f : Π i, κ i → set E} (h : ∀ i j, balanced 𝕜 (f i j)) :
balanced 𝕜 (⋃ i j, f i j) | balanced_Union $ λ _, balanced_Union $ h _ | lemma | balanced_Union₂ | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"balanced_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_Inter {f : ι → set E} (h : ∀ i, balanced 𝕜 (f i)) : balanced 𝕜 (⋂ i, f i) | λ a ha, (smul_set_Inter_subset _ _).trans $ Inter_mono $ λ _, h _ _ ha | lemma | balanced_Inter | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_Inter₂ {f : Π i, κ i → set E} (h : ∀ i j, balanced 𝕜 (f i j)) :
balanced 𝕜 (⋂ i j, f i j) | balanced_Inter $ λ _, balanced_Inter $ h _ | lemma | balanced_Inter₂ | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"balanced_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.smul (a : 𝕝) (hs : balanced 𝕜 s) : balanced 𝕜 (a • s) | λ b hb, (smul_comm _ _ _).subset.trans $ smul_set_mono $ hs _ hb | lemma | balanced.smul | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.neg : absorbs 𝕜 s t → absorbs 𝕜 (-s) (-t) | Exists.imp $ λ r, and.imp_right $ forall₂_imp $ λ _ _ h,
(neg_subset_neg.2 h).trans (smul_set_neg _ _).superset | lemma | absorbs.neg | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"Exists.imp",
"absorbs",
"and.imp_right",
"forall₂_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.neg : balanced 𝕜 s → balanced 𝕜 (-s) | forall₂_imp $ λ _ _ h, (smul_set_neg _ _).subset.trans $ neg_subset_neg.2 h | lemma | balanced.neg | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"forall₂_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.add : absorbs 𝕜 s₁ t₁ → absorbs 𝕜 s₂ t₂ → absorbs 𝕜 (s₁ + s₂) (t₁ + t₂) | λ ⟨r₁, hr₁, h₁⟩ ⟨r₂, hr₂, h₂⟩, ⟨max r₁ r₂, lt_max_of_lt_left hr₁, λ a ha, (add_subset_add
(h₁ _ $ le_of_max_le_left ha) $ h₂ _ $ le_of_max_le_right ha).trans (smul_add _ _ _).superset⟩ | lemma | absorbs.add | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"le_of_max_le_left",
"le_of_max_le_right",
"lt_max_of_lt_left",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.add (hs : balanced 𝕜 s) (ht : balanced 𝕜 t) : balanced 𝕜 (s + t) | λ a ha, (smul_add _ _ _).subset.trans $ add_subset_add (hs _ ha) $ ht _ ha | lemma | balanced.add | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.sub (h₁ : absorbs 𝕜 s₁ t₁) (h₂ : absorbs 𝕜 s₂ t₂) : absorbs 𝕜 (s₁ - s₂) (t₁ - t₂) | by { simp_rw sub_eq_add_neg, exact h₁.add h₂.neg } | lemma | absorbs.sub | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.sub (hs : balanced 𝕜 s) (ht : balanced 𝕜 t) : balanced 𝕜 (s - t) | by { simp_rw sub_eq_add_neg, exact hs.add ht.neg } | lemma | balanced.sub | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_zero : balanced 𝕜 (0 : set E) | λ a ha, (smul_zero _).subset | lemma | balanced_zero | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.smul_mono (hs : balanced 𝕝 s) {a : 𝕝} {b : 𝕜} (h : ‖a‖ ≤ ‖b‖) : a • s ⊆ b • s | begin
obtain rfl | hb := eq_or_ne b 0,
{ rw norm_zero at h,
rw norm_eq_zero.1 (h.antisymm $ norm_nonneg _),
obtain rfl | h := s.eq_empty_or_nonempty,
{ simp_rw [smul_set_empty] },
{ simp_rw [zero_smul_set h] } },
rintro _ ⟨x, hx, rfl⟩,
refine ⟨b⁻¹ • a • x, _, smul_inv_smul₀ hb _⟩,
rw ←smul_ass... | lemma | balanced.smul_mono | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"div_le_one_of_le",
"eq_or_ne",
"norm_inv",
"norm_smul",
"smul_inv_smul₀"
] | Scalar multiplication (by possibly different types) of a balanced set is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced.absorbs_self (hA : balanced 𝕜 A) : absorbs 𝕜 A A | begin
refine ⟨1, zero_lt_one, λ a ha x hx, _⟩,
rw mem_smul_set_iff_inv_smul_mem₀ (norm_pos_iff.1 $ zero_lt_one.trans_le ha),
refine hA a⁻¹ _ (smul_mem_smul_set hx),
rw norm_inv,
exact inv_le_one ha,
end | lemma | balanced.absorbs_self | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"balanced",
"inv_le_one",
"norm_inv",
"zero_lt_one"
] | A balanced set absorbs itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced.subset_smul (hA : balanced 𝕜 A) (ha : 1 ≤ ‖a‖) : A ⊆ a • A | begin
refine (subset_set_smul_iff₀ _).2 (hA (a⁻¹) _),
{ rintro rfl,
rw norm_zero at ha,
exact zero_lt_one.not_le ha },
{ rw norm_inv,
exact inv_le_one ha }
end | lemma | balanced.subset_smul | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"inv_le_one",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.smul_eq (hA : balanced 𝕜 A) (ha : ‖a‖ = 1) : a • A = A | (hA _ ha.le).antisymm $ hA.subset_smul ha.ge | lemma | balanced.smul_eq | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.mem_smul_iff (hs : balanced 𝕜 s) (h : ‖a‖ = ‖b‖) : a • x ∈ s ↔ b • x ∈ s | begin
obtain rfl | hb := eq_or_ne b 0,
{ rw [norm_zero, norm_eq_zero] at h,
rw h },
have ha : a ≠ 0 := norm_ne_zero_iff.1 (ne_of_eq_of_ne h $ norm_ne_zero_iff.2 hb),
split; intro h'; [rw ←inv_mul_cancel_right₀ ha b, rw ←inv_mul_cancel_right₀ hb a];
{ rw [←smul_eq_mul, smul_assoc],
refine hs.smul_mem _... | lemma | balanced.mem_smul_iff | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"eq_or_ne",
"norm_eq_zero",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced.neg_mem_iff (hs : balanced 𝕜 s) : -x ∈ s ↔ x ∈ s | by convert hs.mem_smul_iff (norm_neg 1); simp only [neg_smul, one_smul] | lemma | balanced.neg_mem_iff | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"neg_smul",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs.inter (hs : absorbs 𝕜 s u) (ht : absorbs 𝕜 t u) : absorbs 𝕜 (s ∩ t) u | begin
obtain ⟨a, ha, hs⟩ := hs,
obtain ⟨b, hb, ht⟩ := ht,
have h : 0 < max a b := lt_max_of_lt_left ha,
refine ⟨max a b, lt_max_of_lt_left ha, λ c hc, _⟩,
rw smul_set_inter₀ (norm_pos_iff.1 $ h.trans_le hc),
exact subset_inter (hs _ $ le_of_max_le_left hc) (ht _ $ le_of_max_le_right hc),
end | lemma | absorbs.inter | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"le_of_max_le_left",
"le_of_max_le_right",
"lt_max_of_lt_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs_inter : absorbs 𝕜 (s ∩ t) u ↔ absorbs 𝕜 s u ∧ absorbs 𝕜 t u | ⟨λ h, ⟨h.mono_left $ inter_subset_left _ _, h.mono_left $ inter_subset_right _ _⟩,
λ h, h.1.inter h.2⟩ | lemma | absorbs_inter | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent_univ : absorbent 𝕜 (univ : set E) | begin
refine λ x, ⟨1, zero_lt_one, λ a ha, _⟩,
rw smul_set_univ₀ (norm_pos_iff.1 $ zero_lt_one.trans_le ha),
exact trivial,
end | lemma | absorbent_univ | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent_nhds_zero (hA : A ∈ 𝓝 (0 : E)) : absorbent 𝕜 A | begin
intro x,
obtain ⟨w, hw₁, hw₂, hw₃⟩ := mem_nhds_iff.mp hA,
have hc : continuous (λ t : 𝕜, t • x) := continuous_id.smul continuous_const,
obtain ⟨r, hr₁, hr₂⟩ := metric.is_open_iff.mp (hw₂.preimage hc) 0
(by rwa [mem_preimage, zero_smul]),
have hr₃ := inv_pos.mpr (half_pos hr₁),
refine ⟨(r / 2)⁻¹, ... | lemma | absorbent_nhds_zero | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent",
"continuous",
"continuous_const",
"half_pos",
"inv_le",
"metric.mem_ball",
"norm_inv",
"zero_smul"
] | Every neighbourhood of the origin is absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_zero_union_interior (hA : balanced 𝕜 A) : balanced 𝕜 ((0 : set E) ∪ interior A) | begin
intros a ha,
obtain rfl | h := eq_or_ne a 0,
{ rw zero_smul_set,
exacts [subset_union_left _ _, ⟨0, or.inl rfl⟩] },
{ rw [←image_smul, image_union],
apply union_subset_union,
{ rw [image_zero, smul_zero],
refl },
{ calc a • interior A ⊆ interior (a • A) : (is_open_map_smul₀ h).image_... | lemma | balanced_zero_union_interior | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"eq_or_ne",
"interior",
"interior_mono",
"is_open_map_smul₀",
"smul_zero"
] | The union of `{0}` with the interior of a balanced set is balanced. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced.interior (hA : balanced 𝕜 A) (h : (0 : E) ∈ interior A) : balanced 𝕜 (interior A) | begin
rw ←union_eq_self_of_subset_left (singleton_subset_iff.2 h),
exact balanced_zero_union_interior hA,
end | lemma | balanced.interior | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"balanced_zero_union_interior",
"interior"
] | The interior of a balanced set is balanced if it contains the origin. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced.closure (hA : balanced 𝕜 A) : balanced 𝕜 (closure A) | λ a ha,
(image_closure_subset_closure_image $ continuous_id.const_smul _).trans $ closure_mono $ hA _ ha | lemma | balanced.closure | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"closure",
"closure_mono",
"image_closure_subset_closure_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbs_zero_iff : absorbs 𝕜 s 0 ↔ (0 : E) ∈ s | begin
refine ⟨_, λ h, ⟨1, zero_lt_one, λ a _, zero_subset.2 $ zero_mem_smul_set h⟩⟩,
rintro ⟨r, hr, h⟩,
obtain ⟨a, ha⟩ := normed_space.exists_lt_norm 𝕜 𝕜 r,
have := h _ ha.le,
rwa [zero_subset, zero_mem_smul_set_iff] at this,
exact norm_ne_zero_iff.1 (hr.trans ha).ne',
end | lemma | absorbs_zero_iff | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbs",
"normed_space.exists_lt_norm",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent.zero_mem (hs : absorbent 𝕜 s) : (0 : E) ∈ s | absorbs_zero_iff.1 $ absorbent_iff_forall_absorbs_singleton.1 hs _ | lemma | absorbent.zero_mem | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"absorbent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_convex_hull_of_balanced (hs : balanced 𝕜 s) : balanced 𝕜 (convex_hull ℝ s) | begin
suffices : convex ℝ {x | ∀ a : 𝕜, ‖a‖ ≤ 1 → a • x ∈ convex_hull ℝ s},
{ rw balanced_iff_smul_mem at hs ⊢,
refine λ a ha x hx, convex_hull_min _ this hx a ha,
exact λ y hy a ha, subset_convex_hull ℝ s (hs ha hy) },
intros x hx y hy u v hu hv huv a ha,
simp only [smul_add, ← smul_comm],
exact con... | lemma | balanced_convex_hull_of_balanced | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"balanced",
"balanced_iff_smul_mem",
"convex",
"convex_convex_hull",
"convex_hull",
"convex_hull_min",
"smul_add",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_iff_neg_mem (hs : convex ℝ s) : balanced ℝ s ↔ ∀ ⦃x⦄, x ∈ s → -x ∈ s | begin
refine ⟨λ h x, h.neg_mem_iff.2, λ h a ha, smul_set_subset_iff.2 $ λ x hx, _⟩,
rw [real.norm_eq_abs, abs_le] at ha,
rw [show a = -((1 - a) / 2) + (a - -1)/2, by ring, add_smul, neg_smul, ←smul_neg],
exact hs (h hx) hx (div_nonneg (sub_nonneg_of_le ha.2) zero_le_two)
(div_nonneg (sub_nonneg_of_le ha.1) ... | lemma | balanced_iff_neg_mem | analysis.locally_convex | src/analysis/locally_convex/basic.lean | [
"analysis.convex.basic",
"analysis.convex.hull",
"analysis.normed_space.basic"
] | [
"abs_le",
"add_smul",
"balanced",
"convex",
"div_nonneg",
"neg_smul",
"real.norm_eq_abs",
"ring",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded (s : set E) : Prop | ∀ ⦃V⦄, V ∈ 𝓝 (0 : E) → absorbs 𝕜 V s | def | bornology.is_vonN_bounded | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbs"
] | A set `s` is von Neumann bounded if every neighborhood of 0 absorbs `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded_empty : is_vonN_bounded 𝕜 (∅ : set E) | λ _ _, absorbs_empty | lemma | bornology.is_vonN_bounded_empty | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbs_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_iff (s : set E) : is_vonN_bounded 𝕜 s ↔ ∀ V ∈ 𝓝 (0 : E), absorbs 𝕜 V s | iff.rfl | lemma | bornology.is_vonN_bounded_iff | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.has_basis.is_vonN_bounded_basis_iff {q : ι → Prop} {s : ι → set E} {A : set E}
(h : (𝓝 (0 : E)).has_basis q s) :
is_vonN_bounded 𝕜 A ↔ ∀ i (hi : q i), absorbs 𝕜 (s i) A | begin
refine ⟨λ hA i hi, hA (h.mem_of_mem hi), λ hA V hV, _⟩,
rcases h.mem_iff.mp hV with ⟨i, hi, hV⟩,
exact (hA i hi).mono_left hV,
end | lemma | filter.has_basis.is_vonN_bounded_basis_iff | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded.subset {s₁ s₂ : set E} (h : s₁ ⊆ s₂) (hs₂ : is_vonN_bounded 𝕜 s₂) :
is_vonN_bounded 𝕜 s₁ | λ V hV, (hs₂ hV).mono_right h | lemma | bornology.is_vonN_bounded.subset | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [] | Subsets of bounded sets are bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded.union {s₁ s₂ : set E} (hs₁ : is_vonN_bounded 𝕜 s₁)
(hs₂ : is_vonN_bounded 𝕜 s₂) :
is_vonN_bounded 𝕜 (s₁ ∪ s₂) | λ V hV, (hs₁ hV).union (hs₂ hV) | lemma | bornology.is_vonN_bounded.union | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [] | The union of two bounded sets is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded.of_topological_space_le {t t' : topological_space E} (h : t ≤ t') {s : set E}
(hs : @is_vonN_bounded 𝕜 E _ _ _ t s) : @is_vonN_bounded 𝕜 E _ _ _ t' s | λ V hV, hs $ (le_iff_nhds t t').mp h 0 hV | lemma | bornology.is_vonN_bounded.of_topological_space_le | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"le_iff_nhds",
"topological_space"
] | 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'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded.image {σ : 𝕜₁ →+* 𝕜₂} [ring_hom_surjective σ] [ring_hom_isometric σ]
{s : set E} (hs : is_vonN_bounded 𝕜₁ s) (f : E →SL[σ] F) :
is_vonN_bounded 𝕜₂ (f '' s) | begin
let σ' := ring_equiv.of_bijective σ ⟨σ.injective, σ.is_surjective⟩,
have σ_iso : isometry σ := add_monoid_hom_class.isometry_of_norm σ
(λ x, ring_hom_isometric.is_iso),
have σ'_symm_iso : isometry σ'.symm := σ_iso.right_inv σ'.right_inv,
have f_tendsto_zero := f.continuous.tendsto 0,
rw map_zero at ... | lemma | bornology.is_vonN_bounded.image | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"isometry",
"map_ne_zero",
"preimage_smul_setₛₗ",
"ring_equiv.of_bijective",
"ring_hom_isometric",
"ring_hom_surjective",
"set.image_subset_iff"
] | A continuous linear image of a bounded set is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded.smul_tendsto_zero {S : set E} {ε : ι → 𝕜} {x : ι → E} {l : filter ι}
(hS : is_vonN_bounded 𝕜 S) (hxS : ∀ᶠ n in l, x n ∈ S) (hε : tendsto ε l (𝓝 0)) :
tendsto (ε • x) l (𝓝 0) | begin
rw tendsto_def at *,
intros V hV,
rcases hS hV with ⟨r, r_pos, hrS⟩,
filter_upwards [hxS, hε _ (metric.ball_mem_nhds 0 $ inv_pos.mpr r_pos)] with n hnS hnr,
by_cases this : ε n = 0,
{ simp [this, mem_of_mem_nhds hV] },
{ rw [mem_preimage, mem_ball_zero_iff, lt_inv (norm_pos_iff.mpr this) r_pos, ← no... | lemma | bornology.is_vonN_bounded.smul_tendsto_zero | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"filter",
"lt_inv",
"mem_of_mem_nhds",
"metric.ball_mem_nhds",
"norm_inv",
"pi.smul_apply'",
"set.mem_inv_smul_set_iff₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_of_smul_tendsto_zero {ε : ι → 𝕝} {l : filter ι} [l.ne_bot]
(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)) :
is_vonN_bounded 𝕝 S | begin
rw (nhds_basis_balanced 𝕝 E).is_vonN_bounded_basis_iff,
by_contra' H',
rcases H' with ⟨V, ⟨hV, hVb⟩, hVS⟩,
have : ∀ᶠ n in l, ∃ x : S, (ε n) • (x : E) ∉ V,
{ filter_upwards [hε] with n hn,
rw absorbs at hVS,
push_neg at hVS,
rcases hVS _ (norm_pos_iff.mpr $ inv_ne_zero hn) with ⟨a, haε, haS⟩... | lemma | bornology.is_vonN_bounded_of_smul_tendsto_zero | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbs",
"filter",
"filter.eventually.frequently",
"filter.frequently_false",
"inv_ne_zero",
"nhds_basis_balanced",
"set.mem_inv_smul_set_iff₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_iff_smul_tendsto_zero {ε : ι → 𝕝} {l : filter ι} [l.ne_bot]
(hε : tendsto ε l (𝓝[≠] 0)) {S : set E} :
is_vonN_bounded 𝕝 S ↔ ∀ x : ι → E, (∀ n, x n ∈ S) → tendsto (ε • x) l (𝓝 0) | ⟨λ hS x hxS, hS.smul_tendsto_zero (eventually_of_forall hxS) (le_trans hε nhds_within_le_nhds),
is_vonN_bounded_of_smul_tendsto_zero (hε self_mem_nhds_within)⟩ | lemma | bornology.is_vonN_bounded_iff_smul_tendsto_zero | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"filter",
"nhds_within_le_nhds",
"self_mem_nhds_within"
] | 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 bound... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded_singleton (x : E) : is_vonN_bounded 𝕜 ({x} : set E) | λ V hV, (absorbent_nhds_zero hV).absorbs | lemma | bornology.is_vonN_bounded_singleton | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbent_nhds_zero",
"absorbs"
] | Singletons are bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_vonN_bounded_covers : ⋃₀ (set_of (is_vonN_bounded 𝕜)) = (set.univ : set E) | set.eq_univ_iff_forall.mpr (λ x, set.mem_sUnion.mpr
⟨{x}, is_vonN_bounded_singleton _, set.mem_singleton _⟩) | lemma | bornology.is_vonN_bounded_covers | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"set.mem_singleton"
] | The union of all bounded set is the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vonN_bornology : bornology E | bornology.of_bounded (set_of (is_vonN_bounded 𝕜)) (is_vonN_bounded_empty 𝕜 E)
(λ _ hs _ ht, hs.subset ht) (λ _ hs _, hs.union) is_vonN_bounded_singleton | def | bornology.vonN_bornology | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology",
"bornology.of_bounded"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_iff_is_vonN_bounded {s : set E} :
@is_bounded _ (vonN_bornology 𝕜 E) s ↔ is_vonN_bounded 𝕜 s | is_bounded_of_bounded_iff _ | lemma | bornology.is_bounded_iff_is_vonN_bounded | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded.is_vonN_bounded {s : set E} (hs : totally_bounded s) :
bornology.is_vonN_bounded 𝕜 s | begin
rw totally_bounded_iff_subset_finite_Union_nhds_zero at hs,
intros U hU,
have h : filter.tendsto (λ (x : E × E), x.fst + x.snd) (𝓝 (0,0)) (𝓝 ((0 : E) + (0 : E))) :=
tendsto_add,
rw add_zero at h,
have h' := (nhds_basis_balanced 𝕜 E).prod (nhds_basis_balanced 𝕜 E),
simp_rw [←nhds_prod_eq, id.de... | lemma | totally_bounded.is_vonN_bounded | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"absorbent_nhds_zero",
"absorbs.add",
"absorbs.mono_left",
"absorbs.mono_right",
"bornology.is_vonN_bounded",
"filter.tendsto",
"nhds_basis_balanced",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_ball (r : ℝ) :
bornology.is_vonN_bounded 𝕜 (metric.ball (0 : E) r) | begin
rw [metric.nhds_basis_ball.is_vonN_bounded_basis_iff, ← ball_norm_seminorm 𝕜 E],
exact λ ε hε, (norm_seminorm 𝕜 E).ball_zero_absorbs_ball_zero hε
end | lemma | normed_space.is_vonN_bounded_ball | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"ball_norm_seminorm",
"bornology.is_vonN_bounded",
"metric.ball",
"norm_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_closed_ball (r : ℝ) :
bornology.is_vonN_bounded 𝕜 (metric.closed_ball (0 : E) r) | (is_vonN_bounded_ball 𝕜 E (r+1)).subset (metric.closed_ball_subset_ball $ by linarith) | lemma | normed_space.is_vonN_bounded_closed_ball | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology.is_vonN_bounded",
"metric.closed_ball",
"metric.closed_ball_subset_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_iff (s : set E) :
bornology.is_vonN_bounded 𝕜 s ↔ bornology.is_bounded s | begin
rw [← metric.bounded_iff_is_bounded, metric.bounded_iff_subset_ball (0 : E)],
split,
{ intros h,
rcases h (metric.ball_mem_nhds 0 zero_lt_one) with ⟨ρ, hρ, hρball⟩,
rcases normed_field.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩,
specialize hρball a ha.le,
rw [← ball_norm_seminorm 𝕜 E, seminorm.smul_b... | lemma | normed_space.is_vonN_bounded_iff | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"ball_norm_seminorm",
"bornology.is_bounded",
"bornology.is_vonN_bounded",
"metric.ball_mem_nhds",
"metric.bounded_iff_is_bounded",
"metric.bounded_iff_subset_ball",
"mul_one",
"normed_field.exists_lt_norm",
"seminorm.smul_ball_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_vonN_bounded_iff' (s : set E) :
bornology.is_vonN_bounded 𝕜 s ↔ ∃ r : ℝ, ∀ (x : E) (hx : x ∈ s), ‖x‖ ≤ r | by rw [normed_space.is_vonN_bounded_iff, ←metric.bounded_iff_is_bounded, bounded_iff_forall_norm_le] | lemma | normed_space.is_vonN_bounded_iff' | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology.is_vonN_bounded",
"normed_space.is_vonN_bounded_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_is_vonN_bounded_iff (f : E' → E) (s : set E') :
bornology.is_vonN_bounded 𝕜 (f '' s) ↔ ∃ r : ℝ, ∀ (x : E') (hx : x ∈ s), ‖f x‖ ≤ r | by simp_rw [is_vonN_bounded_iff', set.ball_image_iff] | lemma | normed_space.image_is_vonN_bounded_iff | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology.is_vonN_bounded",
"set.ball_image_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vonN_bornology_eq : bornology.vonN_bornology 𝕜 E = pseudo_metric_space.to_bornology | begin
rw bornology.ext_iff_is_bounded,
intro s,
rw bornology.is_bounded_iff_is_vonN_bounded,
exact is_vonN_bounded_iff 𝕜 E s
end | lemma | normed_space.vonN_bornology_eq | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology.ext_iff_is_bounded",
"bornology.is_bounded_iff_is_vonN_bounded",
"bornology.vonN_bornology"
] | In a normed space, the von Neumann bornology (`bornology.vonN_bornology`) is equal to the
metric bornology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_iff_subset_smul_ball {s : set E} :
bornology.is_bounded s ↔ ∃ a : 𝕜, s ⊆ a • metric.ball 0 1 | begin
rw ← is_vonN_bounded_iff 𝕜,
split,
{ intros h,
rcases h (metric.ball_mem_nhds 0 zero_lt_one) with ⟨ρ, hρ, hρball⟩,
rcases normed_field.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩,
exact ⟨a, hρball a ha.le⟩ },
{ rintros ⟨a, ha⟩,
exact ((is_vonN_bounded_ball 𝕜 E 1).image (a • 1 : E →L[𝕜] E)).subset ... | lemma | normed_space.is_bounded_iff_subset_smul_ball | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology.is_bounded",
"metric.ball",
"metric.ball_mem_nhds",
"normed_field.exists_lt_norm",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_iff_subset_smul_closed_ball {s : set E} :
bornology.is_bounded s ↔ ∃ a : 𝕜, s ⊆ a • metric.closed_ball 0 1 | begin
split,
{ rw is_bounded_iff_subset_smul_ball 𝕜,
exact exists_imp_exists
(λ a ha, ha.trans $ set.smul_set_mono $ metric.ball_subset_closed_ball) },
{ rw ← is_vonN_bounded_iff 𝕜,
rintros ⟨a, ha⟩,
exact ((is_vonN_bounded_closed_ball 𝕜 E 1).image (a • 1 : E →L[𝕜] E)).subset ha }
end | lemma | normed_space.is_bounded_iff_subset_smul_closed_ball | analysis.locally_convex | src/analysis/locally_convex/bounded.lean | [
"analysis.locally_convex.basic",
"analysis.locally_convex.balanced_core_hull",
"analysis.seminorm",
"topology.bornology.basic",
"topology.algebra.uniform_group",
"topology.uniform_space.cauchy"
] | [
"bornology.is_bounded",
"metric.ball_subset_closed_ball",
"metric.closed_ball",
"set.smul_set_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.clm_of_exists_bounded_image (f : E →ₗ[𝕜] F)
(h : ∃ (V : set E) (hV : V ∈ 𝓝 (0 : E)), bornology.is_vonN_bounded 𝕜 (f '' V)) : E →L[𝕜] F | ⟨f, begin
-- It suffices to show that `f` is continuous at `0`.
refine continuous_of_continuous_at_zero f _,
rw [continuous_at_def, f.map_zero],
intros U hU,
-- Continuity means that `U ∈ 𝓝 0` implies that `f ⁻¹' U ∈ 𝓝 0`.
rcases h with ⟨V, hV, h⟩,
rcases h hU with ⟨r, hr, h⟩,
rcases normed_field.exis... | def | linear_map.clm_of_exists_bounded_image | analysis.locally_convex | src/analysis/locally_convex/continuous_of_bounded.lean | [
"analysis.locally_convex.bounded",
"data.is_R_or_C.basic"
] | [
"bornology.is_vonN_bounded",
"continuous_at_def",
"inv_ne_zero",
"inv_smul_smul₀",
"linear_map.map_smul",
"normed_field.exists_lt_norm",
"set.mem_inv_smul_set_iff₀",
"set.mem_preimage",
"set.preimage_mono",
"set.smul_set_mono",
"set.subset_preimage_image",
"set_smul_mem_nhds_smul",
"smul_zer... | Construct a continuous linear map from a linear map `f : E →ₗ[𝕜] F` and the existence of a
neighborhood of zero that gets mapped into a bounded set in `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.clm_of_exists_bounded_image_coe {f : E →ₗ[𝕜] F}
{h : ∃ (V : set E) (hV : V ∈ 𝓝 (0 : E)), bornology.is_vonN_bounded 𝕜 (f '' V)} :
(f.clm_of_exists_bounded_image h : E →ₗ[𝕜] F) = f | rfl | lemma | linear_map.clm_of_exists_bounded_image_coe | analysis.locally_convex | src/analysis/locally_convex/continuous_of_bounded.lean | [
"analysis.locally_convex.bounded",
"data.is_R_or_C.basic"
] | [
"bornology.is_vonN_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.clm_of_exists_bounded_image_apply {f : E →ₗ[𝕜] F}
{h : ∃ (V : set E) (hV : V ∈ 𝓝 (0 : E)), bornology.is_vonN_bounded 𝕜 (f '' V)} {x : E} :
f.clm_of_exists_bounded_image h x = f x | rfl | lemma | linear_map.clm_of_exists_bounded_image_apply | analysis.locally_convex | src/analysis/locally_convex/continuous_of_bounded.lean | [
"analysis.locally_convex.bounded",
"data.is_R_or_C.basic"
] | [
"bornology.is_vonN_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.continuous_at_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ (s : set E) (hs : is_vonN_bounded 𝕜 s), is_vonN_bounded 𝕜' (f '' s)) :
continuous_at f 0 | begin
-- Assume that f is not continuous at 0
by_contradiction,
-- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
-- and reformulate non-continuity in terms of these bases
rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩,
simp only [id.def] at bE,
h... | lemma | linear_map.continuous_at_zero_of_locally_bounded | analysis.locally_convex | src/analysis/locally_convex/continuous_of_bounded.lean | [
"analysis.locally_convex.bounded",
"data.is_R_or_C.basic"
] | [
"by_contradiction",
"continuous_at",
"exists_nat_gt",
"forall_true_left",
"inv_le",
"inv_smul_smul₀",
"is_R_or_C.norm_nat_cast",
"linear_map.map_smulₛₗ",
"map_nat_cast",
"mem_of_mem_nhds",
"mul_inv_cancel",
"nat.cast_add",
"nat.cast_eq_zero",
"nat.cast_one",
"nat.cast_zero",
"nhds_basi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.continuous_of_locally_bounded [uniform_add_group F] (f : E →ₛₗ[σ] F)
(hf : ∀ (s : set E) (hs : is_vonN_bounded 𝕜 s), is_vonN_bounded 𝕜' (f '' s)) :
continuous f | (uniform_continuous_of_continuous_at_zero f $ f.continuous_at_zero_of_locally_bounded hf).continuous | lemma | linear_map.continuous_of_locally_bounded | analysis.locally_convex | src/analysis/locally_convex/continuous_of_bounded.lean | [
"analysis.locally_convex.bounded",
"data.is_R_or_C.basic"
] | [
"continuous",
"uniform_add_group"
] | If `E` is first countable, then every locally bounded linear map `E →ₛₗ[σ] F` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar (s : set E) : set F | {y : F | ∀ x ∈ s, ‖B x y‖ ≤ 1 } | def | linear_map.polar | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | The (absolute) polar of `s : set E` is given by the set of all `y : F` such that `‖B x y‖ ≤ 1`
for all `x ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar_mem_iff (s : set E) (y : F) :
y ∈ B.polar s ↔ ∀ x ∈ s, ‖B x y‖ ≤ 1 | iff.rfl | lemma | linear_map.polar_mem_iff | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_mem (s : set E) (y : F) (hy : y ∈ B.polar s) :
∀ x ∈ s, ‖B x y‖ ≤ 1 | hy | lemma | linear_map.polar_mem | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem_polar (s : set E) :
(0 : F) ∈ B.polar s | λ _ _, by simp only [map_zero, norm_zero, zero_le_one] | lemma | linear_map.zero_mem_polar | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_eq_Inter {s : set E} :
B.polar s = ⋂ x ∈ s, {y : F | ‖B x y‖ ≤ 1} | by { ext, simp only [polar_mem_iff, set.mem_Inter, set.mem_set_of_eq] } | lemma | linear_map.polar_eq_Inter | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"set.mem_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_gc : galois_connection (order_dual.to_dual ∘ B.polar)
(B.flip.polar ∘ order_dual.of_dual) | λ s t, ⟨λ h _ hx _ hy, h hy _ hx, λ h _ hx _ hy, h hy _ hx⟩ | lemma | linear_map.polar_gc | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"galois_connection",
"order_dual.of_dual",
"order_dual.to_dual"
] | The map `B.polar : set E → set F` forms an order-reversing Galois connection with
`B.flip.polar : set F → set E`. We use `order_dual.to_dual` and `order_dual.of_dual` to express
that `polar` is order-reversing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar_Union {ι} {s : ι → set E} : B.polar (⋃ i, s i) = ⋂ i, B.polar (s i) | B.polar_gc.l_supr | lemma | linear_map.polar_Union | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_union {s t : set E} : B.polar (s ∪ t) = B.polar s ∩ B.polar t | B.polar_gc.l_sup | lemma | linear_map.polar_union | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_antitone : antitone (B.polar : set E → set F) | B.polar_gc.monotone_l | lemma | linear_map.polar_antitone | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_empty : B.polar ∅ = set.univ | B.polar_gc.l_bot | lemma | linear_map.polar_empty | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_zero : B.polar ({0} : set E) = set.univ | begin
refine set.eq_univ_iff_forall.mpr (λ y x hx, _),
rw [set.mem_singleton_iff.mp hx, map_zero, linear_map.zero_apply, norm_zero],
exact zero_le_one,
end | lemma | linear_map.polar_zero | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"linear_map.zero_apply",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_bipolar (s : set E) : s ⊆ B.flip.polar (B.polar s) | λ x hx y hy, by { rw B.flip_apply, exact hy x hx } | lemma | linear_map.subset_bipolar | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tripolar_eq_polar (s : set E) : B.polar (B.flip.polar (B.polar s)) = B.polar s | begin
refine (B.polar_antitone (B.subset_bipolar s)).antisymm _,
convert subset_bipolar B.flip (B.polar s),
exact B.flip_flip.symm,
end | lemma | linear_map.tripolar_eq_polar | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_weak_closed (s : set E) :
is_closed[weak_bilin.topological_space B.flip] (B.polar s) | begin
rw polar_eq_Inter,
refine is_closed_Inter (λ x, is_closed_Inter (λ _, _)),
exact is_closed_le (weak_bilin.eval_continuous B.flip x).norm continuous_const,
end | lemma | linear_map.polar_weak_closed | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"continuous_const",
"is_closed",
"is_closed_Inter",
"is_closed_le",
"weak_bilin.eval_continuous"
] | The polar set is closed in the weak topology induced by `B.flip`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar_univ (h : separating_right B) :
B.polar set.univ = {(0 : F)} | begin
rw set.eq_singleton_iff_unique_mem,
refine ⟨by simp only [zero_mem_polar], λ y hy, h _ (λ x, _)⟩,
refine norm_le_zero_iff.mp (le_of_forall_le_of_dense $ λ ε hε, _),
rcases normed_field.exists_norm_lt 𝕜 hε with ⟨c, hc, hcε⟩,
calc ‖B x y‖ = ‖c‖ * ‖B (c⁻¹ • x) y‖ :
by rw [B.map_smul, linear_map.smul_a... | lemma | linear_map.polar_univ | analysis.locally_convex | src/analysis/locally_convex/polar.lean | [
"analysis.normed.field.basic",
"linear_algebra.sesquilinear_form",
"topology.algebra.module.weak_dual"
] | [
"algebra.id.smul_eq_mul",
"le_of_forall_le_of_dense",
"linear_map.smul_apply",
"mul_inv_cancel_left₀",
"mul_le_mul",
"mul_one",
"norm_inv",
"norm_mul",
"normed_field.exists_norm_lt",
"set.eq_singleton_iff_unique_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.locally_convex_space (𝔖 : set (set E)) (h𝔖₁ : 𝔖.nonempty)
(h𝔖₂ : directed_on (⊆) 𝔖) :
@locally_convex_space R (E →SL[σ] F) _ _ _ (strong_topology σ F 𝔖) | begin
letI : topological_space (E →SL[σ] F) := strong_topology σ F 𝔖,
haveI : topological_add_group (E →SL[σ] F) := strong_topology.topological_add_group _ _ _,
refine locally_convex_space.of_basis_zero _ _ _ _
(strong_topology.has_basis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂
(locally_convex_space.convex_b... | lemma | continuous_linear_map.strong_topology.locally_convex_space | analysis.locally_convex | src/analysis/locally_convex/strong_topology.lean | [
"topology.algebra.module.strong_topology",
"topology.algebra.module.locally_convex"
] | [
"directed_on",
"locally_convex_space",
"locally_convex_space.convex_basis_zero",
"locally_convex_space.of_basis_zero",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminorm (f : E →ₗ[𝕜] 𝕜) : seminorm 𝕜 E | (norm_seminorm 𝕜 𝕜).comp f | def | linear_map.to_seminorm | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [
"norm_seminorm",
"seminorm"
] | Construct a seminorm from a linear form `f : E →ₗ[𝕜] 𝕜` over a normed field `𝕜` by
`λ x, ‖f x‖` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_seminorm {f : E →ₗ[𝕜] 𝕜} :
⇑f.to_seminorm = λ x, ‖f x‖ | rfl | lemma | linear_map.coe_to_seminorm | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminorm_apply {f : E →ₗ[𝕜] 𝕜} {x : E} :
f.to_seminorm x = ‖f x‖ | rfl | lemma | linear_map.to_seminorm_apply | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminorm_ball_zero {f : E →ₗ[𝕜] 𝕜} {r : ℝ} :
seminorm.ball f.to_seminorm 0 r = { x : E | ‖f x‖ < r} | by simp only [seminorm.ball_zero_eq, to_seminorm_apply] | lemma | linear_map.to_seminorm_ball_zero | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [
"seminorm.ball",
"seminorm.ball_zero_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminorm_comp (f : F →ₗ[𝕜] 𝕜) (g : E →ₗ[𝕜] F) :
f.to_seminorm.comp g = (f.comp g).to_seminorm | by { ext, simp only [seminorm.comp_apply, to_seminorm_apply, coe_comp] } | lemma | linear_map.to_seminorm_comp | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [
"seminorm.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminorm_family (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : seminorm_family 𝕜 E F | λ y, (B.flip y).to_seminorm | def | linear_map.to_seminorm_family | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [
"seminorm_family"
] | Construct a family of seminorms from a bilinear form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_seminorm_family_apply {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} {x y} :
(B.to_seminorm_family y) x = ‖B x y‖ | rfl | lemma | linear_map.to_seminorm_family_apply | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.has_basis_weak_bilin (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
(𝓝 (0 : weak_bilin B)).has_basis B.to_seminorm_family.basis_sets id | begin
let p := B.to_seminorm_family,
rw [nhds_induced, nhds_pi],
simp only [map_zero, linear_map.zero_apply],
have h := @metric.nhds_basis_ball 𝕜 _ 0,
have h' := filter.has_basis_pi (λ (i : F), h),
have h'' := filter.has_basis.comap (λ x y, B x y) h',
refine h''.to_has_basis _ _,
{ rintros (U : set F ×... | lemma | linear_map.has_basis_weak_bilin | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [
"filter.has_basis.comap",
"filter.has_basis_pi",
"finset.inf'_le",
"finset.le_sup",
"finset.lt_inf'_iff",
"finset.mem_coe",
"linear_map.to_seminorm_family_apply",
"linear_map.zero_apply",
"metric.nhds_basis_ball",
"nhds_induced",
"nhds_pi",
"seminorm.finset_sup_apply_lt",
"seminorm.mem_ball"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.weak_bilin_with_seminorms (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
with_seminorms (linear_map.to_seminorm_family B : F → seminorm 𝕜 (weak_bilin B)) | seminorm_family.with_seminorms_of_has_basis _ B.has_basis_weak_bilin | lemma | linear_map.weak_bilin_with_seminorms | analysis.locally_convex | src/analysis/locally_convex/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed.field.basic",
"analysis.locally_convex.with_seminorms"
] | [
"linear_map.to_seminorm_family",
"seminorm",
"seminorm_family.with_seminorms_of_has_basis",
"weak_bilin",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_family | ι → seminorm 𝕜 E | abbreviation | seminorm_family | analysis.locally_convex | src/analysis/locally_convex/with_seminorms.lean | [
"analysis.seminorm",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis",
"topology.algebra.module.locally_convex"
] | [
"seminorm"
] | An abbreviation for indexed families of seminorms. This is mainly to allow for dot-notation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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