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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basis_sets (p : seminorm_family 𝕜 E ι) : set (set E) | ⋃ (s : finset ι) r (hr : 0 < r), singleton $ ball (s.sup p) (0 : E) r | def | seminorm_family.basis_sets | 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"
] | [
"finset",
"seminorm_family"
] | The sets of a filter basis for the neighborhood filter of 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_sets_iff {U : set E} :
U ∈ p.basis_sets ↔ ∃ (i : finset ι) r (hr : 0 < r), U = ball (i.sup p) 0 r | by simp only [basis_sets, mem_Union, mem_singleton_iff] | lemma | seminorm_family.basis_sets_iff | 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"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_mem (i : finset ι) {r : ℝ} (hr : 0 < r) :
(i.sup p).ball 0 r ∈ p.basis_sets | (basis_sets_iff _).mpr ⟨i,_,hr,rfl⟩ | lemma | seminorm_family.basis_sets_mem | 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"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_singleton_mem (i : ι) {r : ℝ} (hr : 0 < r) :
(p i).ball 0 r ∈ p.basis_sets | (basis_sets_iff _).mpr ⟨{i},_,hr, by rw finset.sup_singleton⟩ | lemma | seminorm_family.basis_sets_singleton_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_nonempty [nonempty ι] : p.basis_sets.nonempty | begin
let i := classical.arbitrary ι,
refine set.nonempty_def.mpr ⟨(p i).ball 0 1, _⟩,
exact p.basis_sets_singleton_mem i zero_lt_one,
end | lemma | seminorm_family.basis_sets_nonempty | 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"
] | [
"classical.arbitrary",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_intersect
(U V : set E) (hU : U ∈ p.basis_sets) (hV : V ∈ p.basis_sets) :
∃ (z : set E) (H : z ∈ p.basis_sets), z ⊆ U ∩ V | begin
classical,
rcases p.basis_sets_iff.mp hU with ⟨s, r₁, hr₁, hU⟩,
rcases p.basis_sets_iff.mp hV with ⟨t, r₂, hr₂, hV⟩,
use ((s ∪ t).sup p).ball 0 (min r₁ r₂),
refine ⟨p.basis_sets_mem (s ∪ t) (lt_min_iff.mpr ⟨hr₁, hr₂⟩), _⟩,
rw [hU, hV, ball_finset_sup_eq_Inter _ _ _ (lt_min_iff.mpr ⟨hr₁, hr₂⟩),
bal... | lemma | seminorm_family.basis_sets_intersect | 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"
] | [
"ball_mono",
"finset.subset_union_left",
"finset.subset_union_right",
"set.Inter₂_mono'",
"set.subset_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_zero (U) (hU : U ∈ p.basis_sets) :
(0 : E) ∈ U | begin
rcases p.basis_sets_iff.mp hU with ⟨ι', r, hr, hU⟩,
rw [hU, mem_ball_zero, map_zero],
exact hr,
end | lemma | seminorm_family.basis_sets_zero | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_add (U) (hU : U ∈ p.basis_sets) :
∃ (V : set E) (H : V ∈ p.basis_sets), V + V ⊆ U | begin
rcases p.basis_sets_iff.mp hU with ⟨s, r, hr, hU⟩,
use (s.sup p).ball 0 (r/2),
refine ⟨p.basis_sets_mem s (div_pos hr zero_lt_two), _⟩,
refine set.subset.trans (ball_add_ball_subset (s.sup p) (r/2) (r/2) 0 0) _,
rw [hU, add_zero, add_halves'],
end | lemma | seminorm_family.basis_sets_add | 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"
] | [
"add_halves'",
"div_pos",
"set.subset.trans",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_neg (U) (hU' : U ∈ p.basis_sets) :
∃ (V : set E) (H : V ∈ p.basis_sets), V ⊆ (λ (x : E), -x) ⁻¹' U | begin
rcases p.basis_sets_iff.mp hU' with ⟨s, r, hr, hU⟩,
rw [hU, neg_preimage, neg_ball (s.sup p), neg_zero],
exact ⟨U, hU', eq.subset hU⟩,
end | lemma | seminorm_family.basis_sets_neg | 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"
] | [
"eq.subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group_filter_basis [nonempty ι] : add_group_filter_basis E | add_group_filter_basis_of_comm p.basis_sets p.basis_sets_nonempty
p.basis_sets_intersect p.basis_sets_zero p.basis_sets_add p.basis_sets_neg | def | seminorm_family.add_group_filter_basis | 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"
] | [
"add_group_filter_basis"
] | The `add_group_filter_basis` induced by the filter basis `seminorm_basis_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_sets_smul_right (v : E) (U : set E)
(hU : U ∈ p.basis_sets) : ∀ᶠ (x : 𝕜) in 𝓝 0, x • v ∈ U | begin
rcases p.basis_sets_iff.mp hU with ⟨s, r, hr, hU⟩,
rw [hU, filter.eventually_iff],
simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul],
by_cases h : 0 < (s.sup p) v,
{ simp_rw (lt_div_iff h).symm,
rw ←_root_.ball_zero_eq,
exact metric.ball_mem_nhds 0 (div_pos hr h) },
simp_rw [le_antisymm (not_... | lemma | seminorm_family.basis_sets_smul_right | 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"
] | [
"div_pos",
"filter.eventually_iff",
"is_open.mem_nhds",
"is_open_univ",
"lt_div_iff",
"map_nonneg",
"metric.ball_mem_nhds",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_smul (U) (hU : U ∈ p.basis_sets) :
∃ (V : set 𝕜) (H : V ∈ 𝓝 (0 : 𝕜)) (W : set E)
(H : W ∈ p.add_group_filter_basis.sets), V • W ⊆ U | begin
rcases p.basis_sets_iff.mp hU with ⟨s, r, hr, hU⟩,
refine ⟨metric.ball 0 r.sqrt, metric.ball_mem_nhds 0 (real.sqrt_pos.mpr hr), _⟩,
refine ⟨(s.sup p).ball 0 r.sqrt, p.basis_sets_mem s (real.sqrt_pos.mpr hr), _⟩,
refine set.subset.trans (ball_smul_ball (s.sup p) r.sqrt r.sqrt) _,
rw [hU, real.mul_self_sq... | lemma | seminorm_family.basis_sets_smul | 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"
] | [
"metric.ball_mem_nhds",
"real.mul_self_sqrt",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_sets_smul_left (x : 𝕜) (U : set E)
(hU : U ∈ p.basis_sets) : ∃ (V : set E)
(H : V ∈ p.add_group_filter_basis.sets), V ⊆ (λ (y : E), x • y) ⁻¹' U | begin
rcases p.basis_sets_iff.mp hU with ⟨s, r, hr, hU⟩,
rw hU,
by_cases h : x ≠ 0,
{ rw [(s.sup p).smul_ball_preimage 0 r x h, smul_zero],
use (s.sup p).ball 0 (r / ‖x‖),
exact ⟨p.basis_sets_mem s (div_pos hr (norm_pos_iff.mpr h)), subset.rfl⟩ },
refine ⟨(s.sup p).ball 0 r, p.basis_sets_mem s hr, _⟩,... | lemma | seminorm_family.basis_sets_smul_left | 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"
] | [
"div_pos",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_filter_basis : module_filter_basis 𝕜 E | { to_add_group_filter_basis := p.add_group_filter_basis,
smul' := p.basis_sets_smul,
smul_left' := p.basis_sets_smul_left,
smul_right' := p.basis_sets_smul_right } | def | seminorm_family.module_filter_basis | 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"
] | [
"module_filter_basis"
] | The `module_filter_basis` induced by the filter basis `seminorm_basis_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter_eq_infi (p : seminorm_family 𝕜 E ι) :
p.module_filter_basis.to_filter_basis.filter = ⨅ i, (𝓝 0).comap (p i) | begin
refine le_antisymm (le_infi $ λ i, _) _,
{ rw p.module_filter_basis.to_filter_basis.has_basis.le_basis_iff
(metric.nhds_basis_ball.comap _),
intros ε hε,
refine ⟨(p i).ball 0 ε, _, _⟩,
{ rw ← (finset.sup_singleton : _ = p i),
exact p.basis_sets_mem {i} hε, },
{ rw [id, (p i).ball_z... | lemma | seminorm_family.filter_eq_infi | 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"
] | [
"eq.subset",
"filter.mem_infi_of_mem",
"finset.sup_singleton",
"le_infi",
"metric.ball_mem_nhds",
"seminorm.ball_finset_sup_eq_Inter",
"seminorm_family"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded (p : ι → seminorm 𝕜 E) (q : ι' → seminorm 𝕜₂ F) (f : E →ₛₗ[σ₁₂] F) : Prop | ∀ i, ∃ s : finset ι, ∃ C : ℝ≥0, (q i).comp f ≤ C • s.sup p | def | seminorm.is_bounded | 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"
] | [
"finset",
"seminorm"
] | The proposition that a linear map is bounded between spaces with families of seminorms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_const (ι' : Type*) [nonempty ι']
{p : ι → seminorm 𝕜 E} {q : seminorm 𝕜₂ F} (f : E →ₛₗ[σ₁₂] F) :
is_bounded p (λ _ : ι', q) f ↔ ∃ (s : finset ι) C : ℝ≥0, q.comp f ≤ C • s.sup p | by simp only [is_bounded, forall_const] | lemma | seminorm.is_bounded_const | 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"
] | [
"finset",
"forall_const",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_is_bounded (ι : Type*) [nonempty ι]
{p : seminorm 𝕜 E} {q : ι' → seminorm 𝕜₂ F} (f : E →ₛₗ[σ₁₂] F) :
is_bounded (λ _ : ι, p) q f ↔ ∀ i, ∃ C : ℝ≥0, (q i).comp f ≤ C • p | begin
split; intros h i,
{ rcases h i with ⟨s, C, h⟩,
exact ⟨C, le_trans h (smul_le_smul (finset.sup_le (λ _ _, le_rfl)) le_rfl)⟩ },
use [{classical.arbitrary ι}],
simp only [h, finset.sup_singleton],
end | lemma | seminorm.const_is_bounded | 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"
] | [
"classical.arbitrary",
"finset.sup_singleton",
"le_rfl",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_sup {p : ι → seminorm 𝕜 E} {q : ι' → seminorm 𝕜₂ F}
{f : E →ₛₗ[σ₁₂] F} (hf : is_bounded p q f) (s' : finset ι') :
∃ (C : ℝ≥0) (s : finset ι), (s'.sup q).comp f ≤ C • (s.sup p) | begin
classical,
obtain rfl | hs' := s'.eq_empty_or_nonempty,
{ exact ⟨1, ∅, by simp [seminorm.bot_eq_zero]⟩ },
choose fₛ fC hf using hf,
use [s'.card • s'.sup fC, finset.bUnion s' fₛ],
have hs : ∀ i : ι', i ∈ s' → (q i).comp f ≤ s'.sup fC • ((finset.bUnion s' fₛ).sup p) :=
begin
intros i hi,
refi... | lemma | seminorm.is_bounded_sup | 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"
] | [
"finset",
"finset.bUnion",
"finset.le_sup",
"finset.subset_bUnion_of_mem",
"finset.sup_mono",
"le_rfl",
"seminorm",
"seminorm.bot_eq_zero",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms (p : seminorm_family 𝕜 E ι) [t : topological_space E] : Prop | (topology_eq_with_seminorms : t = p.module_filter_basis.topology) | structure | with_seminorms | 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_family",
"topological_space"
] | The proposition that the topology of `E` is induced by a family of seminorms `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.with_seminorms_eq {p : seminorm_family 𝕜 E ι} [t : topological_space E]
(hp : with_seminorms p) : t = p.module_filter_basis.topology | hp.1 | lemma | with_seminorms.with_seminorms_eq | 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_family",
"topological_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.topological_add_group (hp : with_seminorms p) : topological_add_group E | begin
rw hp.with_seminorms_eq,
exact add_group_filter_basis.is_topological_add_group _
end | lemma | with_seminorms.topological_add_group | 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"
] | [
"topological_add_group",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.has_basis (hp : with_seminorms p) : (𝓝 (0 : E)).has_basis
(λ (s : set E), s ∈ p.basis_sets) id | begin
rw (congr_fun (congr_arg (@nhds E) hp.1) 0),
exact add_group_filter_basis.nhds_zero_has_basis _,
end | lemma | with_seminorms.has_basis | 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"
] | [
"nhds",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.has_basis_zero_ball (hp : with_seminorms p) : (𝓝 (0 : E)).has_basis
(λ sr : finset ι × ℝ, 0 < sr.2) (λ sr, (sr.1.sup p).ball 0 sr.2) | begin
refine ⟨λ V, _⟩,
simp only [hp.has_basis.mem_iff, seminorm_family.basis_sets_iff, prod.exists],
split,
{ rintros ⟨-, ⟨s, r, hr, rfl⟩, hV⟩,
exact ⟨s, r, hr, hV⟩ },
{ rintros ⟨s, r, hr, hV⟩,
exact ⟨_, ⟨s, r, hr, rfl⟩, hV⟩ }
end | lemma | with_seminorms.has_basis_zero_ball | 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"
] | [
"finset",
"seminorm_family.basis_sets_iff",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.has_basis_ball (hp : with_seminorms p) {x : E} : (𝓝 (x : E)).has_basis
(λ sr : finset ι × ℝ, 0 < sr.2) (λ sr, (sr.1.sup p).ball x sr.2) | begin
haveI : topological_add_group E := hp.topological_add_group,
rw [← map_add_left_nhds_zero],
convert (hp.has_basis_zero_ball.map ((+) x)),
ext sr : 1,
have : (sr.fst.sup p).ball (x +ᵥ 0) sr.snd = x +ᵥ (sr.fst.sup p).ball 0 sr.snd
:= eq.symm (seminorm.vadd_ball (sr.fst.sup p)),
rwa [vadd_eq_add, add... | lemma | with_seminorms.has_basis_ball | 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"
] | [
"finset",
"seminorm.vadd_ball",
"topological_add_group",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.mem_nhds_iff (hp : with_seminorms p) (x : E) (U : set E) :
U ∈ nhds x ↔ ∃ (s : finset ι) (r > 0), (s.sup p).ball x r ⊆ U | by rw [hp.has_basis_ball.mem_iff, prod.exists] | lemma | with_seminorms.mem_nhds_iff | 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"
] | [
"finset",
"nhds",
"with_seminorms"
] | The `x`-neighbourhoods of a space whose topology is induced by a family of seminorms
are exactly the sets which contain seminorm balls around `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.is_open_iff_mem_balls (hp : with_seminorms p) (U : set E) :
is_open U ↔ ∀ x ∈ U, ∃ (s : finset ι) (r > 0), (s.sup p).ball x r ⊆ U | by simp_rw [←with_seminorms.mem_nhds_iff hp _ U, is_open_iff_mem_nhds] | lemma | with_seminorms.is_open_iff_mem_balls | 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"
] | [
"finset",
"is_open",
"is_open_iff_mem_nhds",
"with_seminorms"
] | The open sets of a space whose topology is induced by a family of seminorms
are exactly the sets which contain seminorm balls around all of their points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.t1_of_separating (hp : with_seminorms p) (h : ∀ x ≠ 0, ∃ i, p i x ≠ 0) :
t1_space E | begin
haveI := hp.topological_add_group,
refine topological_add_group.t1_space _ _,
rw [← is_open_compl_iff, hp.is_open_iff_mem_balls],
rintros x (hx : x ≠ 0),
cases h x hx with i pi_nonzero,
refine ⟨{i}, p i x, by positivity, subset_compl_singleton_iff.mpr _⟩,
rw [finset.sup_singleton, mem_ball, zero_sub... | lemma | with_seminorms.t1_of_separating | 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"
] | [
"finset.sup_singleton",
"is_open_compl_iff",
"t1_space",
"with_seminorms"
] | A separating family of seminorms induces a T₁ topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.separating_of_t1 [t1_space E] (hp : with_seminorms p) (x : E) (hx : x ≠ 0) :
∃ i, p i x ≠ 0 | begin
have := ((t1_space_tfae E).out 0 9).mp infer_instance,
by_contra' h,
refine hx (this _),
rw hp.has_basis_zero_ball.specializes_iff,
rintros ⟨s, r⟩ (hr : 0 < r),
simp only [ball_finset_sup_eq_Inter _ _ _ hr, mem_Inter₂, mem_ball_zero, h, hr, forall_true_iff],
end | lemma | with_seminorms.separating_of_t1 | 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"
] | [
"forall_true_iff",
"t1_space",
"t1_space_tfae",
"with_seminorms"
] | A family of seminorms inducing a T₁ topology is separating. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.separating_iff_t1 (hp : with_seminorms p) :
(∀ x ≠ 0, ∃ i, p i x ≠ 0) ↔ t1_space E | begin
refine ⟨with_seminorms.t1_of_separating hp, _⟩,
introI,
exact with_seminorms.separating_of_t1 hp,
end | lemma | with_seminorms.separating_iff_t1 | 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"
] | [
"t1_space",
"with_seminorms",
"with_seminorms.separating_of_t1"
] | A family of seminorms is separating iff it induces a T₁ topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.tendsto_nhds' (hp : with_seminorms p) (u : F → E) {f : filter F} (y₀ : E) :
filter.tendsto u f (𝓝 y₀) ↔ ∀ (s : finset ι) ε, 0 < ε → ∀ᶠ x in f, s.sup p (u x - y₀) < ε | by simp [hp.has_basis_ball.tendsto_right_iff] | lemma | with_seminorms.tendsto_nhds' | 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"
] | [
"filter",
"filter.tendsto",
"finset",
"with_seminorms"
] | Convergence along filters for `with_seminorms`.
Variant with `finset.sup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.tendsto_nhds (hp : with_seminorms p) (u : F → E) {f : filter F} (y₀ : E) :
filter.tendsto u f (𝓝 y₀) ↔ ∀ i ε, 0 < ε → ∀ᶠ x in f, p i (u x - y₀) < ε | begin
rw hp.tendsto_nhds' u y₀,
exact ⟨λ h i, by simpa only [finset.sup_singleton] using h {i},
λ h s ε hε, (s.eventually_all.2 $ λ i _, h i ε hε).mono (λ _, finset_sup_apply_lt hε)⟩,
end | lemma | with_seminorms.tendsto_nhds | 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"
] | [
"filter",
"filter.tendsto",
"finset.sup_singleton",
"with_seminorms"
] | Convergence along filters for `with_seminorms`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.tendsto_nhds_at_top (hp : with_seminorms p) (u : F → E) (y₀ : E) :
filter.tendsto u filter.at_top (𝓝 y₀) ↔ ∀ i ε, 0 < ε → ∃ x₀, ∀ x, x₀ ≤ x → p i (u x - y₀) < ε | begin
rw hp.tendsto_nhds u y₀,
exact forall₃_congr (λ _ _ _, filter.eventually_at_top),
end | lemma | with_seminorms.tendsto_nhds_at_top | 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"
] | [
"filter.at_top",
"filter.eventually_at_top",
"filter.tendsto",
"forall₃_congr",
"with_seminorms"
] | Limit `→ ∞` for `with_seminorms`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_family.with_seminorms_of_nhds (p : seminorm_family 𝕜 E ι)
(h : 𝓝 (0 : E) = p.module_filter_basis.to_filter_basis.filter) :
with_seminorms p | begin
refine ⟨topological_add_group.ext infer_instance
(p.add_group_filter_basis.is_topological_add_group) _⟩,
rw add_group_filter_basis.nhds_zero_eq,
exact h,
end | lemma | seminorm_family.with_seminorms_of_nhds | 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_family",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_family.with_seminorms_of_has_basis (p : seminorm_family 𝕜 E ι)
(h : (𝓝 (0 : E)).has_basis (λ (s : set E), s ∈ p.basis_sets) id) :
with_seminorms p | p.with_seminorms_of_nhds $ filter.has_basis.eq_of_same_basis h
p.add_group_filter_basis.to_filter_basis.has_basis | lemma | seminorm_family.with_seminorms_of_has_basis | 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"
] | [
"filter.has_basis.eq_of_same_basis",
"seminorm_family",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_family.with_seminorms_iff_nhds_eq_infi (p : seminorm_family 𝕜 E ι) :
with_seminorms p ↔ (𝓝 0 : filter E) = ⨅ i, (𝓝 0).comap (p i) | begin
rw ← p.filter_eq_infi,
refine ⟨λ h, _, p.with_seminorms_of_nhds⟩,
rw h.topology_eq_with_seminorms,
exact add_group_filter_basis.nhds_zero_eq _,
end | lemma | seminorm_family.with_seminorms_iff_nhds_eq_infi | 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"
] | [
"filter",
"seminorm_family",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.continuous_seminorm [nontrivially_normed_field 𝕝]
[module 𝕝 E] [has_continuous_const_smul 𝕝 E] {p : seminorm_family 𝕝 E ι} (hp : with_seminorms p)
(i : ι) : continuous (p i) | begin
refine seminorm.continuous one_pos _,
rw [p.with_seminorms_iff_nhds_eq_infi.mp hp, ball_zero_eq_preimage_ball],
exact filter.mem_infi_of_mem i (filter.preimage_mem_comap $ metric.ball_mem_nhds _ one_pos)
end | lemma | with_seminorms.continuous_seminorm | 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"
] | [
"continuous",
"filter.mem_infi_of_mem",
"filter.preimage_mem_comap",
"has_continuous_const_smul",
"metric.ball_mem_nhds",
"module",
"nontrivially_normed_field",
"seminorm.continuous",
"seminorm_family",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_family.with_seminorms_iff_topological_space_eq_infi (p : seminorm_family 𝕜 E ι) :
with_seminorms p ↔ t = ⨅ i, (p i).to_add_group_seminorm.to_seminormed_add_comm_group
.to_uniform_space.to_topological_space | begin
rw [p.with_seminorms_iff_nhds_eq_infi, topological_add_group.ext_iff infer_instance
(topological_add_group_infi $ λ i, infer_instance), nhds_infi],
congrm (_ = ⨅ i, _),
exact @comap_norm_nhds_zero _ (p i).to_add_group_seminorm.to_seminormed_add_group,
all_goals {apply_instance}
end | lemma | seminorm_family.with_seminorms_iff_topological_space_eq_infi | 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"
] | [
"nhds_infi",
"seminorm_family",
"with_seminorms"
] | The topology induced by a family of seminorms is exactly the infimum of the ones induced by
each seminorm individually. We express this as a characterization of `with_seminorms p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_family.with_seminorms_iff_uniform_space_eq_infi [u : uniform_space E]
[uniform_add_group E] (p : seminorm_family 𝕜 E ι) :
with_seminorms p ↔ u = ⨅ i, (p i).to_add_group_seminorm.to_seminormed_add_comm_group
.to_uniform_space | begin
rw [p.with_seminorms_iff_nhds_eq_infi, uniform_add_group.ext_iff infer_instance
(uniform_add_group_infi $ λ i, infer_instance), to_topological_space_infi, nhds_infi],
congrm (_ = ⨅ i, _),
exact @comap_norm_nhds_zero _ (p i).to_add_group_seminorm.to_seminormed_add_group,
all_goals {apply_instance}
... | lemma | seminorm_family.with_seminorms_iff_uniform_space_eq_infi | 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"
] | [
"nhds_infi",
"seminorm_family",
"to_topological_space_infi",
"uniform_add_group",
"uniform_space",
"with_seminorms"
] | The uniform structure induced by a family of seminorms is exactly the infimum of the ones
induced by each seminorm individually. We express this as a characterization of
`with_seminorms p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_with_seminorms (𝕜 E) [normed_field 𝕜] [seminormed_add_comm_group E] [normed_space 𝕜 E] :
with_seminorms (λ (_ : fin 1), norm_seminorm 𝕜 E) | begin
let p : seminorm_family 𝕜 E (fin 1) := λ _, norm_seminorm 𝕜 E,
refine ⟨seminormed_add_comm_group.to_topological_add_group.ext
p.add_group_filter_basis.is_topological_add_group _⟩,
refine filter.has_basis.eq_of_same_basis metric.nhds_basis_ball _,
rw ←ball_norm_seminorm 𝕜 E,
refine filter.has_basi... | lemma | norm_with_seminorms | 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"
] | [
"filter.has_basis.eq_of_same_basis",
"filter.has_basis.to_has_basis",
"finset.sup_const",
"finset.sup_empty",
"metric.nhds_basis_ball",
"norm_seminorm",
"normed_field",
"normed_space",
"seminorm_family",
"seminormed_add_comm_group",
"set.subset_univ",
"with_seminorms"
] | The topology of a `normed_space 𝕜 E` is induced by the seminorm `norm_seminorm 𝕜 E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_seminorms.is_vonN_bounded_iff_finset_seminorm_bounded {s : set E}
(hp : with_seminorms p) :
bornology.is_vonN_bounded 𝕜 s ↔ ∀ I : finset ι, ∃ r (hr : 0 < r), ∀ (x ∈ s), I.sup p x < r | begin
rw (hp.has_basis).is_vonN_bounded_basis_iff,
split,
{ intros h I,
simp only [id.def] at h,
specialize h ((I.sup p).ball 0 1) (p.basis_sets_mem I zero_lt_one),
rcases h with ⟨r, hr, h⟩,
cases normed_field.exists_lt_norm 𝕜 r with a ha,
specialize h a (le_of_lt ha),
rw [seminorm.smul_b... | lemma | with_seminorms.is_vonN_bounded_iff_finset_seminorm_bounded | 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"
] | [
"absorbs.mono_right",
"bornology.is_vonN_bounded",
"finset",
"finset.sup",
"mul_one",
"normed_field.exists_lt_norm",
"seminorm.smul_ball_zero",
"with_seminorms",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.image_is_vonN_bounded_iff_finset_seminorm_bounded (f : G → E) {s : set G}
(hp : with_seminorms p) : bornology.is_vonN_bounded 𝕜 (f '' s) ↔
∀ I : finset ι, ∃ r (hr : 0 < r), ∀ (x ∈ s), I.sup p (f x) < r | by simp_rw [hp.is_vonN_bounded_iff_finset_seminorm_bounded, set.ball_image_iff] | lemma | with_seminorms.image_is_vonN_bounded_iff_finset_seminorm_bounded | 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"
] | [
"bornology.is_vonN_bounded",
"finset",
"set.ball_image_iff",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.is_vonN_bounded_iff_seminorm_bounded {s : set E} (hp : with_seminorms p) :
bornology.is_vonN_bounded 𝕜 s ↔ ∀ i : ι, ∃ r (hr : 0 < r), ∀ (x ∈ s), p i x < r | begin
rw hp.is_vonN_bounded_iff_finset_seminorm_bounded,
split,
{ intros hI i,
convert hI {i},
rw [finset.sup_singleton] },
intros hi I,
by_cases hI : I.nonempty,
{ choose r hr h using hi,
have h' : 0 < I.sup' hI r :=
by { rcases hI.bex with ⟨i, hi⟩, exact lt_of_lt_of_le (hr i) (finset.le_su... | lemma | with_seminorms.is_vonN_bounded_iff_seminorm_bounded | 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"
] | [
"bornology.is_vonN_bounded",
"exists_prop",
"finset.le_sup'",
"finset.le_sup'_iff",
"finset.sup_empty",
"finset.sup_singleton",
"with_seminorms",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.image_is_vonN_bounded_iff_seminorm_bounded (f : G → E) {s : set G}
(hp : with_seminorms p) :
bornology.is_vonN_bounded 𝕜 (f '' s) ↔ ∀ i : ι, ∃ r (hr : 0 < r), ∀ (x ∈ s), p i (f x) < r | by simp_rw [hp.is_vonN_bounded_iff_seminorm_bounded, set.ball_image_iff] | lemma | with_seminorms.image_is_vonN_bounded_iff_seminorm_bounded | 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"
] | [
"bornology.is_vonN_bounded",
"set.ball_image_iff",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_continuous_comp {q : seminorm_family 𝕝₂ F ι'}
[topological_space E] [topological_add_group E]
[topological_space F] [topological_add_group F] (hq : with_seminorms q)
(f : E →ₛₗ[τ₁₂] F) (hf : ∀ i, continuous ((q i).comp f)) : continuous f | begin
refine continuous_of_continuous_at_zero f _,
simp_rw [continuous_at, f.map_zero, q.with_seminorms_iff_nhds_eq_infi.mp hq, filter.tendsto_infi,
filter.tendsto_comap_iff],
intros i,
convert (hf i).continuous_at,
exact (map_zero _).symm
end | lemma | seminorm.continuous_of_continuous_comp | 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"
] | [
"continuous",
"continuous_at",
"filter.tendsto_comap_iff",
"filter.tendsto_infi",
"seminorm_family",
"topological_add_group",
"topological_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_continuous_comp
{q : seminorm_family 𝕜₂ F ι'} [topological_space E] [topological_add_group E]
[topological_space F] [topological_add_group F] [has_continuous_const_smul 𝕜₂ F]
(hq : with_seminorms q) (f : E →ₛₗ[σ₁₂] F) :
continuous f ↔ ∀ i, continuous ((q i).comp f) | ⟨λ h i, continuous.comp (hq.continuous_seminorm i) h, continuous_of_continuous_comp hq f⟩ | lemma | seminorm.continuous_iff_continuous_comp | 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"
] | [
"continuous",
"continuous.comp",
"has_continuous_const_smul",
"seminorm_family",
"topological_add_group",
"topological_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_from_bounded {p : seminorm_family 𝕝 E ι} {q : seminorm_family 𝕝₂ F ι'}
[topological_space E] [topological_add_group E] (hp : with_seminorms p)
[topological_space F] [topological_add_group F] (hq : with_seminorms q)
(f : E →ₛₗ[τ₁₂] F) (hf : seminorm.is_bounded p q f) : continuous f | begin
refine continuous_of_continuous_comp hq _ (λ i, seminorm.continuous_of_continuous_at_zero _),
rw [metric.continuous_at_iff', map_zero],
intros r hr,
rcases hf i with ⟨s₁, C, hf⟩,
have hC' : 0 < C + 1 := by positivity,
rw hp.has_basis.eventually_iff,
refine ⟨(s₁.sup p).ball 0 (r/(C + 1)), p.basis_set... | lemma | seminorm.continuous_from_bounded | 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"
] | [
"continuous",
"le_rfl",
"metric.continuous_at_iff'",
"seminorm.continuous_of_continuous_at_zero",
"seminorm.is_bounded",
"seminorm_family",
"topological_add_group",
"topological_space",
"with_seminorms",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_with_seminorms_normed_space (F) [seminormed_add_comm_group F] [normed_space 𝕝₂ F]
[uniform_space E] [uniform_add_group E]
{p : ι → seminorm 𝕝 E} (hp : with_seminorms p) (f : E →ₛₗ[τ₁₂] F)
(hf : ∃ (s : finset ι) C : ℝ≥0, (norm_seminorm 𝕝₂ F).comp f ≤ C • s.sup p) :
continuous f | begin
rw ←seminorm.is_bounded_const (fin 1) at hf,
exact continuous_from_bounded hp (norm_with_seminorms 𝕝₂ F) f hf,
end | lemma | seminorm.cont_with_seminorms_normed_space | 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"
] | [
"continuous",
"finset",
"norm_seminorm",
"norm_with_seminorms",
"normed_space",
"seminorm",
"seminormed_add_comm_group",
"uniform_add_group",
"uniform_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_normed_space_to_with_seminorms (E) [seminormed_add_comm_group E] [normed_space 𝕝 E]
[uniform_space F] [uniform_add_group F]
{q : ι → seminorm 𝕝₂ F} (hq : with_seminorms q) (f : E →ₛₗ[τ₁₂] F)
(hf : ∀ i : ι, ∃ C : ℝ≥0, (q i).comp f ≤ C • (norm_seminorm 𝕝 E)) : continuous f | begin
rw ←seminorm.const_is_bounded (fin 1) at hf,
exact continuous_from_bounded (norm_with_seminorms 𝕝 E) hq f hf,
end | lemma | seminorm.cont_normed_space_to_with_seminorms | 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"
] | [
"continuous",
"norm_seminorm",
"norm_with_seminorms",
"normed_space",
"seminorm",
"seminormed_add_comm_group",
"uniform_add_group",
"uniform_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.to_locally_convex_space {p : seminorm_family 𝕜 E ι} (hp : with_seminorms p) :
locally_convex_space ℝ E | begin
apply of_basis_zero ℝ E id (λ s, s ∈ p.basis_sets),
{ rw [hp.1, add_group_filter_basis.nhds_eq _, add_group_filter_basis.N_zero],
exact filter_basis.has_basis _ },
{ intros s hs,
change s ∈ set.Union _ at hs,
simp_rw [set.mem_Union, set.mem_singleton_iff] at hs,
rcases hs with ⟨I, r, hr, rfl... | lemma | with_seminorms.to_locally_convex_space | 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"
] | [
"convex_ball",
"filter_basis.has_basis",
"locally_convex_space",
"seminorm_family",
"set.Union",
"set.mem_Union",
"set.mem_singleton_iff",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space.to_locally_convex_space' [normed_space 𝕜 E] [module ℝ E]
[is_scalar_tower ℝ 𝕜 E] : locally_convex_space ℝ E | (norm_with_seminorms 𝕜 E).to_locally_convex_space | lemma | normed_space.to_locally_convex_space' | 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"
] | [
"is_scalar_tower",
"locally_convex_space",
"module",
"norm_with_seminorms",
"normed_space"
] | Not an instance since `𝕜` can't be inferred. See `normed_space.to_locally_convex_space` for a
slightly weaker instance version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.to_locally_convex_space [normed_space ℝ E] :
locally_convex_space ℝ E | normed_space.to_locally_convex_space' ℝ | instance | normed_space.to_locally_convex_space | 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"
] | [
"locally_convex_space",
"normed_space",
"normed_space.to_locally_convex_space'"
] | See `normed_space.to_locally_convex_space'` for a slightly stronger version which is not an
instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_family.comp (q : seminorm_family 𝕜₂ F ι) (f : E →ₛₗ[σ₁₂] F) :
seminorm_family 𝕜 E ι | λ i, (q i).comp f | def | seminorm_family.comp | 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_family"
] | The family of seminorms obtained by composing each seminorm by a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_family.comp_apply (q : seminorm_family 𝕜₂ F ι) (i : ι) (f : E →ₛₗ[σ₁₂] F) :
q.comp f i = (q i).comp f | rfl | lemma | seminorm_family.comp_apply | 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_family"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_family.finset_sup_comp (q : seminorm_family 𝕜₂ F ι) (s : finset ι)
(f : E →ₛₗ[σ₁₂] F) : (s.sup q).comp f = s.sup (q.comp f) | begin
ext x,
rw [seminorm.comp_apply, seminorm.finset_sup_apply, seminorm.finset_sup_apply],
refl
end | lemma | seminorm_family.finset_sup_comp | 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"
] | [
"finset",
"seminorm.comp_apply",
"seminorm.finset_sup_apply",
"seminorm_family"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.with_seminorms_induced [hι : nonempty ι] {q : seminorm_family 𝕜₂ F ι}
(hq : with_seminorms q) (f : E →ₛₗ[σ₁₂] F) :
@with_seminorms 𝕜 E ι _ _ _ _ (q.comp f) (induced f infer_instance) | begin
letI : topological_space E := induced f infer_instance,
letI : topological_add_group E := topological_add_group_induced f,
rw [(q.comp f).with_seminorms_iff_nhds_eq_infi, nhds_induced, map_zero,
q.with_seminorms_iff_nhds_eq_infi.mp hq, filter.comap_infi],
refine infi_congr (λ i, _),
exact filter.c... | lemma | linear_map.with_seminorms_induced | 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"
] | [
"filter.comap_comap",
"filter.comap_infi",
"infi_congr",
"nhds_induced",
"seminorm_family",
"topological_add_group",
"topological_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.with_seminorms [hι : nonempty ι] {q : seminorm_family 𝕜₂ F ι}
(hq : with_seminorms q) [topological_space E] {f : E →ₛₗ[σ₁₂] F} (hf : inducing f) :
with_seminorms (q.comp f) | begin
rw hf.induced,
exact f.with_seminorms_induced hq
end | lemma | inducing.with_seminorms | 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"
] | [
"inducing",
"seminorm_family",
"topological_space",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_seminorms.first_countable (hp : with_seminorms p) :
topological_space.first_countable_topology E | begin
haveI : (𝓝 (0 : E)).is_countably_generated,
{ rw p.with_seminorms_iff_nhds_eq_infi.mp hp,
exact filter.infi.is_countably_generated _ },
haveI : (uniformity E).is_countably_generated := uniform_add_group.uniformity_countably_generated,
exact uniform_space.first_countable_topology E,
end | lemma | with_seminorms.first_countable | 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"
] | [
"filter.infi.is_countably_generated",
"topological_space.first_countable_topology",
"uniform_space.first_countable_topology",
"uniformity",
"with_seminorms"
] | If the topology of a space is induced by a countable family of seminorms, then the topology
is first countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_smul_le (r : α) (x : β) : ‖r • x‖ ≤ ‖r‖ * ‖x‖ | by simpa [smul_zero] using dist_smul_pair r 0 x | lemma | norm_smul_le | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"dist_smul_pair",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_smul_le (r : α) (x : β) : ‖r • x‖₊ ≤ ‖r‖₊ * ‖x‖₊ | norm_smul_le _ _ | lemma | nnnorm_smul_le | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"norm_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_smul_le (s : α) (x y : β) : dist (s • x) (s • y) ≤ ‖s‖ * dist x y | by simpa only [dist_eq_norm, sub_zero] using dist_smul_pair s x y | lemma | dist_smul_le | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"dist_smul_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_smul_le (s : α) (x y : β) : nndist (s • x) (s • y) ≤ ‖s‖₊ * nndist x y | dist_smul_le s x y | lemma | nndist_smul_le | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"dist_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_smul_le (s : α) (x y : β) : edist (s • x) (s • y) ≤ ‖s‖₊ • edist x y | by simpa only [edist_nndist, ennreal.coe_mul] using ennreal.coe_le_coe.mpr (nndist_smul_le s x y) | lemma | edist_smul_le | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"edist_nndist",
"ennreal.coe_mul",
"nndist_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_smul (s : α) : lipschitz_with ‖s‖₊ ((•) s : β → β) | lipschitz_with_iff_dist_le_mul.2 $ dist_smul_le _ | lemma | lipschitz_with_smul | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"dist_smul_le",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_semi_normed_ring.to_has_bounded_smul [non_unital_semi_normed_ring α] :
has_bounded_smul α α | { dist_smul_pair' := λ x y₁ y₂, by simpa [mul_sub, dist_eq_norm] using norm_mul_le x (y₁ - y₂),
dist_pair_smul' := λ x₁ x₂ y, by simpa [sub_mul, dist_eq_norm] using norm_mul_le (x₁ - x₂) y, } | instance | non_unital_semi_normed_ring.to_has_bounded_smul | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"has_bounded_smul",
"non_unital_semi_normed_ring",
"norm_mul_le"
] | Left multiplication is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_semi_normed_ring.to_has_bounded_op_smul [non_unital_semi_normed_ring α] :
has_bounded_smul αᵐᵒᵖ α | { dist_smul_pair' := λ x y₁ y₂,
by simpa [sub_mul, dist_eq_norm, mul_comm] using norm_mul_le (y₁ - y₂) x.unop,
dist_pair_smul' := λ x₁ x₂ y,
by simpa [mul_sub, dist_eq_norm, mul_comm] using norm_mul_le y (x₁ - x₂).unop, } | instance | non_unital_semi_normed_ring.to_has_bounded_op_smul | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"has_bounded_smul",
"mul_comm",
"non_unital_semi_normed_ring",
"norm_mul_le"
] | Right multiplication is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_bounded_smul.of_norm_smul_le (h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖) :
has_bounded_smul α β | { dist_smul_pair' := λ a b₁ b₂, by simpa [smul_sub, dist_eq_norm] using h a (b₁ - b₂),
dist_pair_smul' := λ a₁ a₂ b, by simpa [sub_smul, dist_eq_norm] using h (a₁ - a₂) b } | lemma | has_bounded_smul.of_norm_smul_le | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"has_bounded_smul",
"smul_sub",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul (r : α) (x : β) : ‖r • x‖ = ‖r‖ * ‖x‖ | begin
by_cases h : r = 0,
{ simp [h, zero_smul α x] },
{ refine le_antisymm (norm_smul_le r x) _,
calc ‖r‖ * ‖x‖ = ‖r‖ * ‖r⁻¹ • r • x‖ : by rw [inv_smul_smul₀ h]
... ≤ ‖r‖ * (‖r⁻¹‖ * ‖r • x‖) :
mul_le_mul_of_nonneg_left (norm_smul_le _ _) (norm_nonneg _)
... = ‖r • x‖ ... | lemma | norm_smul | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"inv_smul_smul₀",
"mul_assoc",
"mul_inv_cancel",
"mul_le_mul_of_nonneg_left",
"norm_inv",
"norm_smul_le",
"one_mul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_smul (r : α) (x : β) : ‖r • x‖₊ = ‖r‖₊ * ‖x‖₊ | nnreal.eq $ norm_smul r x | lemma | nnnorm_smul | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"nnreal.eq",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_smul₀ (s : α) (x y : β) : dist (s • x) (s • y) = ‖s‖ * dist x y | by simp_rw [dist_eq_norm, (norm_smul _ _).symm, smul_sub] | lemma | dist_smul₀ | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"norm_smul",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_smul₀ (s : α) (x y : β) : nndist (s • x) (s • y) = ‖s‖₊ * nndist x y | nnreal.eq $ dist_smul₀ s x y | lemma | nndist_smul₀ | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"dist_smul₀",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_smul₀ (s : α) (x y : β) : edist (s • x) (s • y) = ‖s‖₊ • edist x y | by simp only [edist_nndist, nndist_smul₀, ennreal.coe_mul, ennreal.smul_def, smul_eq_mul] | lemma | edist_smul₀ | analysis.normed | src/analysis/normed/mul_action.lean | [
"topology.metric_space.algebra",
"analysis.normed.field.basic"
] | [
"edist_nndist",
"ennreal.coe_mul",
"ennreal.smul_def",
"nndist_smul₀",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_semi_normed_ring (α : Type*)
extends has_norm α, non_unital_ring α, pseudo_metric_space α | (dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b) | class | non_unital_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_norm",
"non_unital_ring",
"norm_mul",
"pseudo_metric_space"
] | A non-unital seminormed ring is a not-necessarily-unital ring
endowed with a seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semi_normed_ring (α : Type*) extends has_norm α, ring α, pseudo_metric_space α | (dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b) | class | semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_norm",
"norm_mul",
"pseudo_metric_space",
"ring"
] | A seminormed ring is a ring endowed with a seminorm which satisfies the inequality
`‖x y‖ ≤ ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semi_normed_ring.to_non_unital_semi_normed_ring [β : semi_normed_ring α] :
non_unital_semi_normed_ring α | { ..β } | instance | semi_normed_ring.to_non_unital_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_semi_normed_ring",
"semi_normed_ring"
] | A seminormed ring is a non-unital seminormed ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_normed_ring (α : Type*) extends has_norm α, non_unital_ring α, metric_space α | (dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b) | class | non_unital_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_norm",
"metric_space",
"non_unital_ring",
"norm_mul"
] | A non-unital normed ring is a not-necessarily-unital ring
endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_normed_ring.to_non_unital_semi_normed_ring [β : non_unital_normed_ring α] :
non_unital_semi_normed_ring α | { ..β } | instance | non_unital_normed_ring.to_non_unital_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_normed_ring",
"non_unital_semi_normed_ring"
] | A non-unital normed ring is a non-unital seminormed ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ring (α : Type*) extends has_norm α, ring α, metric_space α | (dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b) | class | normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_norm",
"metric_space",
"norm_mul",
"ring"
] | A normed ring is a ring endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_division_ring (α : Type*) extends has_norm α, division_ring α, metric_space α | (dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul' : ∀ a b, norm (a * b) = norm a * norm b) | class | normed_division_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"division_ring",
"has_norm",
"metric_space"
] | A normed division ring is a division ring endowed with a seminorm which satisfies the equality
`‖x y‖ = ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_division_ring.to_normed_ring [β : normed_division_ring α] : normed_ring α | { norm_mul := λ a b, (normed_division_ring.norm_mul' a b).le,
..β } | instance | normed_division_ring.to_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul",
"normed_division_ring",
"normed_ring"
] | A normed division ring is a normed ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ring.to_semi_normed_ring [β : normed_ring α] : semi_normed_ring α | { ..β } | instance | normed_ring.to_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"normed_ring",
"semi_normed_ring"
] | A normed ring is a seminormed ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ring.to_non_unital_normed_ring [β : normed_ring α] : non_unital_normed_ring α | { ..β } | instance | normed_ring.to_non_unital_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_normed_ring",
"normed_ring"
] | A normed ring is a non-unital normed ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semi_normed_comm_ring (α : Type*) extends semi_normed_ring α | (mul_comm : ∀ x y : α, x * y = y * x) | class | semi_normed_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_comm",
"semi_normed_ring"
] | A seminormed commutative ring is a commutative ring endowed with a seminorm which satisfies
the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_ring (α : Type*) extends normed_ring α | (mul_comm : ∀ x y : α, x * y = y * x) | class | normed_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_comm",
"normed_ring"
] | A normed commutative ring is a commutative ring endowed with a norm which satisfies
the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_ring.to_semi_normed_comm_ring [β : normed_comm_ring α] :
semi_normed_comm_ring α | { ..β } | instance | normed_comm_ring.to_semi_normed_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"normed_comm_ring",
"semi_normed_comm_ring"
] | A normed commutative ring is a seminormed commutative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_one_class (α : Type*) [has_norm α] [has_one α] : Prop | (norm_one : ‖(1:α)‖ = 1) | class | norm_one_class | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_norm"
] | A mixin class with the axiom `‖1‖ = 1`. Many `normed_ring`s and all `normed_field`s satisfy this
axiom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_one [seminormed_add_comm_group α] [has_one α] [norm_one_class α] :
‖(1 : α)‖₊ = 1 | nnreal.eq norm_one | lemma | nnnorm_one | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnreal.eq",
"norm_one_class",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_one_class.nontrivial (α : Type*) [seminormed_add_comm_group α] [has_one α]
[norm_one_class α] :
nontrivial α | nontrivial_of_ne 0 1 $ ne_of_apply_ne norm $ by simp | lemma | norm_one_class.nontrivial | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"ne_of_apply_ne",
"nontrivial",
"nontrivial_of_ne",
"norm_one_class",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semi_normed_comm_ring.to_comm_ring [β : semi_normed_comm_ring α] : comm_ring α | { ..β } | instance | semi_normed_comm_ring.to_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"comm_ring",
"semi_normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_normed_ring.to_normed_add_comm_group [β : non_unital_normed_ring α] :
normed_add_comm_group α | { ..β } | instance | non_unital_normed_ring.to_normed_add_comm_group | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_normed_ring",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_semi_normed_ring.to_seminormed_add_comm_group [non_unital_semi_normed_ring α] :
seminormed_add_comm_group α | { ..‹non_unital_semi_normed_ring α› } | instance | non_unital_semi_normed_ring.to_seminormed_add_comm_group | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_semi_normed_ring",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.norm_one_class [seminormed_add_comm_group α] [has_one α] [norm_one_class α]
[seminormed_add_comm_group β] [has_one β] [norm_one_class β] :
norm_one_class (α × β) | ⟨by simp [prod.norm_def]⟩ | instance | prod.norm_one_class | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_one_class",
"prod.norm_def",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.norm_one_class {ι : Type*} {α : ι → Type*} [nonempty ι] [fintype ι]
[Π i, seminormed_add_comm_group (α i)] [Π i, has_one (α i)] [∀ i, norm_one_class (α i)] :
norm_one_class (Π i, α i) | ⟨by simp [pi.norm_def, finset.sup_const finset.univ_nonempty]⟩ | instance | pi.norm_one_class | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset.sup_const",
"finset.univ_nonempty",
"fintype",
"norm_one_class",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_opposite.norm_one_class [seminormed_add_comm_group α] [has_one α] [norm_one_class α] :
norm_one_class αᵐᵒᵖ | ⟨@norm_one α _ _ _⟩ | instance | mul_opposite.norm_one_class | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_one_class",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mul_le (a b : α) : (‖a*b‖) ≤ (‖a‖) * (‖b‖) | non_unital_semi_normed_ring.norm_mul _ _ | lemma | norm_mul_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_mul_le (a b : α) : ‖a * b‖₊ ≤ ‖a‖₊ * ‖b‖₊ | by simpa only [←norm_to_nnreal, ←real.to_nnreal_mul (norm_nonneg _)]
using real.to_nnreal_mono (norm_mul_le _ _) | lemma | nnnorm_mul_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul_le",
"real.to_nnreal_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_norm_one (β) [normed_ring β] [nontrivial β] : 1 ≤ ‖(1 : β)‖ | (le_mul_iff_one_le_left $ norm_pos_iff.mpr (one_ne_zero : (1 : β) ≠ 0)).mp
(by simpa only [mul_one] using norm_mul_le (1 : β) 1) | lemma | one_le_norm_one | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"le_mul_iff_one_le_left",
"mul_one",
"nontrivial",
"norm_mul_le",
"normed_ring",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_nnnorm_one (β) [normed_ring β] [nontrivial β] : 1 ≤ ‖(1 : β)‖₊ | one_le_norm_one β | lemma | one_le_nnnorm_one | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nontrivial",
"normed_ring",
"one_le_norm_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.zero_mul_is_bounded_under_le {f g : ι → α} {l : filter ι}
(hf : tendsto f l (𝓝 0)) (hg : is_bounded_under (≤) l (norm ∘ g)) :
tendsto (λ x, f x * g x) l (𝓝 0) | hf.op_zero_is_bounded_under_le hg (*) norm_mul_le | lemma | filter.tendsto.zero_mul_is_bounded_under_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"filter",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.is_bounded_under_le.mul_tendsto_zero {f g : ι → α} {l : filter ι}
(hf : is_bounded_under (≤) l (norm ∘ f)) (hg : tendsto g l (𝓝 0)) :
tendsto (λ x, f x * g x) l (𝓝 0) | hg.op_zero_is_bounded_under_le hf (flip (*)) (λ x y, ((norm_mul_le y x).trans_eq (mul_comm _ _))) | lemma | filter.is_bounded_under_le.mul_tendsto_zero | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"filter",
"mul_comm",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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