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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|---|---|---|---|---|---|---|---|---|---|---|
nhds (x : Rˣ) : {x : R | is_unit x} ∈ 𝓝 (x : R) | is_open.mem_nhds units.is_open x.is_unit | lemma | units.nhds | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"is_open.mem_nhds",
"is_unit",
"nhds",
"units.is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_compl_ball : nonunits R ⊆ (metric.ball (1 : R) 1)ᶜ | set.subset_compl_comm.mp $ λ x hx, by simpa [sub_sub_self, units.coe_one_sub] using
(units.one_sub (1 - x) (by rwa [metric.mem_ball, dist_eq_norm, norm_sub_rev] at hx)).is_unit | lemma | nonunits.subset_compl_ball | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"is_unit",
"metric.ball",
"metric.mem_ball",
"nonunits",
"units.one_sub"
] | The `nonunits` in a complete normed ring are contained in the complement of the ball of radius
`1` centered at `1 : R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed : is_closed (nonunits R) | units.is_open.is_closed_compl | lemma | nonunits.is_closed | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"is_closed",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_one_sub (t : R) (h : ‖t‖ < 1) : inverse (1 - t) = ↑(units.one_sub t h)⁻¹ | by rw [← inverse_unit (units.one_sub t h), units.coe_one_sub] | lemma | normed_ring.inverse_one_sub | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"units.one_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_add (x : Rˣ) :
∀ᶠ t in (𝓝 0), inverse ((x : R) + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹ | begin
nontriviality R,
rw [eventually_iff, metric.mem_nhds_iff],
have hinv : 0 < ‖(↑x⁻¹ : R)‖⁻¹, by cancel_denoms,
use [‖(↑x⁻¹ : R)‖⁻¹, hinv],
intros t ht,
simp only [mem_ball, dist_zero_right] at ht,
have ht' : ‖-↑x⁻¹ * t‖ < 1,
{ refine lt_of_le_of_lt (norm_mul_le _ _) _,
rw norm_neg,
refine lt... | lemma | normed_ring.inverse_add | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"metric.mem_nhds_iff",
"mul_lt_mul_of_pos_left",
"neg_mul",
"norm_mul_le",
"units.add"
] | The formula `inverse (x + t) = inverse (1 + x⁻¹ * t) * x⁻¹` holds for `t` sufficiently small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_one_sub_nth_order (n : ℕ) :
∀ᶠ t in (𝓝 0), inverse ((1:R) - t) = (∑ i in range n, t ^ i) + (t ^ n) * inverse (1 - t) | begin
simp only [eventually_iff, metric.mem_nhds_iff],
use [1, by norm_num],
intros t ht,
simp only [mem_ball, dist_zero_right] at ht,
simp only [inverse_one_sub t ht, set.mem_set_of_eq],
have h : 1 = ((range n).sum (λ i, t ^ i)) * (units.one_sub t ht) + t ^ n,
{ simp only [units.coe_one_sub],
rw [geo... | lemma | normed_ring.inverse_one_sub_nth_order | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"geom_sum_mul_neg",
"metric.mem_nhds_iff",
"mul_assoc",
"mul_inv",
"one_mul",
"units.one_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_add_nth_order (x : Rˣ) (n : ℕ) :
∀ᶠ t in (𝓝 0), inverse ((x : R) + t)
= (∑ i in range n, (- ↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (- ↑x⁻¹ * t) ^ n * inverse (x + t) | begin
refine (inverse_add x).mp _,
have hzero : tendsto (λ (t : R), - ↑x⁻¹ * t) (𝓝 0) (𝓝 0),
{ convert ((mul_left_continuous (- (↑x⁻¹ : R))).tendsto 0).comp tendsto_id,
simp },
refine (hzero.eventually (inverse_one_sub_nth_order n)).mp (eventually_of_forall _),
simp only [neg_mul, sub_neg_eq_add],
int... | lemma | normed_ring.inverse_add_nth_order | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"mul_assoc",
"mul_left_continuous",
"neg_mul"
] | The formula
`inverse (x + t) = (∑ i in range n, (- x⁻¹ * t) ^ i) * x⁻¹ + (- x⁻¹ * t) ^ n * inverse (x + t)`
holds for `t` sufficiently small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_one_sub_norm : (λ t : R, inverse (1 - t)) =O[𝓝 0] (λ t, 1 : R → ℝ) | begin
simp only [is_O, is_O_with, eventually_iff, metric.mem_nhds_iff],
refine ⟨‖(1:R)‖ + 1, (2:ℝ)⁻¹, by norm_num, _⟩,
intros t ht,
simp only [ball, dist_zero_right, set.mem_set_of_eq] at ht,
have ht' : ‖t‖ < 1,
{ have : (2:ℝ)⁻¹ < 1 := by cancel_denoms,
linarith },
simp only [inverse_one_sub t ht', no... | lemma | normed_ring.inverse_one_sub_norm | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"inv_inv",
"inv_le_inv_of_le",
"metric.mem_nhds_iff",
"mul_one",
"normed_ring.tsum_geometric_of_norm_lt_1",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_add_norm (x : Rˣ) : (λ t : R, inverse (↑x + t)) =O[𝓝 0] (λ t, (1:ℝ)) | begin
simp only [is_O_iff, norm_one, mul_one],
cases is_O_iff.mp (@inverse_one_sub_norm R _ _) with C hC,
use C * ‖((x⁻¹:Rˣ):R)‖,
have hzero : tendsto (λ t, - (↑x⁻¹ : R) * t) (𝓝 0) (𝓝 0),
{ convert ((mul_left_continuous (-↑x⁻¹ : R)).tendsto 0).comp tendsto_id,
simp },
refine (inverse_add x).mp ((hzero... | lemma | normed_ring.inverse_add_norm | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"bound",
"mul_left_continuous",
"mul_one",
"norm_mul_le"
] | The function `λ t, inverse (x + t)` is O(1) as `t → 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_add_norm_diff_nth_order (x : Rˣ) (n : ℕ) :
(λ t : R, inverse (↑x + t) - (∑ i in range n, (- ↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 (0:R)]
(λ t, ‖t‖ ^ n) | begin
by_cases h : n = 0,
{ simpa [h] using inverse_add_norm x },
have hn : 0 < n := nat.pos_of_ne_zero h,
simp [is_O_iff],
cases (is_O_iff.mp (inverse_add_norm x)) with C hC,
use C * ‖(1:ℝ)‖ * ‖(↑x⁻¹ : R)‖ ^ n,
have h : eventually_eq (𝓝 (0:R))
(λ t, inverse (↑x + t) - (∑ i in range n, (- ↑x⁻¹ * t) ^... | lemma | normed_ring.inverse_add_norm_diff_nth_order | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"mul_pow",
"neg_mul",
"norm_mul_le",
"norm_pow_le'",
"pow_le_pow_of_le_left",
"pow_nonneg"
] | The function
`λ t, inverse (x + t) - (∑ i in range n, (- x⁻¹ * t) ^ i) * x⁻¹`
is `O(t ^ n)` as `t → 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_add_norm_diff_first_order (x : Rˣ) :
(λ t : R, inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] (λ t, ‖t‖) | by simpa using inverse_add_norm_diff_nth_order x 1 | lemma | normed_ring.inverse_add_norm_diff_first_order | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [] | The function `λ t, inverse (x + t) - x⁻¹` is `O(t)` as `t → 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_add_norm_diff_second_order (x : Rˣ) :
(λ t : R, inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] (λ t, ‖t‖ ^ 2) | begin
convert inverse_add_norm_diff_nth_order x 2,
ext t,
simp only [range_succ, range_one, sum_insert, mem_singleton, sum_singleton, not_false_iff,
one_ne_zero, pow_zero, add_mul, pow_one, one_mul, neg_mul,
sub_add_eq_sub_sub_swap, sub_neg_eq_add],
end | lemma | normed_ring.inverse_add_norm_diff_second_order | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"neg_mul",
"one_mul",
"one_ne_zero",
"pow_one",
"pow_zero"
] | The function
`λ t, inverse (x + t) - x⁻¹ + x⁻¹ * t * x⁻¹`
is `O(t ^ 2)` as `t → 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_continuous_at (x : Rˣ) : continuous_at inverse (x : R) | begin
have h_is_o : (λ t : R, inverse (↑x + t) - ↑x⁻¹) =o[𝓝 0] (λ _, 1 : R → ℝ) :=
(inverse_add_norm_diff_first_order x).trans_is_o (is_o.norm_left $ is_o_id_const one_ne_zero),
have h_lim : tendsto (λ (y:R), y - x) (𝓝 x) (𝓝 0),
{ refine tendsto_zero_iff_norm_tendsto_zero.mpr _,
exact tendsto_iff_norm_... | lemma | normed_ring.inverse_continuous_at | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"continuous_at",
"one_ne_zero"
] | The function `inverse` is continuous at each unit of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_coe : is_open_map (coe : Rˣ → R) | begin
rw is_open_map_iff_nhds_le,
intros x s,
rw [mem_map, mem_nhds_induced],
rintros ⟨t, ht, hts⟩,
obtain ⟨u, hu, v, hv, huvt⟩ :
∃ (u : set R), u ∈ 𝓝 ↑x ∧ ∃ (v : set Rᵐᵒᵖ), v ∈ 𝓝 (op ↑x⁻¹) ∧ u ×ˢ v ⊆ t,
{ simpa [embed_product, mem_nhds_prod_iff] using ht },
have : u ∩ (op ∘ ring.inverse) ⁻¹' v ∩ (s... | lemma | units.is_open_map_coe | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"is_open_map",
"is_open_map_iff_nhds_le",
"mem_map",
"mem_nhds_induced",
"mem_nhds_prod_iff",
"ring.inverse",
"set.range",
"units.nhds"
] | In a normed ring, the coercion from `Rˣ` (equipped with the induced topology from the
embedding in `R × R`) to `R` is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_embedding_coe : open_embedding (coe : Rˣ → R) | open_embedding_of_continuous_injective_open continuous_coe ext is_open_map_coe | lemma | units.open_embedding_coe | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | In a normed ring, the coercion from `Rˣ` (equipped with the induced topology from the
embedding in `R × R`) to `R` is an open embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_top_of_norm_lt_one (I : ideal R) {x : R} (hxI : x ∈ I) (hx : ‖1 - x‖ < 1) : I = ⊤ | let u := units.one_sub (1 - x) hx in (I.eq_top_iff_one.mpr $
by simpa only [show u.inv * x = 1, by simp] using I.mul_mem_left u.inv hxI) | lemma | ideal.eq_top_of_norm_lt_one | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"ideal",
"units.one_sub"
] | An ideal which contains an element within `1` of `1 : R` is the unit ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_ne_top (I : ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤ | have h : _ := closure_minimal (coe_subset_nonunits hI) nonunits.is_closed,
by simpa only [I.closure.eq_top_iff_one, ne.def] using mt (@h 1) one_not_mem_nonunits | lemma | ideal.closure_ne_top | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"closure_minimal",
"coe_subset_nonunits",
"ideal",
"nonunits.is_closed",
"one_not_mem_nonunits"
] | The `ideal.closure` of a proper ideal in a complete normed ring is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_maximal.closure_eq {I : ideal R} (hI : I.is_maximal) : I.closure = I | (hI.eq_of_le (I.closure_ne_top hI.ne_top) subset_closure).symm | lemma | ideal.is_maximal.closure_eq | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"ideal",
"subset_closure"
] | The `ideal.closure` of a maximal ideal in a complete normed ring is the ideal itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_maximal.is_closed {I : ideal R} [hI : I.is_maximal] : is_closed (I : set R) | is_closed_of_closure_subset $ eq.subset $ congr_arg (coe : ideal R → set R) hI.closure_eq | instance | ideal.is_maximal.is_closed | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"eq.subset",
"ideal",
"is_closed",
"is_closed_of_closure_subset"
] | Maximal ideals in complete normed rings are closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_weak_dual : dual 𝕜 E ≃ₗ[𝕜] weak_dual 𝕜 E | linear_equiv.refl 𝕜 (E →L[𝕜] 𝕜) | def | normed_space.dual.to_weak_dual | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"linear_equiv.refl",
"weak_dual"
] | For normed spaces `E`, there is a canonical map `dual 𝕜 E → weak_dual 𝕜 E` (the "identity"
mapping). It is a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_weak_dual (x' : dual 𝕜 E) : ⇑(x'.to_weak_dual) = x' | rfl | lemma | normed_space.dual.coe_to_weak_dual | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_weak_dual_eq_iff (x' y' : dual 𝕜 E) :
x'.to_weak_dual = y'.to_weak_dual ↔ x' = y' | to_weak_dual.injective.eq_iff | lemma | normed_space.dual.to_weak_dual_eq_iff | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_weak_dual_continuous : continuous (λ (x' : dual 𝕜 E), x'.to_weak_dual) | weak_bilin.continuous_of_continuous_eval _ $ λ z, (inclusion_in_double_dual 𝕜 E z).continuous | theorem | normed_space.dual.to_weak_dual_continuous | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"continuous",
"weak_bilin.continuous_of_continuous_eval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_to_weak_dual : dual 𝕜 E →L[𝕜] weak_dual 𝕜 E | { cont := to_weak_dual_continuous, .. to_weak_dual, } | def | normed_space.dual.continuous_linear_map_to_weak_dual | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"cont",
"weak_dual"
] | For a normed space `E`, according to `to_weak_dual_continuous` the "identity mapping"
`dual 𝕜 E → weak_dual 𝕜 E` is continuous. This definition implements it as a continuous linear
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_norm_topology_le_weak_dual_topology :
(by apply_instance : topological_space (dual 𝕜 E)) ≤
(by apply_instance : topological_space (weak_dual 𝕜 E)) | by { convert to_weak_dual_continuous.le_induced, exact induced_id.symm } | theorem | normed_space.dual.dual_norm_topology_le_weak_dual_topology | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"topological_space",
"weak_dual"
] | The weak-star topology is coarser than the dual-norm topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_dual : weak_dual 𝕜 E ≃ₗ[𝕜] dual 𝕜 E | normed_space.dual.to_weak_dual.symm | def | weak_dual.to_normed_dual | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"weak_dual"
] | For normed spaces `E`, there is a canonical map `weak_dual 𝕜 E → dual 𝕜 E` (the "identity"
mapping). It is a linear equivalence. Here it is implemented as the inverse of the linear
equivalence `normed_space.dual.to_weak_dual` in the other direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_dual_apply (x : weak_dual 𝕜 E) (y : E) : (to_normed_dual x) y = x y | rfl | lemma | weak_dual.to_normed_dual_apply | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_normed_dual (x' : weak_dual 𝕜 E) : ⇑(x'.to_normed_dual) = x' | rfl | lemma | weak_dual.coe_to_normed_dual | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_normed_dual_eq_iff (x' y' : weak_dual 𝕜 E) :
x'.to_normed_dual = y'.to_normed_dual ↔ x' = y' | weak_dual.to_normed_dual.injective.eq_iff | lemma | weak_dual.to_normed_dual_eq_iff | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_closed_ball (x' : dual 𝕜 E) (r : ℝ) :
is_closed (to_normed_dual ⁻¹' closed_ball x' r) | is_closed_induced_iff'.2 (continuous_linear_map.is_weak_closed_closed_ball x' r) | lemma | weak_dual.is_closed_closed_ball | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map.is_weak_closed_closed_ball",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar (s : set E) : set (weak_dual 𝕜 E) | to_normed_dual ⁻¹' polar 𝕜 s | def | weak_dual.polar | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"weak_dual"
] | The polar set `polar 𝕜 s` of `s : set E` seen as a subset of the dual of `E` with the
weak-star topology is `weak_dual.polar 𝕜 s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar_def (s : set E) : polar 𝕜 s = {f : weak_dual 𝕜 E | ∀ x ∈ s, ‖f x‖ ≤ 1} | rfl | lemma | weak_dual.polar_def | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_polar (s : set E) : is_closed (polar 𝕜 s) | begin
simp only [polar_def, set_of_forall],
exact is_closed_bInter (λ x hx, is_closed_Iic.preimage (weak_bilin.eval_continuous _ _).norm)
end | lemma | weak_dual.is_closed_polar | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"is_closed",
"is_closed_bInter",
"weak_bilin.eval_continuous"
] | The polar `polar 𝕜 s` of a set `s : E` is a closed subset when the weak star topology
is used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_image_coe_of_bounded_of_closed {s : set (weak_dual 𝕜 E)}
(hb : bounded (dual.to_weak_dual ⁻¹' s)) (hc : is_closed s) :
is_closed ((coe_fn : weak_dual 𝕜 E → E → 𝕜) '' s) | continuous_linear_map.is_closed_image_coe_of_bounded_of_weak_closed hb (is_closed_induced_iff'.1 hc) | lemma | weak_dual.is_closed_image_coe_of_bounded_of_closed | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map.is_closed_image_coe_of_bounded_of_weak_closed",
"is_closed",
"weak_dual"
] | While the coercion `coe_fn : weak_dual 𝕜 E → (E → 𝕜)` is not a closed map, it sends *bounded*
closed sets to closed sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_of_bounded_of_closed [proper_space 𝕜] {s : set (weak_dual 𝕜 E)}
(hb : bounded (dual.to_weak_dual ⁻¹' s)) (hc : is_closed s) :
is_compact s | (embedding.is_compact_iff_is_compact_image fun_like.coe_injective.embedding_induced).mpr $
continuous_linear_map.is_compact_image_coe_of_bounded_of_closed_image hb $
is_closed_image_coe_of_bounded_of_closed hb hc | lemma | weak_dual.is_compact_of_bounded_of_closed | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map.is_compact_image_coe_of_bounded_of_closed_image",
"embedding.is_compact_iff_is_compact_image",
"is_closed",
"is_compact",
"proper_space",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_image_polar_of_mem_nhds {s : set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
is_closed ((coe_fn : weak_dual 𝕜 E → E → 𝕜) '' polar 𝕜 s) | is_closed_image_coe_of_bounded_of_closed (bounded_polar_of_mem_nhds_zero 𝕜 s_nhd)
(is_closed_polar _ _) | lemma | weak_dual.is_closed_image_polar_of_mem_nhds | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"is_closed",
"weak_dual"
] | The image under `coe_fn : weak_dual 𝕜 E → (E → 𝕜)` of a polar `weak_dual.polar 𝕜 s` of a
neighborhood `s` of the origin is a closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.normed_space.dual.is_closed_image_polar_of_mem_nhds {s : set E}
(s_nhd : s ∈ 𝓝 (0 : E)) : is_closed ((coe_fn : dual 𝕜 E → E → 𝕜) '' normed_space.polar 𝕜 s) | is_closed_image_polar_of_mem_nhds 𝕜 s_nhd | lemma | normed_space.dual.is_closed_image_polar_of_mem_nhds | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"is_closed",
"normed_space.polar"
] | The image under `coe_fn : normed_space.dual 𝕜 E → (E → 𝕜)` of a polar `polar 𝕜 s` of a
neighborhood `s` of the origin is a closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_polar [proper_space 𝕜] {s : set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
is_compact (polar 𝕜 s) | is_compact_of_bounded_of_closed (bounded_polar_of_mem_nhds_zero 𝕜 s_nhd) (is_closed_polar _ _) | theorem | weak_dual.is_compact_polar | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"is_compact",
"proper_space"
] | The **Banach-Alaoglu theorem**: the polar set of a neighborhood `s` of the origin in a
normed space `E` is a compact subset of `weak_dual 𝕜 E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_closed_ball [proper_space 𝕜] (x' : dual 𝕜 E) (r : ℝ) :
is_compact (to_normed_dual ⁻¹' (closed_ball x' r)) | is_compact_of_bounded_of_closed bounded_closed_ball (is_closed_closed_ball x' r) | theorem | weak_dual.is_compact_closed_ball | analysis.normed_space | src/analysis/normed_space/weak_dual.lean | [
"topology.algebra.module.weak_dual",
"analysis.normed_space.dual",
"analysis.normed_space.operator_norm"
] | [
"is_compact",
"proper_space"
] | The **Banach-Alaoglu theorem**: closed balls of the dual of a normed space `E` are compact in
the weak-star topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_norm_eq (p : subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ | begin
rcases exists_extension_of_le_sublinear ⟨p, f⟩ (λ x, ‖f‖ * ‖x‖)
(λ c hc x, by simp only [norm_smul c x, real.norm_eq_abs, abs_of_pos hc, mul_left_comm])
(λ x y, _) (λ x, le_trans (le_abs_self _) (f.le_op_norm _))
with ⟨g, g_eq, g_le⟩,
set g' := g.mk_continuous (‖f‖)
(λ x, abs_le.2 ⟨neg_le.1 $ ... | theorem | real.exists_extension_norm_eq | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/extension.lean | [
"analysis.convex.cone.basic",
"analysis.normed_space.is_R_or_C",
"analysis.normed_space.extend",
"data.is_R_or_C.lemmas"
] | [
"abs_of_pos",
"exists_extension_norm_eq",
"exists_extension_of_le_sublinear",
"le_abs_self",
"mul_le_mul_of_nonneg_left",
"mul_left_comm",
"norm_smul",
"real.norm_eq_abs",
"subspace"
] | Hahn-Banach theorem for continuous linear functions over `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_extension_norm_eq (p : subspace 𝕜 F) (f : p →L[𝕜] 𝕜) :
∃ g : F →L[𝕜] 𝕜, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ | begin
letI : module ℝ F := restrict_scalars.module ℝ 𝕜 F,
letI : is_scalar_tower ℝ 𝕜 F := restrict_scalars.is_scalar_tower _ _ _,
letI : normed_space ℝ F := normed_space.restrict_scalars _ 𝕜 _,
-- Let `fr: p →L[ℝ] ℝ` be the real part of `f`.
let fr := re_clm.comp (f.restrict_scalars ℝ),
have fr_apply : ∀... | theorem | exists_extension_norm_eq | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/extension.lean | [
"analysis.convex.cone.basic",
"analysis.normed_space.is_R_or_C",
"analysis.normed_space.extend",
"data.is_R_or_C.lemmas"
] | [
"algebra.id.smul_eq_mul",
"continuous_linear_map.extend_to_𝕜_apply",
"continuous_linear_map.map_smul",
"continuous_linear_map.op_norm_comp_le",
"is_scalar_tower",
"module",
"mul_neg",
"mul_zero",
"normed_space",
"normed_space.restrict_scalars",
"one_mul",
"real.exists_extension_norm_eq",
"s... | Hahn-Banach theorem for continuous linear functions over `𝕜` satisyfing `is_R_or_C 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coord_norm' {x : E} (h : x ≠ 0) : ‖(‖x‖ : 𝕜) • coord 𝕜 x h‖ = 1 | by rw [norm_smul, is_R_or_C.norm_coe_norm, coord_norm, mul_inv_cancel (mt norm_eq_zero.mp h)] | lemma | coord_norm' | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/extension.lean | [
"analysis.convex.cone.basic",
"analysis.normed_space.is_R_or_C",
"analysis.normed_space.extend",
"data.is_R_or_C.lemmas"
] | [
"is_R_or_C.norm_coe_norm",
"mul_inv_cancel",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dual_vector (x : E) (h : x ≠ 0) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖ | begin
let p : submodule 𝕜 E := 𝕜 ∙ x,
let f := (‖x‖ : 𝕜) • coord 𝕜 x h,
obtain ⟨g, hg⟩ := exists_extension_norm_eq p f,
refine ⟨g, _, _⟩,
{ rw [hg.2, coord_norm'] },
{ calc g x = g (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) : by rw coe_mk
... = ((‖x‖ : 𝕜) • coord 𝕜 x h) (⟨x, mem_span_singleton_self... | theorem | exists_dual_vector | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/extension.lean | [
"analysis.convex.cone.basic",
"analysis.normed_space.is_R_or_C",
"analysis.normed_space.extend",
"data.is_R_or_C.lemmas"
] | [
"coord_norm'",
"exists_extension_norm_eq",
"submodule"
] | Corollary of Hahn-Banach. Given a nonzero element `x` of a normed space, there exists an
element of the dual space, of norm `1`, whose value on `x` is `‖x‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_dual_vector' [nontrivial E] (x : E) :
∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖ | begin
by_cases hx : x = 0,
{ obtain ⟨y, hy⟩ := exists_ne (0 : E),
obtain ⟨g, hg⟩ : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g y = ‖y‖ := exists_dual_vector 𝕜 y hy,
refine ⟨g, hg.left, _⟩,
simp [hx] },
{ exact exists_dual_vector 𝕜 x hx }
end | theorem | exists_dual_vector' | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/extension.lean | [
"analysis.convex.cone.basic",
"analysis.normed_space.is_R_or_C",
"analysis.normed_space.extend",
"data.is_R_or_C.lemmas"
] | [
"exists_dual_vector",
"exists_ne",
"nontrivial"
] | Variant of Hahn-Banach, eliminating the hypothesis that `x` be nonzero, and choosing
the dual element arbitrarily when `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_dual_vector'' (x : E) :
∃ g : E →L[𝕜] 𝕜, ‖g‖ ≤ 1 ∧ g x = ‖x‖ | begin
by_cases hx : x = 0,
{ refine ⟨0, by simp, _⟩,
symmetry,
simp [hx], },
{ rcases exists_dual_vector 𝕜 x hx with ⟨g, g_norm, g_eq⟩,
exact ⟨g, g_norm.le, g_eq⟩ }
end | theorem | exists_dual_vector'' | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/extension.lean | [
"analysis.convex.cone.basic",
"analysis.normed_space.is_R_or_C",
"analysis.normed_space.extend",
"data.is_R_or_C.lemmas"
] | [
"exists_dual_vector"
] | Variant of Hahn-Banach, eliminating the hypothesis that `x` be nonzero, but only ensuring that
the dual element has norm at most `1` (this can not be improved for the trivial
vector space). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separate_convex_open_set [topological_space E] [add_comm_group E] [topological_add_group E]
[module ℝ E] [has_continuous_smul ℝ E] {s : set E}
(hs₀ : (0 : E) ∈ s) (hs₁ : convex ℝ s) (hs₂ : is_open s) {x₀ : E} (hx₀ : x₀ ∉ s) :
∃ f : E →L[ℝ] ℝ, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1 | begin
let f : E →ₗ.[ℝ] ℝ :=
linear_pmap.mk_span_singleton x₀ 1 (ne_of_mem_of_not_mem hs₀ hx₀).symm,
obtain ⟨φ, hφ₁, hφ₂⟩ := exists_extension_of_le_sublinear f (gauge s)
(λ c hc, gauge_smul_of_nonneg hc.le)
(gauge_add_le hs₁ $ absorbent_nhds_zero $ hs₂.mem_nhds hs₀) _,
have hφ₃ : φ x₀ = 1,
{ rw [←sub... | lemma | separate_convex_open_set | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"absorbent_nhds_zero",
"absorbs",
"add_comm_group",
"algebra.id.smul_eq_mul",
"convex",
"exists_extension_of_le_sublinear",
"gauge",
"gauge_add_le",
"gauge_lt_one_of_mem_of_open",
"gauge_nonneg",
"gauge_smul_of_nonneg",
"has_continuous_smul",
"is_open",
"le_mul_iff_one_le_right",
"linear... | Given a set `s` which is a convex neighbourhood of `0` and a point `x₀` outside of it, there is
a continuous linear functional `f` separating `x₀` and `s`, in the sense that it sends `x₀` to 1 and
all of `s` to values strictly below `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geometric_hahn_banach_open (hs₁ : convex ℝ s) (hs₂ : is_open s) (ht : convex ℝ t)
(disj : disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b | begin
obtain rfl | ⟨a₀, ha₀⟩ := s.eq_empty_or_nonempty,
{ exact ⟨0, 0, by simp, λ b hb, le_rfl⟩ },
obtain rfl | ⟨b₀, hb₀⟩ := t.eq_empty_or_nonempty,
{ exact ⟨0, 1, λ a ha, zero_lt_one, by simp⟩ },
let x₀ := b₀ - a₀,
let C := x₀ +ᵥ (s - t),
have : (0:E) ∈ C := ⟨a₀ - b₀, sub_mem_sub ha₀ hb₀,
by rw [vadd... | theorem | geometric_hahn_banach_open | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"cInf_le",
"convex",
"disjoint",
"interior_maximal",
"is_open",
"le_cInf",
"separate_convex_open_set",
"zero_lt_one"
] | A version of the **Hahn-Banach theorem**: given disjoint convex sets `s`, `t` where `s` is open,
there is a continuous linear functional which separates them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geometric_hahn_banach_open_point (hs₁ : convex ℝ s) (hs₂ : is_open s) (disj : x ∉ s) :
∃ f : E →L[ℝ] ℝ, ∀ a ∈ s, f a < f x | let ⟨f, s, hs, hx⟩ := geometric_hahn_banach_open hs₁ hs₂ (convex_singleton x)
(disjoint_singleton_right.2 disj)
in ⟨f, λ a ha, lt_of_lt_of_le (hs a ha) (hx x (mem_singleton _))⟩ | theorem | geometric_hahn_banach_open_point | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"convex_singleton",
"geometric_hahn_banach_open",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geometric_hahn_banach_point_open (ht₁ : convex ℝ t) (ht₂ : is_open t) (disj : x ∉ t) :
∃ f : E →L[ℝ] ℝ, ∀ b ∈ t, f x < f b | let ⟨f, hf⟩ := geometric_hahn_banach_open_point ht₁ ht₂ disj in ⟨-f, by simpa⟩ | theorem | geometric_hahn_banach_point_open | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"geometric_hahn_banach_open_point",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geometric_hahn_banach_open_open (hs₁ : convex ℝ s) (hs₂ : is_open s) (ht₁ : convex ℝ t)
(ht₃ : is_open t) (disj : disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b | begin
obtain (rfl | ⟨a₀, ha₀⟩) := s.eq_empty_or_nonempty,
{ exact ⟨0, -1, by simp, λ b hb, by norm_num⟩ },
obtain (rfl | ⟨b₀, hb₀⟩) := t.eq_empty_or_nonempty,
{ exact ⟨0, 1, λ a ha, by norm_num, by simp⟩ },
obtain ⟨f, s, hf₁, hf₂⟩ := geometric_hahn_banach_open hs₁ hs₂ ht₁ disj,
have hf : is_open_map f,
{ ... | theorem | geometric_hahn_banach_open_open | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"disjoint",
"geometric_hahn_banach_open",
"interior_maximal",
"is_open",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geometric_hahn_banach_compact_closed (hs₁ : convex ℝ s) (hs₂ : is_compact s)
(ht₁ : convex ℝ t) (ht₂ : is_closed t) (disj : disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b | begin
obtain rfl | hs := s.eq_empty_or_nonempty,
{ exact ⟨0, -2, -1, by simp, by norm_num, λ b hb, by norm_num⟩ },
unfreezingI { obtain rfl | ht := t.eq_empty_or_nonempty },
{ exact ⟨0, 1, 2, λ a ha, by norm_num, by norm_num, by simp⟩ },
obtain ⟨U, V, hU, hV, hU₁, hV₁, sU, tV, disj'⟩ := disj.exists_open_conve... | theorem | geometric_hahn_banach_compact_closed | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"disjoint",
"geometric_hahn_banach_open_open",
"is_closed",
"is_compact"
] | A version of the **Hahn-Banach theorem**: given disjoint convex sets `s`, `t` where `s` is
compact and `t` is closed, there is a continuous linear functional which strongly separates them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geometric_hahn_banach_closed_compact (hs₁ : convex ℝ s) (hs₂ : is_closed s)
(ht₁ : convex ℝ t) (ht₂ : is_compact t) (disj : disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b | let ⟨f, s, t, hs, st, ht⟩ := geometric_hahn_banach_compact_closed ht₁ ht₂ hs₁ hs₂ disj.symm in
⟨-f, -t, -s, by simpa using ht, by simpa using st, by simpa using hs⟩ | theorem | geometric_hahn_banach_closed_compact | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"disjoint",
"geometric_hahn_banach_compact_closed",
"is_closed",
"is_compact"
] | A version of the **Hahn-Banach theorem**: given disjoint convex sets `s`, `t` where `s` is
closed, and `t` is compact, there is a continuous linear functional which strongly separates them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geometric_hahn_banach_point_closed (ht₁ : convex ℝ t) (ht₂ : is_closed t) (disj : x ∉ t) :
∃ (f : E →L[ℝ] ℝ) (u : ℝ), f x < u ∧ ∀ b ∈ t, u < f b | let ⟨f, u, v, ha, hst, hb⟩ := geometric_hahn_banach_compact_closed (convex_singleton x)
is_compact_singleton ht₁ ht₂ (disjoint_singleton_left.2 disj)
in ⟨f, v, hst.trans' $ ha x $ mem_singleton _, hb⟩ | theorem | geometric_hahn_banach_point_closed | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"convex_singleton",
"geometric_hahn_banach_compact_closed",
"is_closed",
"is_compact_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geometric_hahn_banach_closed_point (hs₁ : convex ℝ s) (hs₂ : is_closed s) (disj : x ∉ s) :
∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ u < f x | let ⟨f, s, t, ha, hst, hb⟩ := geometric_hahn_banach_closed_compact hs₁ hs₂ (convex_singleton x)
is_compact_singleton (disjoint_singleton_right.2 disj)
in ⟨f, s, ha, hst.trans $ hb x $ mem_singleton _⟩ | theorem | geometric_hahn_banach_closed_point | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex",
"convex_singleton",
"geometric_hahn_banach_closed_compact",
"is_closed",
"is_compact_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geometric_hahn_banach_point_point [t1_space E] (hxy : x ≠ y) :
∃ (f : E →L[ℝ] ℝ), f x < f y | begin
obtain ⟨f, s, t, hs, st, ht⟩ :=
geometric_hahn_banach_compact_closed (convex_singleton x) is_compact_singleton
(convex_singleton y) is_closed_singleton (disjoint_singleton.2 hxy),
exact ⟨f, by linarith [hs x rfl, ht y rfl]⟩,
end | theorem | geometric_hahn_banach_point_point | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"convex_singleton",
"geometric_hahn_banach_compact_closed",
"is_closed_singleton",
"is_compact_singleton",
"t1_space"
] | See also `normed_space.eq_iff_forall_dual_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Inter_halfspaces_eq (hs₁ : convex ℝ s) (hs₂ : is_closed s) :
(⋂ (l : E →L[ℝ] ℝ), {x | ∃ y ∈ s, l x ≤ l y}) = s | begin
rw set.Inter_set_of,
refine set.subset.antisymm (λ x hx, _) (λ x hx l, ⟨x, hx, le_rfl⟩),
by_contra,
obtain ⟨l, s, hlA, hl⟩ := geometric_hahn_banach_closed_point hs₁ hs₂ h,
obtain ⟨y, hy, hxy⟩ := hx l,
exact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl,
end | lemma | Inter_halfspaces_eq | analysis.normed_space.hahn_banach | src/analysis/normed_space/hahn_banach/separation.lean | [
"analysis.convex.cone.basic",
"analysis.convex.gauge",
"topology.algebra.module.finite_dimension",
"topology.algebra.module.locally_convex"
] | [
"by_contra",
"convex",
"geometric_hahn_banach_closed_point",
"is_closed",
"le_rfl",
"set.Inter_set_of",
"set.subset.antisymm"
] | A closed convex set is the intersection of the halfspaces containing it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_star_group (E : Type*) [seminormed_add_comm_group E] [star_add_monoid E] : Prop | (norm_star : ∀ x : E, ‖x⋆‖ = ‖x‖) | class | normed_star_group | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"seminormed_add_comm_group",
"star_add_monoid"
] | A normed star group is a normed group with a compatible `star` which is isometric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_star (x : E) : ‖star x‖₊ = ‖x‖₊ | subtype.ext $ norm_star _ | lemma | nnnorm_star | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_normed_add_group_hom : normed_add_group_hom E E | { bound' := ⟨1, λ v, le_trans (norm_star _).le (one_mul _).symm.le⟩,
.. star_add_equiv } | def | star_normed_add_group_hom | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"bound'",
"normed_add_group_hom",
"one_mul",
"star_add_equiv"
] | The `star` map in a normed star group is a normed group homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_isometry : isometry (star : E → E) | show isometry star_add_equiv,
by exact add_monoid_hom_class.isometry_of_norm star_add_equiv
(show ∀ x, ‖x⋆‖ = ‖x‖, from norm_star) | lemma | star_isometry | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"isometry",
"star_add_equiv"
] | The `star` map in a normed star group is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_star_group.to_has_continuous_star : has_continuous_star E | ⟨star_isometry.continuous⟩ | instance | normed_star_group.to_has_continuous_star | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"has_continuous_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_isometric.star_ring_end [normed_comm_ring E] [star_ring E]
[normed_star_group E] : ring_hom_isometric (star_ring_end E) | ⟨norm_star⟩ | instance | ring_hom_isometric.star_ring_end | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"normed_comm_ring",
"normed_star_group",
"ring_hom_isometric",
"star_ring",
"star_ring_end"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cstar_ring (E : Type*) [non_unital_normed_ring E] [star_ring E] : Prop | (norm_star_mul_self : ∀ {x : E}, ‖x⋆ * x‖ = ‖x‖ * ‖x‖) | class | cstar_ring | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"non_unital_normed_ring",
"star_ring"
] | A C*-ring is a normed star ring that satifies the stronger condition `‖x⋆ * x‖ = ‖x‖^2`
for every `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_star_group : normed_star_group E | ⟨begin
intro x,
by_cases htriv : x = 0,
{ simp only [htriv, star_zero] },
{ have hnt : 0 < ‖x‖ := norm_pos_iff.mpr htriv,
have hnt_star : 0 < ‖x⋆‖ :=
norm_pos_iff.mpr ((add_equiv.map_ne_zero_iff star_add_equiv).mpr htriv),
have h₁ := calc
‖x‖ * ‖x‖ = ‖x⋆ * x‖ : norm_star_mul_self.symm... | instance | cstar_ring.to_normed_star_group | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"le_of_mul_le_mul_right",
"norm_mul_le",
"normed_star_group",
"star_add_equiv",
"star_star",
"star_zero"
] | In a C*-ring, star preserves the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_self_mul_star {x : E} : ‖x * x⋆‖ = ‖x‖ * ‖x‖ | by { nth_rewrite 0 [←star_star x], simp only [norm_star_mul_self, norm_star] } | lemma | cstar_ring.norm_self_mul_star | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_star_mul_self' {x : E} : ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖ | by rw [norm_star_mul_self, norm_star] | lemma | cstar_ring.norm_star_mul_self' | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_self_mul_star {x : E} : ‖x * star x‖₊ = ‖x‖₊ * ‖x‖₊ | subtype.ext norm_self_mul_star | lemma | cstar_ring.nnnorm_self_mul_star | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_star_mul_self {x : E} : ‖x⋆ * x‖₊ = ‖x‖₊ * ‖x‖₊ | subtype.ext norm_star_mul_self | lemma | cstar_ring.nnnorm_star_mul_self | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self_eq_zero_iff (x : E) : star x * x = 0 ↔ x = 0 | by { rw [←norm_eq_zero, norm_star_mul_self], exact mul_self_eq_zero.trans norm_eq_zero } | lemma | cstar_ring.star_mul_self_eq_zero_iff | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"norm_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self_ne_zero_iff (x : E) : star x * x ≠ 0 ↔ x ≠ 0 | by simp only [ne.def, star_mul_self_eq_zero_iff] | lemma | cstar_ring.star_mul_self_ne_zero_iff | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_self_eq_zero_iff (x : E) : x * star x = 0 ↔ x = 0 | by simpa only [star_eq_zero, star_star] using @star_mul_self_eq_zero_iff _ _ _ _ (star x) | lemma | cstar_ring.mul_star_self_eq_zero_iff | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"star_eq_zero",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_self_ne_zero_iff (x : E) : x * star x ≠ 0 ↔ x ≠ 0 | by simp only [ne.def, mul_star_self_eq_zero_iff] | lemma | cstar_ring.mul_star_self_ne_zero_iff | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pi.star_ring' : star_ring (Π i, R i) | infer_instance | instance | pi.star_ring' | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"star_ring"
] | This instance exists to short circuit type class resolution because of problems with
inference involving Π-types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.prod.cstar_ring : cstar_ring (R₁ × R₂) | { norm_star_mul_self := λ x,
begin
unfold norm,
simp only [prod.fst_mul, prod.fst_star, prod.snd_mul, prod.snd_star, norm_star_mul_self, ←sq],
refine le_antisymm _ _,
{ refine max_le _ _;
rw [sq_le_sq, abs_of_nonneg (norm_nonneg _)],
exact (le_max_left _ _).trans (le_abs_self _),
exa... | instance | prod.cstar_ring | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"abs_of_nonneg",
"cstar_ring",
"le_abs_self",
"le_sup_iff",
"prod.fst_mul",
"prod.fst_star",
"prod.snd_mul",
"prod.snd_star",
"sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pi.cstar_ring : cstar_ring (Π i, R i) | { norm_star_mul_self := λ x,
begin
simp only [norm, pi.mul_apply, pi.star_apply, nnnorm_star_mul_self, ←sq],
norm_cast,
exact (finset.comp_sup_eq_sup_comp_of_is_total (λ x : nnreal, x ^ 2)
(λ x y h, by simpa only [sq] using mul_le_mul' h h) (by simp)).symm,
end } | instance | pi.cstar_ring | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"cstar_ring",
"finset.comp_sup_eq_sup_comp_of_is_total",
"mul_le_mul'",
"nnreal",
"pi.mul_apply",
"pi.star_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pi.cstar_ring' : cstar_ring (ι → R₁) | pi.cstar_ring | instance | pi.cstar_ring' | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"cstar_ring",
"pi.cstar_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_one [nontrivial E] : ‖(1 : E)‖ = 1 | begin
have : 0 < ‖(1 : E)‖ := norm_pos_iff.mpr one_ne_zero,
rw [←mul_left_inj' this.ne', ←norm_star_mul_self, mul_one, star_one, one_mul],
end | lemma | cstar_ring.norm_one | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"mul_one",
"nontrivial",
"one_mul",
"one_ne_zero",
"star_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_unitary [nontrivial E] (U : unitary E) : ‖(U : E)‖ = 1 | begin
rw [←sq_eq_sq (norm_nonneg _) zero_le_one, one_pow 2, sq, ←cstar_ring.norm_star_mul_self,
unitary.coe_star_mul_self, cstar_ring.norm_one],
end | lemma | cstar_ring.norm_coe_unitary | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"cstar_ring.norm_one",
"nontrivial",
"one_pow",
"unitary",
"unitary.coe_star_mul_self",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_mem_unitary [nontrivial E] {U : E} (hU : U ∈ unitary E) : ‖U‖ = 1 | norm_coe_unitary ⟨U, hU⟩ | lemma | cstar_ring.norm_of_mem_unitary | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"nontrivial",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_unitary_mul (U : unitary E) (A : E) : ‖(U : E) * A‖ = ‖A‖ | begin
nontriviality E,
refine le_antisymm _ _,
{ calc _ ≤ ‖(U : E)‖ * ‖A‖ : norm_mul_le _ _
... = ‖A‖ : by rw [norm_coe_unitary, one_mul] },
{ calc _ = ‖(U : E)⋆ * U * A‖ : by rw [unitary.coe_star_mul_self U, one_mul]
... ≤ ‖(U : E)⋆‖ * ‖(U : E) * A‖ : by { rw [mul_as... | lemma | cstar_ring.norm_coe_unitary_mul | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"mul_assoc",
"norm_mul_le",
"one_mul",
"unitary",
"unitary.coe_star_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_unitary_smul (U : unitary E) (A : E) : ‖U • A‖ = ‖A‖ | norm_coe_unitary_mul U A | lemma | cstar_ring.norm_unitary_smul | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mem_unitary_mul {U : E} (A : E) (hU : U ∈ unitary E) : ‖U * A‖ = ‖A‖ | norm_coe_unitary_mul ⟨U, hU⟩ A | lemma | cstar_ring.norm_mem_unitary_mul | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mul_coe_unitary (A : E) (U : unitary E) : ‖A * U‖ = ‖A‖ | calc _ = ‖((U : E)⋆ * A⋆)⋆‖ : by simp only [star_star, star_mul]
... = ‖(U : E)⋆ * A⋆‖ : by rw [norm_star]
... = ‖A⋆‖ : norm_mem_unitary_mul (star A) (unitary.star_mem U.prop)
... = ‖A‖ : norm_star _ | lemma | cstar_ring.norm_mul_coe_unitary | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"star_star",
"unitary",
"unitary.star_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mul_mem_unitary (A : E) {U : E} (hU : U ∈ unitary E) : ‖A * U‖ = ‖A‖ | norm_mul_coe_unitary A ⟨U, hU⟩ | lemma | cstar_ring.norm_mul_mem_unitary | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint.nnnorm_pow_two_pow [normed_ring E] [star_ring E]
[cstar_ring E] {x : E} (hx : is_self_adjoint x) (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ (2 ^ n) | begin
induction n with k hk,
{ simp only [pow_zero, pow_one] },
{ rw [pow_succ, pow_mul', sq],
nth_rewrite 0 ←(self_adjoint.mem_iff.mp hx),
rw [←star_pow, cstar_ring.nnnorm_star_mul_self, ←sq, hk, pow_mul'] },
end | lemma | is_self_adjoint.nnnorm_pow_two_pow | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"cstar_ring",
"cstar_ring.nnnorm_star_mul_self",
"is_self_adjoint",
"normed_ring",
"pow_mul'",
"pow_one",
"pow_succ",
"pow_zero",
"star_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_adjoint.nnnorm_pow_two_pow [normed_ring E] [star_ring E] [cstar_ring E]
(x : self_adjoint E) (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ (2 ^ n) | x.prop.nnnorm_pow_two_pow _ | lemma | self_adjoint.nnnorm_pow_two_pow | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"cstar_ring",
"normed_ring",
"self_adjoint",
"star_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
starₗᵢ : E ≃ₗᵢ⋆[𝕜] E | { map_smul' := star_smul,
norm_map' := norm_star,
.. star_add_equiv } | def | starₗᵢ | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"star_add_equiv"
] | `star` bundled as a linear isometric equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_starₗᵢ : (starₗᵢ 𝕜 : E → E) = star | rfl | lemma | coe_starₗᵢ | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"starₗᵢ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
starₗᵢ_apply {x : E} : starₗᵢ 𝕜 x = star x | rfl | lemma | starₗᵢ_apply | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"starₗᵢ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
starₗᵢ_to_continuous_linear_equiv :
(starₗᵢ 𝕜 : E ≃ₗᵢ⋆[𝕜] E).to_continuous_linear_equiv = (starL 𝕜 : E ≃L⋆[𝕜] E) | continuous_linear_equiv.ext rfl | lemma | starₗᵢ_to_continuous_linear_equiv | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"continuous_linear_equiv.ext",
"starL",
"starₗᵢ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_normed_algebra {𝕜 A : Type*} [normed_field 𝕜] [star_ring 𝕜]
[semi_normed_ring A] [star_ring A] [normed_algebra 𝕜 A] [star_module 𝕜 A]
(S : star_subalgebra 𝕜 A) : normed_algebra 𝕜 S | @normed_algebra.induced _ 𝕜 S A _ (subring_class.to_ring S) S.algebra _ _ _ S.subtype | instance | star_subalgebra.to_normed_algebra | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"normed_algebra",
"normed_algebra.induced",
"normed_field",
"semi_normed_ring",
"star_module",
"star_ring",
"star_subalgebra",
"subring_class.to_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cstar_ring {R A} [comm_ring R] [star_ring R] [normed_ring A]
[star_ring A] [cstar_ring A] [algebra R A] [star_module R A] (S : star_subalgebra R A) :
cstar_ring S | { norm_star_mul_self := λ x, @cstar_ring.norm_star_mul_self A _ _ _ x } | instance | star_subalgebra.to_cstar_ring | analysis.normed_space.star | src/analysis/normed_space/star/basic.lean | [
"analysis.normed.group.hom",
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry",
"algebra.star.self_adjoint",
"algebra.star.unitary",
"topology.algebra.star_subalgebra",
"topology.algebra.module.star"
] | [
"algebra",
"comm_ring",
"cstar_ring",
"normed_ring",
"star_module",
"star_ring",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
elemental_star_algebra.complex.normed_algebra (a : A) :
normed_algebra ℂ (elemental_star_algebra ℂ a) | infer_instance | instance | elemental_star_algebra.complex.normed_algebra | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"elemental_star_algebra",
"normed_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectrum_star_mul_self_of_is_star_normal :
spectrum ℂ (star a * a) ⊆ set.Icc (0 : ℂ) (‖star a * a‖) | begin
-- this instance should be found automatically, but without providing it Lean goes on a wild
-- goose chase when trying to apply `spectrum.gelfand_transform_eq`.
letI := elemental_star_algebra.complex.normed_algebra a,
unfreezingI { rcases subsingleton_or_nontrivial A },
{ simp only [spectrum.of_subsing... | lemma | spectrum_star_mul_self_of_is_star_normal | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"alg_hom.norm_apply_le_self",
"complex.eq_coe_norm_of_nonneg",
"continuous_map.spectrum_eq_range",
"elemental_star_algebra",
"elemental_star_algebra.complex.normed_algebra",
"map_mul",
"set.Icc",
"set.empty_subset",
"spectrum",
"spectrum.of_subsingleton",
"spectrum.subset_star_subalgebra",
"st... | This lemma is used in the proof of `star_subalgebra.is_unit_of_is_unit_of_is_star_normal`,
which in turn is the key to spectral permanence `star_subalgebra.spectrum_eq`, which is itself
necessary for the continuous functional calculus. Using the continuous functional calculus, this
lemma can be superseded by one that o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
elemental_star_algebra.is_unit_of_is_unit_of_is_star_normal (h : is_unit a) :
is_unit (⟨a, self_mem ℂ a⟩ : elemental_star_algebra ℂ a) | begin
/- Sketch of proof: Because `a` is normal, it suffices to prove that `star a * a` is invertible
in `elemental_star_algebra ℂ a`. For this it suffices to prove that it is sufficiently close to a
unit, namely `algebra_map ℂ _ ‖star a * a‖`, and in this case the required distance is
`‖star a * a‖`. So one mu... | lemma | elemental_star_algebra.is_unit_of_is_unit_of_is_star_normal | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"abs_norm",
"algebra_map",
"coe_coe",
"commute",
"complex.abs_of_real",
"complex.eq_coe_norm_of_nonneg",
"complex.norm_eq_abs",
"elemental_star_algebra",
"inv_inv",
"is_R_or_C.conj_of_real",
"is_R_or_C.star_def",
"is_self_adjoint",
"is_self_adjoint.spectral_radius_eq_nnnorm",
"is_unit",
... | This is the key lemma on the way to establishing spectral permanence for C⋆-algebras, which is
established in `star_subalgebra.spectrum_eq`. This lemma is superseded by
`star_subalgebra.coe_is_unit`, which does not require an `is_star_normal` hypothesis and holds for
any closed star subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_subalgebra.is_unit_coe_inv_mem {S : star_subalgebra ℂ A} (hS : is_closed (S : set A))
{x : A} (h : is_unit x) (hxS : x ∈ S) : ↑h.unit⁻¹ ∈ S | begin
have hx := h.star.mul h,
suffices this : (↑hx.unit⁻¹ : A) ∈ S,
{ rw [←one_mul (↑h.unit⁻¹ : A), ←hx.unit.inv_mul, mul_assoc, is_unit.unit_spec, mul_assoc,
h.mul_coe_inv, mul_one],
exact mul_mem this (star_mem hxS) },
refine le_of_is_closed_of_mem ℂ hS (mul_mem (star_mem hxS) hxS) _,
haveI := (i... | lemma | star_subalgebra.is_unit_coe_inv_mem | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"elemental_star_algebra",
"elemental_star_algebra.is_unit_of_is_unit_of_is_star_normal",
"is_closed",
"is_self_adjoint.star_mul_self",
"is_star_normal",
"is_unit",
"is_unit.unit_spec",
"left_inv_eq_right_inv",
"mul_assoc",
"mul_one",
"star_subalgebra"
] | For `x : A` which is invertible in `A`, the inverse lies in any unital C⋆-subalgebra `S`
containing `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_subalgebra.coe_is_unit {S : star_subalgebra ℂ A} (hS : is_closed (S : set A)) {x : S} :
is_unit (x : A) ↔ is_unit x | begin
refine ⟨λ hx, ⟨⟨x, ⟨(↑hx.unit⁻¹ : A), star_subalgebra.is_unit_coe_inv_mem hS hx x.prop⟩, _, _⟩,
rfl⟩, λ hx, hx.map S.subtype⟩,
exacts [subtype.coe_injective hx.mul_coe_inv, subtype.coe_injective hx.coe_inv_mul],
end | lemma | star_subalgebra.coe_is_unit | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"is_closed",
"is_unit",
"star_subalgebra",
"star_subalgebra.is_unit_coe_inv_mem",
"subtype.coe_injective"
] | For a unital C⋆-subalgebra `S` of `A` and `x : S`, if `↑x : A` is invertible in `A`, then
`x` is invertible in `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_subalgebra.mem_spectrum_iff {S : star_subalgebra ℂ A} (hS : is_closed (S : set A))
{x : S} {z : ℂ} : z ∈ spectrum ℂ x ↔ z ∈ spectrum ℂ (x : A) | not_iff_not.2 (star_subalgebra.coe_is_unit hS).symm | lemma | star_subalgebra.mem_spectrum_iff | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"is_closed",
"spectrum",
"star_subalgebra",
"star_subalgebra.coe_is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_subalgebra.spectrum_eq {S : star_subalgebra ℂ A} (hS : is_closed (S : set A)) (x : S) :
spectrum ℂ x = spectrum ℂ (x : A) | set.ext $ λ z, star_subalgebra.mem_spectrum_iff hS | lemma | star_subalgebra.spectrum_eq | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"is_closed",
"set.ext",
"spectrum",
"star_subalgebra",
"star_subalgebra.mem_spectrum_iff"
] | **Spectral permanence.** The spectrum of an element is invariant of the (closed)
`star_subalgebra` in which it is contained. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
elemental_star_algebra.character_space_to_spectrum (x : A)
(φ : character_space ℂ (elemental_star_algebra ℂ x)) : spectrum ℂ x | { val := φ ⟨x, self_mem ℂ x⟩,
property := by simpa only [star_subalgebra.spectrum_eq (elemental_star_algebra.is_closed ℂ x)
⟨x, self_mem ℂ x⟩] using alg_hom.apply_mem_spectrum φ (⟨x, self_mem ℂ x⟩) } | def | elemental_star_algebra.character_space_to_spectrum | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"alg_hom.apply_mem_spectrum",
"elemental_star_algebra",
"elemental_star_algebra.is_closed",
"spectrum",
"star_subalgebra.spectrum_eq"
] | The natural map from `character_space ℂ (elemental_star_algebra ℂ x)` to `spectrum ℂ x` given
by evaluating `φ` at `x`. This is essentially just evaluation of the `gelfand_transform` of `x`,
but because we want something in `spectrum ℂ x`, as opposed to
`spectrum ℂ ⟨x, elemental_star_algebra.self_mem ℂ x⟩` there is sli... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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