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supr_edist_ne_top_aux {ι : Type*} [finite ι] {α : ι → Type*} [Π i, pseudo_metric_space (α i)]
(f g : pi_Lp ∞ α) : (⨆ i, edist (f i) (g i)) ≠ ⊤ | begin
casesI nonempty_fintype ι,
obtain ⟨M, hM⟩ := fintype.exists_le (λ i, (⟨dist (f i) (g i), dist_nonneg⟩ : ℝ≥0)),
refine ne_of_lt ((supr_le $ λ i, _).trans_lt (@ennreal.coe_lt_top M)),
simp only [edist, pseudo_metric_space.edist_dist, ennreal.of_real_eq_coe_nnreal dist_nonneg],
exact_mod_cast hM i,
end | lemma | pi_Lp.supr_edist_ne_top_aux | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"dist_nonneg",
"ennreal.coe_lt_top",
"ennreal.of_real_eq_coe_nnreal",
"finite",
"fintype.exists_le",
"nonempty_fintype",
"pi_Lp",
"pseudo_metric_space",
"supr_le"
] | An auxiliary lemma used twice in the proof of `pi_Lp.pseudo_metric_aux` below. Not intended for
use outside this file. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_aux : pseudo_metric_space (pi_Lp p α) | pseudo_emetric_space.to_pseudo_metric_space_of_dist dist
(λ f g,
begin
unfreezingI { rcases p.dichotomy with (rfl | h) },
{ exact supr_edist_ne_top_aux f g },
{ rw edist_eq_sum (zero_lt_one.trans_le h),
exact ennreal.rpow_ne_top_of_nonneg (one_div_nonneg.2 (zero_le_one.trans h)) (ne_of_lt $
... | def | pi_Lp.pseudo_metric_aux | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"csupr_le",
"csupr_of_empty",
"dist_edist",
"dist_nonneg",
"edist_ne_top",
"ennreal.bot_eq_zero",
"ennreal.of_real_le_of_real",
"ennreal.rpow_ne_top_of_nonneg",
"ennreal.sum_lt_top",
"ennreal.to_real_le_of_le_of_real",
"ennreal.to_real_rpow",
"ennreal.to_real_sum",
"ennreal.zero_to_real",
... | Endowing the space `pi_Lp p α` with the `L^p` pseudometric structure. This definition is not
satisfactory, as it does not register the fact that the topology, the uniform structure, and the
bornology coincide with the product ones. Therefore, we do not register it as an instance. Using
this as a temporary pseudoemetric... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_with_equiv_aux : lipschitz_with 1 (pi_Lp.equiv p β) | begin
intros x y,
unfreezingI { rcases p.dichotomy with (rfl | h) },
{ simpa only [ennreal.coe_one, one_mul, edist_eq_supr, edist, finset.sup_le_iff,
finset.mem_univ, forall_true_left] using le_supr (λ i, edist (x i) (y i)), },
{ have cancel : p.to_real * (1/p.to_real) = 1 := mul_div_cancel' 1 (zero_lt_on... | lemma | pi_Lp.lipschitz_with_equiv_aux | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"bot_le",
"ennreal.coe_one",
"ennreal.rpow_le_rpow",
"ennreal.rpow_mul",
"finset.mem_univ",
"finset.sup_le_iff",
"forall_prop_of_true",
"forall_true_left",
"le_supr",
"lipschitz_with",
"mul_div_cancel'",
"one_div",
"one_mul",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with_equiv_aux :
antilipschitz_with ((fintype.card ι : ℝ≥0) ^ (1 / p).to_real) (pi_Lp.equiv p β) | begin
intros x y,
unfreezingI { rcases p.dichotomy with (rfl | h) },
{ simp only [edist_eq_supr, ennreal.div_top, ennreal.zero_to_real, nnreal.rpow_zero,
ennreal.coe_one, one_mul, supr_le_iff],
exact λ i, finset.le_sup (finset.mem_univ i), },
{ have pos : 0 < p.to_real := zero_lt_one.trans_le h,
h... | lemma | pi_Lp.antilipschitz_with_equiv_aux | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"antilipschitz_with",
"ennreal.coe_nat",
"ennreal.coe_one",
"ennreal.coe_rpow_of_nonneg",
"ennreal.div_top",
"ennreal.mul_rpow_of_nonneg",
"ennreal.one_to_real",
"ennreal.rpow_le_rpow",
"ennreal.rpow_one",
"ennreal.to_real_div",
"ennreal.zero_to_real",
"finset.card_univ",
"finset.le_sup",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aux_uniformity_eq :
𝓤 (pi_Lp p β) = 𝓤[Pi.uniform_space _] | begin
have A : uniform_inducing (pi_Lp.equiv p β) :=
(antilipschitz_with_equiv_aux p β).uniform_inducing
(lipschitz_with_equiv_aux p β).uniform_continuous,
have : (λ (x : pi_Lp p β × pi_Lp p β),
((pi_Lp.equiv p β) x.fst, (pi_Lp.equiv p β) x.snd)) = id,
by ext i; refl,
rw [← A.comap_uniformity, thi... | lemma | pi_Lp.aux_uniformity_eq | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"Pi.uniform_space",
"pi_Lp",
"pi_Lp.equiv",
"uniform_continuous",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aux_cobounded_eq :
cobounded (pi_Lp p α) = @cobounded _ pi.bornology | calc cobounded (pi_Lp p α) = comap (pi_Lp.equiv p α) (cobounded _) :
le_antisymm (antilipschitz_with_equiv_aux p α).tendsto_cobounded.le_comap
(lipschitz_with_equiv_aux p α).comap_cobounded_le
... = _ : comap_id | lemma | pi_Lp.aux_cobounded_eq | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space [Π i, uniform_space (β i)] : uniform_space (pi_Lp p β) | Pi.uniform_space _ | instance | pi_Lp.uniform_space | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"Pi.uniform_space",
"pi_Lp",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_equiv [Π i, uniform_space (β i)] :
uniform_continuous (pi_Lp.equiv p β) | uniform_continuous_id | lemma | pi_Lp.uniform_continuous_equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv",
"uniform_continuous",
"uniform_continuous_id",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_equiv_symm [Π i, uniform_space (β i)] :
uniform_continuous (pi_Lp.equiv p β).symm | uniform_continuous_id | lemma | pi_Lp.uniform_continuous_equiv_symm | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv",
"uniform_continuous",
"uniform_continuous_id",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_equiv [Π i, uniform_space (β i)] : continuous (pi_Lp.equiv p β) | continuous_id | lemma | pi_Lp.continuous_equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"continuous",
"continuous_id",
"pi_Lp.equiv",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_equiv_symm [Π i, uniform_space (β i)] : continuous (pi_Lp.equiv p β).symm | continuous_id | lemma | pi_Lp.continuous_equiv_symm | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"continuous",
"continuous_id",
"pi_Lp.equiv",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bornology [Π i, bornology (β i)] : bornology (pi_Lp p β) | pi.bornology | instance | pi_Lp.bornology | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"bornology",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq_sum {p : ℝ≥0∞} [fact (1 ≤ p)] {β : ι → Type*}
[Π i, pseudo_metric_space (β i)] (hp : p ≠ ∞) (x y : pi_Lp p β) :
nndist x y = (∑ i : ι, nndist (x i) (y i) ^ p.to_real) ^ (1 / p.to_real) | subtype.ext $ by { push_cast, exact dist_eq_sum (p.to_real_pos_iff_ne_top.mpr hp) _ _ } | lemma | pi_Lp.nndist_eq_sum | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"fact",
"pi_Lp",
"pseudo_metric_space",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq_supr {β : ι → Type*} [Π i, pseudo_metric_space (β i)] (x y : pi_Lp ∞ β) :
nndist x y = ⨆ i, nndist (x i) (y i) | subtype.ext $ by { push_cast, exact dist_eq_csupr _ _ } | lemma | pi_Lp.nndist_eq_supr | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"pseudo_metric_space",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_equiv [Π i, pseudo_emetric_space (β i)] :
lipschitz_with 1 (pi_Lp.equiv p β) | lipschitz_with_equiv_aux p β | lemma | pi_Lp.lipschitz_with_equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"lipschitz_with",
"pi_Lp.equiv",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with_equiv [Π i, pseudo_emetric_space (β i)] :
antilipschitz_with ((fintype.card ι : ℝ≥0) ^ (1 / p).to_real) (pi_Lp.equiv p β) | antilipschitz_with_equiv_aux p β | lemma | pi_Lp.antilipschitz_with_equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"antilipschitz_with",
"fintype.card",
"pi_Lp.equiv",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_equiv_isometry [Π i, pseudo_emetric_space (β i)] :
isometry (pi_Lp.equiv ∞ β) | λ x y, le_antisymm (by simpa only [ennreal.coe_one, one_mul] using lipschitz_with_equiv ∞ β x y)
(by simpa only [ennreal.div_top, ennreal.zero_to_real, nnreal.rpow_zero, ennreal.coe_one, one_mul]
using antilipschitz_with_equiv ∞ β x y) | lemma | pi_Lp.infty_equiv_isometry | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.coe_one",
"ennreal.div_top",
"ennreal.zero_to_real",
"isometry",
"nnreal.rpow_zero",
"one_mul",
"pi_Lp.equiv",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_add_comm_group [Π i, seminormed_add_comm_group (β i)] :
seminormed_add_comm_group (pi_Lp p β) | { dist_eq := λ x y,
begin
unfreezingI { rcases p.dichotomy with (rfl | h) },
{ simpa only [dist_eq_csupr, norm_eq_csupr, dist_eq_norm] },
{ have : p ≠ ∞, { intros hp, rw [hp, ennreal.top_to_real] at h, linarith,} ,
simpa only [dist_eq_sum (zero_lt_one.trans_le h), norm_eq_sum (zero_lt_one.trans_le h... | instance | pi_Lp.seminormed_add_comm_group | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.top_to_real",
"pi_Lp",
"seminormed_add_comm_group"
] | seminormed group instance on the product of finitely many normed groups, using the `L^p`
norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_comm_group [Π i, normed_add_comm_group (α i)] :
normed_add_comm_group (pi_Lp p α) | { ..pi_Lp.seminormed_add_comm_group p α } | instance | pi_Lp.normed_add_comm_group | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"normed_add_comm_group",
"pi_Lp",
"pi_Lp.seminormed_add_comm_group"
] | normed group instance on the product of finitely many normed groups, using the `L^p` norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_eq_sum {p : ℝ≥0∞} [fact (1 ≤ p)] {β : ι → Type*} (hp : p ≠ ∞)
[Π i, seminormed_add_comm_group (β i)] (f : pi_Lp p β) :
‖f‖₊ = (∑ i, ‖f i‖₊ ^ p.to_real) ^ (1 / p.to_real) | by { ext, simp [nnreal.coe_sum, norm_eq_sum (p.to_real_pos_iff_ne_top.mpr hp)] } | lemma | pi_Lp.nnnorm_eq_sum | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"fact",
"nnreal.coe_sum",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_eq_csupr {β : ι → Type*} [Π i, seminormed_add_comm_group (β i)] (f : pi_Lp ∞ β) :
‖f‖₊ = ⨆ i, ‖f i‖₊ | by { ext, simp [nnreal.coe_supr, norm_eq_csupr] } | lemma | pi_Lp.nnnorm_eq_csupr | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"nnreal.coe_supr",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_of_nat {p : ℝ≥0∞} [fact (1 ≤ p)] {β : ι → Type*}
[Π i, seminormed_add_comm_group (β i)] (n : ℕ) (h : p = n) (f : pi_Lp p β) :
‖f‖ = (∑ i, ‖f i‖ ^ n) ^ (1/(n : ℝ)) | begin
have := p.to_real_pos_iff_ne_top.mpr (ne_of_eq_of_ne h $ ennreal.nat_ne_top n),
simp only [one_div, h, real.rpow_nat_cast, ennreal.to_real_nat, eq_self_iff_true,
finset.sum_congr, norm_eq_sum this],
end | lemma | pi_Lp.norm_eq_of_nat | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.nat_ne_top",
"ennreal.to_real_nat",
"fact",
"one_div",
"pi_Lp",
"real.rpow_nat_cast",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_of_L2 {β : ι → Type*} [Π i, seminormed_add_comm_group (β i)] (x : pi_Lp 2 β) :
‖x‖ = sqrt (∑ (i : ι), ‖x i‖ ^ 2) | by { convert norm_eq_of_nat 2 (by norm_cast) _, rw sqrt_eq_rpow, norm_cast } | lemma | pi_Lp.norm_eq_of_L2 | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_eq_of_L2 {β : ι → Type*} [Π i, seminormed_add_comm_group (β i)] (x : pi_Lp 2 β) :
‖x‖₊ = nnreal.sqrt (∑ (i : ι), ‖x i‖₊ ^ 2) | subtype.ext $ by { push_cast, exact norm_eq_of_L2 x } | lemma | pi_Lp.nnnorm_eq_of_L2 | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"nnreal.sqrt",
"pi_Lp",
"seminormed_add_comm_group",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_eq_of_L2 (β : ι → Type*) [Π i, seminormed_add_comm_group (β i)] (x : pi_Lp 2 β) :
‖x‖ ^ 2 = ∑ (i : ι), ‖x i‖ ^ 2 | begin
suffices : ‖x‖₊ ^ 2 = ∑ (i : ι), ‖x i‖₊ ^ 2,
{ simpa only [nnreal.coe_sum] using congr_arg (coe : ℝ≥0 → ℝ) this },
rw [nnnorm_eq_of_L2, nnreal.sq_sqrt],
end | lemma | pi_Lp.norm_sq_eq_of_L2 | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"nnreal.coe_sum",
"nnreal.sq_sqrt",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_of_L2 {β : ι → Type*} [Π i, seminormed_add_comm_group (β i)] (x y : pi_Lp 2 β) :
dist x y = (∑ i, dist (x i) (y i) ^ 2).sqrt | by simp_rw [dist_eq_norm, norm_eq_of_L2, pi.sub_apply] | lemma | pi_Lp.dist_eq_of_L2 | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq_of_L2 {β : ι → Type*} [Π i, seminormed_add_comm_group (β i)] (x y : pi_Lp 2 β) :
nndist x y = (∑ i, nndist (x i) (y i) ^ 2).sqrt | subtype.ext $ by { push_cast, exact dist_eq_of_L2 _ _ } | lemma | pi_Lp.nndist_eq_of_L2 | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"seminormed_add_comm_group",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq_of_L2 {β : ι → Type*} [Π i, seminormed_add_comm_group (β i)] (x y : pi_Lp 2 β) :
edist x y = (∑ i, edist (x i) (y i) ^ 2) ^ (1 / 2 : ℝ) | by simp [pi_Lp.edist_eq_sum] | lemma | pi_Lp.edist_eq_of_L2 | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"pi_Lp.edist_eq_sum",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space [Π i, seminormed_add_comm_group (β i)]
[Π i, normed_space 𝕜 (β i)] : normed_space 𝕜 (pi_Lp p β) | { norm_smul_le := λ c f,
begin
unfreezingI { rcases p.dichotomy with (rfl | hp) },
{ letI : module 𝕜 (pi_Lp ∞ β) := pi.module ι β 𝕜,
suffices : ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊, { exact_mod_cast nnreal.coe_mono this.le },
simpa only [nnnorm_eq_csupr, nnreal.mul_supr, ←nnnorm_smul] },
{ have : p.to_rea... | instance | pi_Lp.normed_space | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"module",
"mul_div_cancel'",
"nnreal.coe_mono",
"nnreal.mul_supr",
"norm_smul",
"norm_smul_le",
"normed_space",
"pi.module",
"pi.smul_apply",
"pi_Lp",
"seminormed_add_comm_group"
] | The product of finitely many normed spaces is a normed space, with the `L^p` norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_scalar_tower [Π i, seminormed_add_comm_group (β i)]
[has_smul 𝕜 𝕜'] [Π i, normed_space 𝕜 (β i)] [Π i, normed_space 𝕜' (β i)]
[Π i, is_scalar_tower 𝕜 𝕜' (β i)] : is_scalar_tower 𝕜 𝕜' (pi_Lp p β) | pi.is_scalar_tower | instance | pi_Lp.is_scalar_tower | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"has_smul",
"is_scalar_tower",
"normed_space",
"pi.is_scalar_tower",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class [Π i, seminormed_add_comm_group (β i)]
[Π i, normed_space 𝕜 (β i)] [Π i, normed_space 𝕜' (β i)]
[Π i, smul_comm_class 𝕜 𝕜' (β i)] : smul_comm_class 𝕜 𝕜' (pi_Lp p β) | pi.smul_comm_class | instance | pi_Lp.smul_comm_class | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"normed_space",
"pi.smul_comm_class",
"pi_Lp",
"seminormed_add_comm_group",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_dimensional [Π i, seminormed_add_comm_group (β i)]
[Π i, normed_space 𝕜 (β i)] [I : ∀ i, finite_dimensional 𝕜 (β i)] :
finite_dimensional 𝕜 (pi_Lp p β) | finite_dimensional.finite_dimensional_pi' _ _ | instance | pi_Lp.finite_dimensional | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"finite_dimensional",
"finite_dimensional.finite_dimensional_pi'",
"normed_space",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply : (0 : pi_Lp p β) i = 0 | rfl | lemma | pi_Lp.zero_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply : (x + y) i = x i + y i | rfl | lemma | pi_Lp.add_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply : (x - y) i = x i - y i | rfl | lemma | pi_Lp.sub_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply : (c • x) i = c • x i | rfl | lemma | pi_Lp.smul_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply : (-x) i = - (x i) | rfl | lemma | pi_Lp.neg_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivₗᵢ : pi_Lp ∞ β ≃ₗᵢ[𝕜] Π i, β i | { map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
norm_map' := λ f,
begin
suffices : finset.univ.sup (λ i, ‖f i‖₊) = ⨆ i, ‖f i‖₊,
{ simpa only [nnreal.coe_supr] using congr_arg (coe : ℝ≥0 → ℝ) this },
refine antisymm (finset.sup_le (λ i _, le_csupr (fintype.bdd_above_range (λ i, ‖f i‖₊)) _)) _,
c... | def | pi_Lp.equivₗᵢ | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"csupr_le",
"csupr_of_empty",
"finset.le_sup",
"finset.mem_univ",
"finset.sup_empty",
"finset.univ_eq_empty",
"fintype.bdd_above_range",
"is_empty_or_nonempty",
"le_csupr",
"nnreal.coe_supr",
"pi_Lp",
"pi_Lp.equiv"
] | The canonical map `pi_Lp.equiv` between `pi_Lp ∞ β` and `Π i, β i` as a linear isometric
equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.linear_isometry_equiv.pi_Lp_congr_left (e : ι ≃ ι') :
pi_Lp p (λ i : ι, E) ≃ₗᵢ[𝕜] pi_Lp p (λ i : ι', E) | { to_linear_equiv := linear_equiv.Pi_congr_left' 𝕜 (λ i : ι, E) e,
norm_map' := λ x,
begin
unfreezingI { rcases p.dichotomy with (rfl | h) },
{ simp_rw [norm_eq_csupr, linear_equiv.Pi_congr_left'_apply 𝕜 (λ i : ι, E) e x _],
exact e.symm.supr_congr (λ i, rfl) },
{ simp only [norm_eq_sum (zero_lt... | def | linear_isometry_equiv.pi_Lp_congr_left | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"linear_equiv.Pi_congr_left'",
"pi_Lp"
] | An equivalence of finite domains induces a linearly isometric equivalence of finitely supported
functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.linear_isometry_equiv.pi_Lp_congr_left_apply
(e : ι ≃ ι') (v : pi_Lp p (λ i : ι, E)) :
linear_isometry_equiv.pi_Lp_congr_left p 𝕜 E e v = equiv.Pi_congr_left' (λ i : ι, E) e v | rfl | lemma | linear_isometry_equiv.pi_Lp_congr_left_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"equiv.Pi_congr_left'",
"linear_isometry_equiv.pi_Lp_congr_left",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_isometry_equiv.pi_Lp_congr_left_symm (e : ι ≃ ι') :
(linear_isometry_equiv.pi_Lp_congr_left p 𝕜 E e).symm
= (linear_isometry_equiv.pi_Lp_congr_left p 𝕜 E e.symm) | linear_isometry_equiv.ext $ λ x, rfl | lemma | linear_isometry_equiv.pi_Lp_congr_left_symm | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"linear_isometry_equiv.ext",
"linear_isometry_equiv.pi_Lp_congr_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_isometry_equiv.pi_Lp_congr_left_single
[decidable_eq ι] [decidable_eq ι'] (e : ι ≃ ι') (i : ι) (v : E) :
linear_isometry_equiv.pi_Lp_congr_left p 𝕜 E e (
(pi_Lp.equiv p (λ _, E)).symm $ pi.single i v) =
(pi_Lp.equiv p (λ _, E)).symm (pi.single (e i) v) | begin
funext x,
simp [linear_isometry_equiv.pi_Lp_congr_left, linear_equiv.Pi_congr_left', equiv.Pi_congr_left',
pi.single, function.update, equiv.symm_apply_eq],
end | lemma | linear_isometry_equiv.pi_Lp_congr_left_single | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"equiv.Pi_congr_left'",
"equiv.symm_apply_eq",
"linear_equiv.Pi_congr_left'",
"linear_isometry_equiv.pi_Lp_congr_left",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_zero : pi_Lp.equiv p β 0 = 0 | rfl | lemma | pi_Lp.equiv_zero | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm_zero : (pi_Lp.equiv p β).symm 0 = 0 | rfl | lemma | pi_Lp.equiv_symm_zero | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_add :
pi_Lp.equiv p β (x + y) = pi_Lp.equiv p β x + pi_Lp.equiv p β y | rfl | lemma | pi_Lp.equiv_add | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm_add :
(pi_Lp.equiv p β).symm (x' + y') = (pi_Lp.equiv p β).symm x' + (pi_Lp.equiv p β).symm y' | rfl | lemma | pi_Lp.equiv_symm_add | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_sub : pi_Lp.equiv p β (x - y) = pi_Lp.equiv p β x - pi_Lp.equiv p β y | rfl | lemma | pi_Lp.equiv_sub | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm_sub :
(pi_Lp.equiv p β).symm (x' - y') = (pi_Lp.equiv p β).symm x' - (pi_Lp.equiv p β).symm y' | rfl | lemma | pi_Lp.equiv_symm_sub | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_neg : pi_Lp.equiv p β (-x) = -pi_Lp.equiv p β x | rfl | lemma | pi_Lp.equiv_neg | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm_neg : (pi_Lp.equiv p β).symm (-x') = -(pi_Lp.equiv p β).symm x' | rfl | lemma | pi_Lp.equiv_symm_neg | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_smul : pi_Lp.equiv p β (c • x) = c • pi_Lp.equiv p β x | rfl | lemma | pi_Lp.equiv_smul | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm_smul :
(pi_Lp.equiv p β).symm (c • x') = c • (pi_Lp.equiv p β).symm x' | rfl | lemma | pi_Lp.equiv_symm_smul | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_equiv_symm_single (i : ι) (b : β i) :
‖(pi_Lp.equiv p β).symm (pi.single i b)‖₊ = ‖b‖₊ | begin
haveI : nonempty ι := ⟨i⟩,
unfreezingI { induction p using with_top.rec_top_coe },
{ simp_rw [nnnorm_eq_csupr, equiv_symm_apply],
refine csupr_eq_of_forall_le_of_forall_lt_exists_gt (λ j, _) (λ n hn, ⟨i, hn.trans_eq _⟩),
{ obtain rfl | hij := decidable.eq_or_ne i j,
{ rw pi.single_eq_same },
... | lemma | pi_Lp.nnnorm_equiv_symm_single | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"csupr_eq_of_forall_le_of_forall_lt_exists_gt",
"decidable.eq_or_ne",
"ennreal.coe_ne_top",
"ennreal.coe_to_real",
"mul_inv_cancel",
"nnreal.rpow_one",
"nnreal.zero_rpow",
"one_div",
"pi_Lp.equiv",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_equiv_symm_single (i : ι) (b : β i) :
‖(pi_Lp.equiv p β).symm (pi.single i b)‖ = ‖b‖ | congr_arg coe $ nnnorm_equiv_symm_single p β i b | lemma | pi_Lp.norm_equiv_symm_single | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_equiv_symm_single_same (i : ι) (b₁ b₂ : β i) :
nndist ((pi_Lp.equiv p β).symm (pi.single i b₁)) ((pi_Lp.equiv p β).symm (pi.single i b₂)) =
nndist b₁ b₂ | by rw [nndist_eq_nnnorm, nndist_eq_nnnorm, ←equiv_symm_sub, ←pi.single_sub,
nnnorm_equiv_symm_single] | lemma | pi_Lp.nndist_equiv_symm_single_same | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_equiv_symm_single_same (i : ι) (b₁ b₂ : β i) :
dist ((pi_Lp.equiv p β).symm (pi.single i b₁)) ((pi_Lp.equiv p β).symm (pi.single i b₂)) =
dist b₁ b₂ | congr_arg coe $ nndist_equiv_symm_single_same p β i b₁ b₂ | lemma | pi_Lp.dist_equiv_symm_single_same | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_equiv_symm_single_same (i : ι) (b₁ b₂ : β i) :
edist ((pi_Lp.equiv p β).symm (pi.single i b₁)) ((pi_Lp.equiv p β).symm (pi.single i b₂)) =
edist b₁ b₂ | by simpa only [edist_nndist] using congr_arg coe (nndist_equiv_symm_single_same p β i b₁ b₂) | lemma | pi_Lp.edist_equiv_symm_single_same | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"edist_nndist",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_equiv_symm_const {β} [seminormed_add_comm_group β] (hp : p ≠ ∞) (b : β) :
‖(pi_Lp.equiv p (λ _ : ι, β)).symm (function.const _ b)‖₊=
fintype.card ι ^ (1 / p).to_real * ‖b‖₊ | begin
rcases p.dichotomy with (h | h),
{ exact false.elim (hp h) },
{ have ne_zero : p.to_real ≠ 0 := (zero_lt_one.trans_le h).ne',
simp_rw [nnnorm_eq_sum hp, equiv_symm_apply, function.const_apply, finset.sum_const,
finset.card_univ, nsmul_eq_mul, nnreal.mul_rpow, ←nnreal.rpow_mul, mul_one_div_cancel n... | lemma | pi_Lp.nnnorm_equiv_symm_const | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.one_to_real",
"ennreal.to_real_div",
"finset.card_univ",
"fintype.card",
"function.const_apply",
"mul_one_div_cancel",
"ne_zero",
"nnreal.mul_rpow",
"nnreal.rpow_one",
"nsmul_eq_mul",
"pi_Lp.equiv",
"seminormed_add_comm_group"
] | When `p = ∞`, this lemma does not hold without the additional assumption `nonempty ι` because
the left-hand side simplifies to `0`, while the right-hand side simplifies to `‖b‖₊`. See
`pi_Lp.nnnorm_equiv_symm_const'` for a version which exchanges the hypothesis `p ≠ ∞` for
`nonempty ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_equiv_symm_const' {β} [seminormed_add_comm_group β] [nonempty ι] (b : β) :
‖(pi_Lp.equiv p (λ _ : ι, β)).symm (function.const _ b)‖₊=
fintype.card ι ^ (1 / p).to_real * ‖b‖₊ | begin
unfreezingI { rcases (em $ p = ∞) with (rfl | hp) },
{ simp only [equiv_symm_apply, ennreal.div_top, ennreal.zero_to_real, nnreal.rpow_zero, one_mul,
nnnorm_eq_csupr, function.const_apply, csupr_const], },
{ exact nnnorm_equiv_symm_const hp b, },
end | lemma | pi_Lp.nnnorm_equiv_symm_const' | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"csupr_const",
"em",
"ennreal.div_top",
"ennreal.zero_to_real",
"fintype.card",
"function.const_apply",
"nnreal.rpow_zero",
"one_mul",
"pi_Lp.equiv",
"seminormed_add_comm_group"
] | When `is_empty ι`, this lemma does not hold without the additional assumption `p ≠ ∞` because
the left-hand side simplifies to `0`, while the right-hand side simplifies to `‖b‖₊`. See
`pi_Lp.nnnorm_equiv_symm_const` for a version which exchanges the hypothesis `nonempty ι`.
for `p ≠ ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_equiv_symm_const {β} [seminormed_add_comm_group β] (hp : p ≠ ∞) (b : β) :
‖(pi_Lp.equiv p (λ _ : ι, β)).symm (function.const _ b)‖ =
fintype.card ι ^ (1 / p).to_real * ‖b‖ | (congr_arg coe $ nnnorm_equiv_symm_const hp b).trans $ by simp | lemma | pi_Lp.norm_equiv_symm_const | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"fintype.card",
"pi_Lp.equiv",
"seminormed_add_comm_group"
] | When `p = ∞`, this lemma does not hold without the additional assumption `nonempty ι` because
the left-hand side simplifies to `0`, while the right-hand side simplifies to `‖b‖₊`. See
`pi_Lp.norm_equiv_symm_const'` for a version which exchanges the hypothesis `p ≠ ∞` for
`nonempty ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_equiv_symm_const' {β} [seminormed_add_comm_group β] [nonempty ι] (b : β) :
‖(pi_Lp.equiv p (λ _ : ι, β)).symm (function.const _ b)‖ =
fintype.card ι ^ (1 / p).to_real * ‖b‖ | (congr_arg coe $ nnnorm_equiv_symm_const' b).trans $ by simp | lemma | pi_Lp.norm_equiv_symm_const' | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"fintype.card",
"pi_Lp.equiv",
"seminormed_add_comm_group"
] | When `is_empty ι`, this lemma does not hold without the additional assumption `p ≠ ∞` because
the left-hand side simplifies to `0`, while the right-hand side simplifies to `‖b‖₊`. See
`pi_Lp.norm_equiv_symm_const` for a version which exchanges the hypothesis `nonempty ι`.
for `p ≠ ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_equiv_symm_one {β} [seminormed_add_comm_group β] (hp : p ≠ ∞) [has_one β] :
‖(pi_Lp.equiv p (λ _ : ι, β)).symm 1‖₊ = fintype.card ι ^ (1 / p).to_real * ‖(1 : β)‖₊ | (nnnorm_equiv_symm_const hp (1 : β)).trans rfl | lemma | pi_Lp.nnnorm_equiv_symm_one | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"fintype.card",
"pi_Lp.equiv",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_equiv_symm_one {β} [seminormed_add_comm_group β] (hp : p ≠ ∞) [has_one β] :
‖(pi_Lp.equiv p (λ _ : ι, β)).symm 1‖ = fintype.card ι ^ (1 / p).to_real * ‖(1 : β)‖ | (norm_equiv_symm_const hp (1 : β)).trans rfl | lemma | pi_Lp.norm_equiv_symm_one | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"fintype.card",
"pi_Lp.equiv",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv : pi_Lp p β ≃ₗ[𝕜] Π i, β i | { to_fun := pi_Lp.equiv _ _,
inv_fun := (pi_Lp.equiv _ _).symm,
..linear_equiv.refl _ _} | def | pi_Lp.linear_equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"inv_fun",
"linear_equiv",
"linear_equiv.refl",
"pi_Lp",
"pi_Lp.equiv"
] | `pi_Lp.equiv` as a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv : pi_Lp p β ≃L[𝕜] Π i, β i | { to_linear_equiv := pi_Lp.linear_equiv _ _ _,
continuous_to_fun := continuous_equiv _ _,
continuous_inv_fun := continuous_equiv_symm _ _ } | def | pi_Lp.continuous_linear_equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"continuous_linear_equiv",
"pi_Lp",
"pi_Lp.linear_equiv"
] | `pi_Lp.equiv` as a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_fun : basis ι 𝕜 (pi_Lp p (λ _, 𝕜)) | basis.of_equiv_fun (pi_Lp.linear_equiv p 𝕜 (λ _ : ι, 𝕜)) | def | pi_Lp.basis_fun | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"basis",
"basis.of_equiv_fun",
"pi_Lp",
"pi_Lp.linear_equiv"
] | A version of `pi.basis_fun` for `pi_Lp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_fun_apply [decidable_eq ι] (i) :
basis_fun p 𝕜 ι i = (pi_Lp.equiv p _).symm (pi.single i 1) | by simp_rw [basis_fun, basis.coe_of_equiv_fun, pi_Lp.linear_equiv_symm_apply, pi.single] | lemma | pi_Lp.basis_fun_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"basis.coe_of_equiv_fun",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_fun_repr (x : pi_Lp p (λ i : ι, 𝕜)) (i : ι) :
(basis_fun p 𝕜 ι).repr x i = x i | rfl | lemma | pi_Lp.basis_fun_repr | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_fun_equiv_fun :
(basis_fun p 𝕜 ι).equiv_fun = pi_Lp.linear_equiv p 𝕜 (λ _ : ι, 𝕜) | basis.equiv_fun_of_equiv_fun _ | lemma | pi_Lp.basis_fun_equiv_fun | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"basis.equiv_fun_of_equiv_fun",
"pi_Lp.linear_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_fun_eq_pi_basis_fun :
basis_fun p 𝕜 ι = (pi.basis_fun 𝕜 ι).map (pi_Lp.linear_equiv p 𝕜 (λ _ : ι, 𝕜)).symm | rfl | lemma | pi_Lp.basis_fun_eq_pi_basis_fun | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi.basis_fun",
"pi_Lp.linear_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_fun_map :
(basis_fun p 𝕜 ι).map (pi_Lp.linear_equiv p 𝕜 (λ _ : ι, 𝕜)) = pi.basis_fun 𝕜 ι | rfl | lemma | pi_Lp.basis_fun_map | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi.basis_fun",
"pi_Lp.linear_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_to_matrix_basis_fun_mul (b : basis ι 𝕜 (pi_Lp p (λ i : ι, 𝕜))) (A : matrix ι ι 𝕜) :
b.to_matrix (pi_Lp.basis_fun _ _ _) ⬝ A =
matrix.of (λ i j, b.repr ((pi_Lp.equiv _ _).symm (Aᵀ j)) i) | begin
have := basis_to_matrix_basis_fun_mul (b.map (pi_Lp.linear_equiv _ 𝕜 _)) A,
simp_rw [←pi_Lp.basis_fun_map p, basis.map_repr, linear_equiv.trans_apply,
pi_Lp.linear_equiv_symm_apply, basis.to_matrix_map, function.comp, basis.map_apply,
linear_equiv.symm_apply_apply] at this,
exact this,
end | lemma | pi_Lp.basis_to_matrix_basis_fun_mul | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"basis",
"basis.map_apply",
"basis.to_matrix_map",
"basis_to_matrix_basis_fun_mul",
"linear_equiv.symm_apply_apply",
"linear_equiv.trans_apply",
"matrix",
"matrix.of",
"pi_Lp",
"pi_Lp.basis_fun",
"pi_Lp.equiv",
"pi_Lp.linear_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_smul_le (c : 𝕜) (s : set E) :
emetric.diam (c • s) ≤ ‖c‖₊ • emetric.diam s | (lipschitz_with_smul c).ediam_image_le s | lemma | ediam_smul_le | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"emetric.diam",
"lipschitz_with_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_smul₀ (c : 𝕜) (s : set E) :
emetric.diam (c • s) = ‖c‖₊ • emetric.diam s | begin
refine le_antisymm (ediam_smul_le c s) _,
obtain rfl | hc := eq_or_ne c 0,
{ obtain rfl | hs := s.eq_empty_or_nonempty,
{ simp },
simp [zero_smul_set hs, ←set.singleton_zero], },
{ have := (lipschitz_with_smul c⁻¹).ediam_image_le (c • s),
rwa [← smul_eq_mul, ←ennreal.smul_def, set.image_smul, ... | lemma | ediam_smul₀ | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ediam_smul_le",
"emetric.diam",
"ennreal.le_inv_smul_iff",
"eq_or_ne",
"inv_smul_smul₀",
"lipschitz_with_smul",
"nnnorm_inv",
"set.image_smul",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_smul₀ (c : 𝕜) (x : set E) : diam (c • x) = ‖c‖ * diam x | by simp_rw [diam, ediam_smul₀, ennreal.to_real_smul, nnreal.smul_def, coe_nnnorm, smul_eq_mul] | lemma | diam_smul₀ | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ediam_smul₀",
"ennreal.to_real_smul",
"nnreal.smul_def",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : set E) (x : E) :
emetric.inf_edist (c • x) (c • s) = ‖c‖₊ • emetric.inf_edist x s | begin
simp_rw [emetric.inf_edist],
have : function.surjective ((•) c : E → E) :=
function.right_inverse.surjective (smul_inv_smul₀ hc),
transitivity ⨅ y (H : y ∈ s), ‖c‖₊ • edist x y,
{ refine (this.infi_congr _ $ λ y, _).symm,
simp_rw [smul_mem_smul_set_iff₀ hc, edist_smul₀] },
{ have : (‖c‖₊ : ennre... | lemma | inf_edist_smul₀ | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"edist_smul₀",
"emetric.inf_edist",
"ennreal",
"ennreal.coe_ne_top",
"ennreal.mul_infi_of_ne",
"ennreal.smul_def",
"smul_eq_mul",
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_dist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : set E) (x : E) :
metric.inf_dist (c • x) (c • s) = ‖c‖ * metric.inf_dist x s | by simp_rw [metric.inf_dist, inf_edist_smul₀ hc, ennreal.to_real_smul, nnreal.smul_def, coe_nnnorm,
smul_eq_mul] | lemma | inf_dist_smul₀ | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ennreal.to_real_smul",
"inf_edist_smul₀",
"metric.inf_dist",
"nnreal.smul_def",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
c • ball x r = ball (c • x) (‖c‖ * r) | begin
ext y,
rw mem_smul_set_iff_inv_smul_mem₀ hc,
conv_lhs { rw ←inv_smul_smul₀ hc x },
simp [← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀],
end | theorem | smul_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"dist_smul₀",
"div_eq_inv_mul",
"div_lt_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_unit_ball {c : 𝕜} (hc : c ≠ 0) : c • ball (0 : E) (1 : ℝ) = ball (0 : E) (‖c‖) | by rw [smul_ball hc, smul_zero, mul_one] | lemma | smul_unit_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"mul_one",
"smul_ball",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sphere' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
c • sphere x r = sphere (c • x) (‖c‖ * r) | begin
ext y,
rw mem_smul_set_iff_inv_smul_mem₀ hc,
conv_lhs { rw ←inv_smul_smul₀ hc x },
simp only [mem_sphere, dist_smul₀, norm_inv, ← div_eq_inv_mul,
div_eq_iff (norm_pos_iff.2 hc).ne', mul_comm r],
end | theorem | smul_sphere' | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"dist_smul₀",
"div_eq_iff",
"div_eq_inv_mul",
"mul_comm",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
c • closed_ball x r = closed_ball (c • x) (‖c‖ * r) | by simp only [← ball_union_sphere, set.smul_set_union, smul_ball hc, smul_sphere' hc] | theorem | smul_closed_ball' | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"set.smul_set_union",
"smul_ball",
"smul_sphere'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.smul {s : set E} (hs : bounded s) (c : 𝕜) :
bounded (c • s) | begin
obtain ⟨R, hR⟩ : ∃ (R : ℝ), ∀ x ∈ s, ‖x‖ ≤ R := hs.exists_norm_le,
refine bounded_iff_forall_norm_le.2 ⟨‖c‖ * R, λ z hz, _⟩,
obtain ⟨y, ys, rfl⟩ : ∃ (y : E), y ∈ s ∧ c • y = z := mem_smul_set.1 hz,
calc ‖c • y‖ = ‖c‖ * ‖y‖ : norm_smul _ _
... ≤ ‖c‖ * R : mul_le_mul_of_nonneg_left (hR y ys) (norm_nonneg ... | lemma | metric.bounded.smul | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"mul_le_mul_of_nonneg_left",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_singleton_add_smul_subset
{x : E} {s : set E} (hs : bounded s) {u : set E} (hu : u ∈ 𝓝 x) :
∀ᶠ r in 𝓝 (0 : 𝕜), {x} + r • s ⊆ u | begin
obtain ⟨ε, εpos, hε⟩ : ∃ ε (hε : 0 < ε), closed_ball x ε ⊆ u :=
nhds_basis_closed_ball.mem_iff.1 hu,
obtain ⟨R, Rpos, hR⟩ : ∃ (R : ℝ), 0 < R ∧ s ⊆ closed_ball 0 R := hs.subset_ball_lt 0 0,
have : metric.closed_ball (0 : 𝕜) (ε / R) ∈ 𝓝 (0 : 𝕜) :=
closed_ball_mem_nhds _ (div_pos εpos Rpos),
filte... | lemma | eventually_singleton_add_smul_subset | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"div_pos",
"metric.closed_ball",
"mul_le_mul",
"norm_smul"
] | If `s` is a bounded set, then for small enough `r`, the set `{x} + r • s` is contained in any
fixed neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_unit_ball_of_pos {r : ℝ} (hr : 0 < r) : r • ball 0 1 = ball (0 : E) r | by rw [smul_unit_ball hr.ne', real.norm_of_nonneg hr.le] | lemma | smul_unit_ball_of_pos | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"real.norm_of_nonneg",
"smul_unit_ball"
] | In a real normed space, the image of the unit ball under scalar multiplication by a positive
constant `r` is the ball of radius `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_dist_eq (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z | begin
use a • x + b • z,
nth_rewrite 0 [←one_smul ℝ x],
nth_rewrite 3 [←one_smul ℝ z],
simp [dist_eq_norm, ←hab, add_smul, ←smul_sub, norm_smul_of_nonneg, ha, hb],
end | lemma | exists_dist_eq | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"add_smul",
"norm_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dist_le_le (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ) :
∃ y, dist x y ≤ δ ∧ dist y z ≤ ε | begin
obtain rfl | hε' := hε.eq_or_lt,
{ exact ⟨z, by rwa zero_add at h, (dist_self _).le⟩ },
have hεδ := add_pos_of_pos_of_nonneg hε' hδ,
refine (exists_dist_eq x z (div_nonneg hε $ add_nonneg hε hδ) (div_nonneg hδ $ add_nonneg hε hδ) $
by rw [←add_div, div_self hεδ.ne']).imp (λ y hy, _),
rw [hy.1, hy.2,... | lemma | exists_dist_le_le | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"dist_self",
"div_mul_comm",
"div_nonneg",
"div_self",
"exists_dist_eq",
"mul_le_of_le_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dist_le_lt (hδ : 0 ≤ δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
∃ y, dist x y ≤ δ ∧ dist y z < ε | begin
refine (exists_dist_eq x z (div_nonneg hε.le $ add_nonneg hε.le hδ) (div_nonneg hδ $ add_nonneg
hε.le hδ) $ by rw [←add_div, div_self (add_pos_of_pos_of_nonneg hε hδ).ne']).imp (λ y hy, _),
rw [hy.1, hy.2, div_mul_comm, div_mul_comm ε],
rw ←div_lt_one (add_pos_of_pos_of_nonneg hε hδ) at h,
exact ⟨mul_... | lemma | exists_dist_le_lt | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"div_mul_comm",
"div_nonneg",
"div_self",
"exists_dist_eq",
"mul_lt_of_lt_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dist_lt_le (hδ : 0 < δ) (hε : 0 ≤ ε) (h : dist x z < ε + δ) :
∃ y, dist x y < δ ∧ dist y z ≤ ε | begin
obtain ⟨y, yz, xy⟩ := exists_dist_le_lt hε hδ
(show dist z x < δ + ε, by simpa only [dist_comm, add_comm] using h),
exact ⟨y, by simp [dist_comm x y, dist_comm y z, *]⟩,
end | lemma | exists_dist_lt_le | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"dist_comm",
"exists_dist_le_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dist_lt_lt (hδ : 0 < δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
∃ y, dist x y < δ ∧ dist y z < ε | begin
refine (exists_dist_eq x z (div_nonneg hε.le $ add_nonneg hε.le hδ.le) (div_nonneg hδ.le $
add_nonneg hε.le hδ.le) $ by rw [←add_div, div_self (add_pos hε hδ).ne']).imp (λ y hy, _),
rw [hy.1, hy.2, div_mul_comm, div_mul_comm ε],
rw ←div_lt_one (add_pos hε hδ) at h,
exact ⟨mul_lt_of_lt_one_left hδ h, m... | lemma | exists_dist_lt_lt | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"div_mul_comm",
"div_nonneg",
"div_self",
"exists_dist_eq",
"mul_lt_of_lt_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_ball_ball_iff (hδ : 0 < δ) (hε : 0 < ε) :
disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y | begin
refine ⟨λ h, le_of_not_lt $ λ hxy, _, ball_disjoint_ball⟩,
rw add_comm at hxy,
obtain ⟨z, hxz, hzy⟩ := exists_dist_lt_lt hδ hε hxy,
rw dist_comm at hxz,
exact h.le_bot ⟨hxz, hzy⟩,
end | lemma | disjoint_ball_ball_iff | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"disjoint",
"dist_comm",
"exists_dist_lt_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_ball_closed_ball_iff (hδ : 0 < δ) (hε : 0 ≤ ε) :
disjoint (ball x δ) (closed_ball y ε) ↔ δ + ε ≤ dist x y | begin
refine ⟨λ h, le_of_not_lt $ λ hxy, _, ball_disjoint_closed_ball⟩,
rw add_comm at hxy,
obtain ⟨z, hxz, hzy⟩ := exists_dist_lt_le hδ hε hxy,
rw dist_comm at hxz,
exact h.le_bot ⟨hxz, hzy⟩,
end | lemma | disjoint_ball_closed_ball_iff | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"disjoint",
"dist_comm",
"exists_dist_lt_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_closed_ball_ball_iff (hδ : 0 ≤ δ) (hε : 0 < ε) :
disjoint (closed_ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y | by rw [disjoint.comm, disjoint_ball_closed_ball_iff hε hδ, add_comm, dist_comm]; apply_instance | lemma | disjoint_closed_ball_ball_iff | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"disjoint",
"disjoint.comm",
"disjoint_ball_closed_ball_iff",
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_closed_ball_closed_ball_iff (hδ : 0 ≤ δ) (hε : 0 ≤ ε) :
disjoint (closed_ball x δ) (closed_ball y ε) ↔ δ + ε < dist x y | begin
refine ⟨λ h, lt_of_not_ge $ λ hxy, _, closed_ball_disjoint_closed_ball⟩,
rw add_comm at hxy,
obtain ⟨z, hxz, hzy⟩ := exists_dist_le_le hδ hε hxy,
rw dist_comm at hxz,
exact h.le_bot ⟨hxz, hzy⟩,
end | lemma | disjoint_closed_ball_closed_ball_iff | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"disjoint",
"dist_comm",
"exists_dist_le_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_thickening (hδ : 0 < δ) (s : set E) (x : E) :
inf_edist x (thickening δ s) = inf_edist x s - ennreal.of_real δ | begin
obtain hs | hs := lt_or_le (inf_edist x s) (ennreal.of_real δ),
{ rw [inf_edist_zero_of_mem, tsub_eq_zero_of_le hs.le], exact hs },
refine (tsub_le_iff_right.2 inf_edist_le_inf_edist_thickening_add).antisymm' _,
refine le_sub_of_add_le_right of_real_ne_top _,
refine le_inf_edist.2 (λ z hz, le_of_forall_... | lemma | inf_edist_thickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"antisymm'",
"edist_lt_coe",
"ennreal.add_lt_add_right",
"ennreal.of_real",
"exists_dist_lt_lt",
"le_of_forall_lt'",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_thickening (hε : 0 < ε) (hδ : 0 < δ) (s : set E) :
thickening ε (thickening δ s) = thickening (ε + δ) s | (thickening_thickening_subset _ _ _).antisymm $ λ x, begin
simp_rw mem_thickening_iff,
rintro ⟨z, hz, hxz⟩,
rw add_comm at hxz,
obtain ⟨y, hxy, hyz⟩ := exists_dist_lt_lt hε hδ hxz,
exact ⟨y, ⟨_, hz, hyz⟩, hxy⟩,
end | lemma | thickening_thickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"exists_dist_lt_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_thickening (hε : 0 ≤ ε) (hδ : 0 < δ) (s : set E) :
cthickening ε (thickening δ s) = cthickening (ε + δ) s | (cthickening_thickening_subset hε _ _).antisymm $ λ x, begin
simp_rw [mem_cthickening_iff, ennreal.of_real_add hε hδ.le, inf_edist_thickening hδ],
exact tsub_le_iff_right.2,
end | lemma | cthickening_thickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ennreal.of_real_add",
"inf_edist_thickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_thickening (hδ : 0 < δ) (s : set E) :
closure (thickening δ s) = cthickening δ s | by { rw [←cthickening_zero, cthickening_thickening le_rfl hδ, zero_add], apply_instance } | lemma | closure_thickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"closure",
"cthickening_thickening",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_cthickening (δ : ℝ) (s : set E) (x : E) :
inf_edist x (cthickening δ s) = inf_edist x s - ennreal.of_real δ | begin
obtain hδ | hδ := le_or_lt δ 0,
{ rw [cthickening_of_nonpos hδ, inf_edist_closure, of_real_of_nonpos hδ, tsub_zero] },
{ rw [←closure_thickening hδ, inf_edist_closure, inf_edist_thickening hδ]; apply_instance }
end | lemma | inf_edist_cthickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ennreal.of_real",
"inf_edist_thickening",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_cthickening (hε : 0 < ε) (hδ : 0 ≤ δ) (s : set E) :
thickening ε (cthickening δ s) = thickening (ε + δ) s | begin
obtain rfl | hδ := hδ.eq_or_lt,
{ rw [cthickening_zero, thickening_closure, add_zero] },
{ rw [←closure_thickening hδ, thickening_closure, thickening_thickening hε hδ]; apply_instance }
end | lemma | thickening_cthickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"thickening_thickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_cthickening (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (s : set E) :
cthickening ε (cthickening δ s) = cthickening (ε + δ) s | (cthickening_cthickening_subset hε hδ _).antisymm $ λ x, begin
simp_rw [mem_cthickening_iff, ennreal.of_real_add hε hδ, inf_edist_cthickening],
exact tsub_le_iff_right.2,
end | lemma | cthickening_cthickening | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ennreal.of_real_add",
"inf_edist_cthickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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