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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
mem_ℓp (f : lp E p) : mem_ℓp f p | f.prop | lemma | lp.mem_ℓp | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_zero : ⇑(0 : lp E p) = 0 | rfl | lemma | lp.coe_fn_zero | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_neg (f : lp E p) : ⇑(-f) = -f | rfl | lemma | lp.coe_fn_neg | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_add (f g : lp E p) : ⇑(f + g) = f + g | rfl | lemma | lp.coe_fn_add | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_sum {ι : Type*} (f : ι → lp E p) (s : finset ι) :
⇑(∑ i in s, f i) = ∑ i in s, ⇑(f i) | begin
classical,
refine finset.induction _ _ s,
{ simp },
intros i s his,
simp [finset.sum_insert his],
end | lemma | lp.coe_fn_sum | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"finset",
"finset.induction",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_sub (f g : lp E p) : ⇑(f - g) = f - g | rfl | lemma | lp.coe_fn_sub | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_card_dsupport (f : lp E 0) : ‖f‖ = (lp.mem_ℓp f).finite_dsupport.to_finset.card | dif_pos rfl | lemma | lp.norm_eq_card_dsupport | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_csupr (f : lp E ∞) : ‖f‖ = ⨆ i, ‖f i‖ | begin
dsimp [norm],
rw [dif_neg ennreal.top_ne_zero, if_pos rfl]
end | lemma | lp.norm_eq_csupr | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"ennreal.top_ne_zero",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_norm [nonempty α] (f : lp E ∞) : is_lub (set.range (λ i, ‖f i‖)) ‖f‖ | begin
rw lp.norm_eq_csupr,
exact is_lub_csupr (lp.mem_ℓp f)
end | lemma | lp.is_lub_norm | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"is_lub",
"is_lub_csupr",
"lp",
"lp.mem_ℓp",
"lp.norm_eq_csupr",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_tsum_rpow (hp : 0 < p.to_real) (f : lp E p) :
‖f‖ = (∑' i, ‖f i‖ ^ p.to_real) ^ (1/p.to_real) | begin
dsimp [norm],
rw ennreal.to_real_pos_iff at hp,
rw [dif_neg hp.1.ne', if_neg hp.2.ne],
end | lemma | lp.norm_eq_tsum_rpow | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"ennreal.to_real_pos_iff",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_rpow_eq_tsum (hp : 0 < p.to_real) (f : lp E p) :
‖f‖ ^ p.to_real = ∑' i, ‖f i‖ ^ p.to_real | begin
rw [norm_eq_tsum_rpow hp, ← real.rpow_mul],
{ field_simp [hp.ne'] },
apply tsum_nonneg,
intros i,
calc (0:ℝ) = 0 ^ p.to_real : by rw real.zero_rpow hp.ne'
... ≤ _ : real.rpow_le_rpow rfl.le (norm_nonneg (f i)) hp.le
end | lemma | lp.norm_rpow_eq_tsum | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"real.rpow_le_rpow",
"real.rpow_mul",
"real.zero_rpow",
"tsum_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_norm (hp : 0 < p.to_real) (f : lp E p) :
has_sum (λ i, ‖f i‖ ^ p.to_real) (‖f‖ ^ p.to_real) | begin
rw norm_rpow_eq_tsum hp,
exact ((lp.mem_ℓp f).summable hp).has_sum
end | lemma | lp.has_sum_norm | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"has_sum",
"lp",
"lp.mem_ℓp",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_nonneg' (f : lp E p) : 0 ≤ ‖f‖ | begin
rcases p.trichotomy with rfl | rfl | hp,
{ simp [lp.norm_eq_card_dsupport f] },
{ cases is_empty_or_nonempty α with _i _i; resetI,
{ rw lp.norm_eq_csupr,
simp [real.csupr_empty] },
inhabit α,
exact (norm_nonneg (f default)).trans ((lp.is_lub_norm f).1 ⟨default, rfl⟩) },
{ rw lp.norm_eq_t... | lemma | lp.norm_nonneg' | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"is_empty_or_nonempty",
"lp",
"lp.is_lub_norm",
"lp.norm_eq_card_dsupport",
"lp.norm_eq_csupr",
"lp.norm_eq_tsum_rpow",
"norm_nonneg'",
"real.csupr_empty",
"real.rpow_nonneg_of_nonneg",
"tsum_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_zero : ‖(0 : lp E p)‖ = 0 | begin
rcases p.trichotomy with rfl | rfl | hp,
{ simp [lp.norm_eq_card_dsupport] },
{ simp [lp.norm_eq_csupr] },
{ rw lp.norm_eq_tsum_rpow hp,
have hp' : 1 / p.to_real ≠ 0 := one_div_ne_zero hp.ne',
simpa [real.zero_rpow hp.ne'] using real.zero_rpow hp' }
end | lemma | lp.norm_zero | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.norm_eq_card_dsupport",
"lp.norm_eq_csupr",
"lp.norm_eq_tsum_rpow",
"one_div_ne_zero",
"real.zero_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_zero_iff {f : lp E p} : ‖f‖ = 0 ↔ f = 0 | begin
classical,
refine ⟨λ h, _, by { rintros rfl, exact norm_zero }⟩,
rcases p.trichotomy with rfl | rfl | hp,
{ ext i,
have : {i : α | ¬f i = 0} = ∅ := by simpa [lp.norm_eq_card_dsupport f] using h,
have : (¬ (f i = 0)) = false := congr_fun this i,
tauto },
{ cases is_empty_or_nonempty α with _i... | lemma | lp.norm_eq_zero_iff | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"has_sum",
"has_sum_zero_iff_of_nonneg",
"is_empty_or_nonempty",
"is_lub",
"lp",
"lp.has_sum_norm",
"lp.is_lub_norm",
"lp.norm_eq_card_dsupport",
"real.rpow_eq_zero_iff_of_nonneg",
"real.rpow_nonneg_of_nonneg",
"real.zero_rpow",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff_coe_fn_eq_zero {f : lp E p} : f = 0 ↔ ⇑f = 0 | by rw [lp.ext_iff, coe_fn_zero] | lemma | lp.eq_zero_iff_coe_fn_eq_zero | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_neg ⦃f : lp E p⦄ : ‖-f‖ = ‖f‖ | begin
rcases p.trichotomy with rfl | rfl | hp,
{ simp [lp.norm_eq_card_dsupport] },
{ cases is_empty_or_nonempty α; resetI,
{ simp [lp.eq_zero' f], },
apply (lp.is_lub_norm (-f)).unique,
simpa using lp.is_lub_norm f },
{ suffices : ‖-f‖ ^ p.to_real = ‖f‖ ^ p.to_real,
{ exact real.rpow_left_inj_o... | lemma | lp.norm_neg | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"is_empty_or_nonempty",
"lp",
"lp.eq_zero'",
"lp.has_sum_norm",
"lp.is_lub_norm",
"lp.norm_eq_card_dsupport",
"norm_nonneg'",
"real.rpow_left_inj_on",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_le_mul_norm {p q : ℝ≥0∞}
(hpq : p.to_real.is_conjugate_exponent q.to_real) (f : lp E p) (g : lp E q) :
summable (λ i, ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ | begin
have hf₁ : ∀ i, 0 ≤ ‖f i‖ := λ i, norm_nonneg _,
have hg₁ : ∀ i, 0 ≤ ‖g i‖ := λ i, norm_nonneg _,
have hf₂ := lp.has_sum_norm hpq.pos f,
have hg₂ := lp.has_sum_norm hpq.symm.pos g,
obtain ⟨C, -, hC', hC⟩ :=
real.inner_le_Lp_mul_Lq_has_sum_of_nonneg hpq (norm_nonneg' _) (norm_nonneg' _) hf₁ hg₁ hf₂ h... | lemma | lp.tsum_mul_le_mul_norm | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.has_sum_norm",
"norm_nonneg'",
"real.inner_le_Lp_mul_Lq_has_sum_of_nonneg",
"summable"
] | Hölder inequality | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_mul {p q : ℝ≥0∞}
(hpq : p.to_real.is_conjugate_exponent q.to_real) (f : lp E p) (g : lp E q) :
summable (λ i, ‖f i‖ * ‖g i‖) | (lp.tsum_mul_le_mul_norm hpq f g).1 | lemma | lp.summable_mul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.tsum_mul_le_mul_norm",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_le_mul_norm' {p q : ℝ≥0∞}
(hpq : p.to_real.is_conjugate_exponent q.to_real) (f : lp E p) (g : lp E q) :
∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ | (lp.tsum_mul_le_mul_norm hpq f g).2 | lemma | lp.tsum_mul_le_mul_norm' | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.tsum_mul_le_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_apply_le_norm (hp : p ≠ 0) (f : lp E p) (i : α) : ‖f i‖ ≤ ‖f‖ | begin
rcases eq_or_ne p ∞ with rfl | hp',
{ haveI : nonempty α := ⟨i⟩,
exact (is_lub_norm f).1 ⟨i, rfl⟩ },
have hp'' : 0 < p.to_real := ennreal.to_real_pos hp hp',
have : ∀ i, 0 ≤ ‖f i‖ ^ p.to_real,
{ exact λ i, real.rpow_nonneg_of_nonneg (norm_nonneg _) _ },
rw ← real.rpow_le_rpow_iff (norm_nonneg _) (... | lemma | lp.norm_apply_le_norm | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"ennreal.to_real_pos",
"eq_or_ne",
"le_has_sum",
"lp",
"norm_nonneg'",
"real.rpow_le_rpow_iff",
"real.rpow_nonneg_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_rpow_le_norm_rpow (hp : 0 < p.to_real) (f : lp E p) (s : finset α) :
∑ i in s, ‖f i‖ ^ p.to_real ≤ ‖f‖ ^ p.to_real | begin
rw lp.norm_rpow_eq_tsum hp f,
have : ∀ i, 0 ≤ ‖f i‖ ^ p.to_real,
{ exact λ i, real.rpow_nonneg_of_nonneg (norm_nonneg _) _ },
refine sum_le_tsum _ (λ i hi, this i) _,
exact (lp.mem_ℓp f).summable hp
end | lemma | lp.sum_rpow_le_norm_rpow | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"finset",
"lp",
"lp.mem_ℓp",
"lp.norm_rpow_eq_tsum",
"real.rpow_nonneg_of_nonneg",
"sum_le_tsum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_of_forall_le' [nonempty α] {f : lp E ∞} (C : ℝ) (hCf : ∀ i, ‖f i‖ ≤ C) : ‖f‖ ≤ C | begin
refine (is_lub_norm f).2 _,
rintros - ⟨i, rfl⟩,
exact hCf i,
end | lemma | lp.norm_le_of_forall_le' | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_of_forall_le {f : lp E ∞} {C : ℝ} (hC : 0 ≤ C) (hCf : ∀ i, ‖f i‖ ≤ C) : ‖f‖ ≤ C | begin
casesI is_empty_or_nonempty α,
{ simpa [eq_zero' f] using hC, },
{ exact norm_le_of_forall_le' C hCf },
end | lemma | lp.norm_le_of_forall_le | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"is_empty_or_nonempty",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_of_tsum_le (hp : 0 < p.to_real) {C : ℝ} (hC : 0 ≤ C) {f : lp E p}
(hf : ∑' i, ‖f i‖ ^ p.to_real ≤ C ^ p.to_real) :
‖f‖ ≤ C | begin
rw [← real.rpow_le_rpow_iff (norm_nonneg' _) hC hp, norm_rpow_eq_tsum hp],
exact hf,
end | lemma | lp.norm_le_of_tsum_le | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"norm_nonneg'",
"real.rpow_le_rpow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_of_forall_sum_le (hp : 0 < p.to_real) {C : ℝ} (hC : 0 ≤ C) {f : lp E p}
(hf : ∀ s : finset α, ∑ i in s, ‖f i‖ ^ p.to_real ≤ C ^ p.to_real) :
‖f‖ ≤ C | norm_le_of_tsum_le hp hC (tsum_le_of_sum_le ((lp.mem_ℓp f).summable hp) hf) | lemma | lp.norm_le_of_forall_sum_le | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"finset",
"lp",
"lp.mem_ℓp",
"summable",
"tsum_le_of_sum_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_lp_const_smul (c : 𝕜) (f : lp E p) : c • (f : pre_lp E) ∈ lp E p | (lp.mem_ℓp f).const_smul c | lemma | lp.mem_lp_const_smul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.mem_ℓp",
"pre_lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lp_submodule : submodule 𝕜 (pre_lp E) | { smul_mem' := λ c f hf, by simpa using mem_lp_const_smul c ⟨f, hf⟩,
.. lp E p } | def | lp_submodule | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"pre_lp",
"submodule"
] | The `𝕜`-submodule of elements of `Π i : α, E i` whose `lp` norm is finite. This is `lp E p`,
with extra structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lp_submodule : (lp_submodule E p 𝕜).to_add_subgroup = lp E p | rfl | lemma | lp.coe_lp_submodule | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_smul (c : 𝕜) (f : lp E p) : ⇑(c • f) = c • f | rfl | lemma | lp.coe_fn_smul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_const_smul_le (hp : p ≠ 0) (c : 𝕜) (f : lp E p) : ‖c • f‖ ≤ ‖c‖ * ‖f‖ | begin
rcases p.trichotomy with rfl | rfl | hp,
{ exact absurd rfl hp },
{ cases is_empty_or_nonempty α; resetI,
{ simp [lp.eq_zero' f], },
have hcf := lp.is_lub_norm (c • f),
have hfc := (lp.is_lub_norm f).mul_left (norm_nonneg c),
simp_rw [←set.range_comp, function.comp] at hfc,
-- TODO: some... | lemma | lp.norm_const_smul_le | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"ennreal.to_real_nonneg",
"forall_apply_eq_imp_iff'",
"forall_exists_index",
"has_nnnorm",
"has_sum_mono",
"is_empty_or_nonempty",
"lp",
"lp.eq_zero'",
"lp.has_sum_norm",
"lp.is_lub_norm",
"mem_upper_bounds",
"nnnorm_smul_le",
"nnreal.has_sum_coe",
"nnreal.mul_rpow",
"nnreal.rpow_le_rpow... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_const_smul (hp : p ≠ 0) {c : 𝕜} (f : lp E p) : ‖c • f‖ = ‖c‖ * ‖f‖ | begin
obtain rfl | hc := eq_or_ne c 0,
{ simp },
refine le_antisymm (norm_const_smul_le hp c f) _,
have := mul_le_mul_of_nonneg_left (norm_const_smul_le hp c⁻¹ (c • f)) (norm_nonneg c),
rwa [inv_smul_smul₀ hc, norm_inv, mul_inv_cancel_left₀ (norm_ne_zero_iff.mpr hc)] at this,
end | lemma | lp.norm_const_smul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"eq_or_ne",
"inv_smul_smul₀",
"lp",
"mul_inv_cancel_left₀",
"mul_le_mul_of_nonneg_left",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mem_ℓp.star_mem {f : Π i, E i}
(hf : mem_ℓp f p) : mem_ℓp (star f) p | begin
rcases p.trichotomy with rfl | rfl | hp,
{ apply mem_ℓp_zero,
simp [hf.finite_dsupport] },
{ apply mem_ℓp_infty,
simpa using hf.bdd_above },
{ apply mem_ℓp_gen,
simpa using hf.summable hp },
end | lemma | mem_ℓp.star_mem | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"mem_ℓp",
"mem_ℓp_gen",
"mem_ℓp_infty",
"mem_ℓp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mem_ℓp.star_iff {f : Π i, E i} : mem_ℓp (star f) p ↔ mem_ℓp f p | ⟨λ h, star_star f ▸ mem_ℓp.star_mem h ,mem_ℓp.star_mem⟩ | lemma | mem_ℓp.star_iff | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"mem_ℓp",
"mem_ℓp.star_mem",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_star (f : lp E p) : ⇑(star f) = star f | rfl | lemma | lp.coe_fn_star | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_apply (f : lp E p) (i : α) : star f i = star (f i) | rfl | theorem | lp.star_apply | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mem_ℓp.infty_mul {f g : Π i, B i} (hf : mem_ℓp f ∞) (hg : mem_ℓp g ∞) :
mem_ℓp (f * g) ∞ | begin
rw mem_ℓp_infty_iff,
obtain ⟨⟨Cf, hCf⟩, ⟨Cg, hCg⟩⟩ := ⟨hf.bdd_above, hg.bdd_above⟩,
refine ⟨Cf * Cg, _⟩,
rintros _ ⟨i, rfl⟩,
calc ‖(f * g) i‖ ≤ ‖f i‖ * ‖g i‖ : norm_mul_le (f i) (g i)
... ≤ Cf * Cg : mul_le_mul (hCf ⟨i, rfl⟩) (hCg ⟨i, rfl⟩) (norm_nonneg _)
... | lemma | mem_ℓp.infty_mul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"mem_ℓp",
"mem_ℓp_infty_iff",
"mul_le_mul",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_coe_fn_mul (f g : lp B ∞) : ⇑(f * g) = f * g | rfl | lemma | lp.infty_coe_fn_mul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_is_scalar_tower
{𝕜} [normed_ring 𝕜] [Π i, module 𝕜 (B i)] [∀ i, has_bounded_smul 𝕜 (B i)]
[Π i, is_scalar_tower 𝕜 (B i) (B i)] :
is_scalar_tower 𝕜 (lp B ∞) (lp B ∞) | ⟨λ r f g, lp.ext $ smul_assoc r ⇑f ⇑g⟩ | instance | lp.infty_is_scalar_tower | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"has_bounded_smul",
"is_scalar_tower",
"lp",
"lp.ext",
"module",
"normed_ring",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_smul_comm_class
{𝕜} [normed_ring 𝕜] [Π i, module 𝕜 (B i)] [∀ i, has_bounded_smul 𝕜 (B i)]
[Π i, smul_comm_class 𝕜 (B i) (B i)] :
smul_comm_class 𝕜 (lp B ∞) (lp B ∞) | ⟨λ r f g, lp.ext $ smul_comm r ⇑f ⇑g⟩ | instance | lp.infty_smul_comm_class | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"has_bounded_smul",
"lp",
"lp.ext",
"module",
"normed_ring",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_star_ring : star_ring (lp B ∞) | { star_mul := λ f g, ext $ star_mul (_ : Π i, B i) _,
.. (show star_add_monoid (lp B ∞),
by { letI : Π i, star_add_monoid (B i) := λ i, infer_instance, apply_instance }) } | instance | lp.infty_star_ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"star_add_monoid",
"star_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_cstar_ring [∀ i, cstar_ring (B i)] : cstar_ring (lp B ∞) | { norm_star_mul_self := λ f,
begin
apply le_antisymm,
{ rw ←sq,
refine lp.norm_le_of_forall_le (sq_nonneg ‖ f ‖) (λ i, _),
simp only [lp.star_apply, cstar_ring.norm_star_mul_self, ←sq, infty_coe_fn_mul, pi.mul_apply],
refine sq_le_sq' _ (lp.norm_apply_le_norm ennreal.top_ne_zero _ _),
... | instance | lp.infty_cstar_ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"cstar_ring",
"ennreal.top_ne_zero",
"lp",
"lp.norm_apply_le_norm",
"lp.norm_le_of_forall_le",
"lp.star_apply",
"pi.mul_apply",
"real.le_sqrt",
"sq_le_sq'",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pre_lp.ring : ring (pre_lp B) | pi.ring | instance | pre_lp.ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"pi.ring",
"pre_lp",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.one_mem_ℓp_infty : mem_ℓp (1 : Π i, B i) ∞ | ⟨1, by { rintros i ⟨i, rfl⟩, exact norm_one.le,}⟩ | lemma | one_mem_ℓp_infty | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lp_infty_subring : subring (pre_lp B) | { carrier := {f | mem_ℓp f ∞},
one_mem' := one_mem_ℓp_infty,
mul_mem' := λ f g hf hg, hf.infty_mul hg,
.. lp B ∞ } | def | lp_infty_subring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"mem_ℓp",
"one_mem_ℓp_infty",
"pre_lp",
"subring"
] | The `𝕜`-subring of elements of `Π i : α, B i` whose `lp` norm is finite. This is `lp E ∞`,
with extra structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infty_ring : ring (lp B ∞) | (lp_infty_subring B).to_ring | instance | lp.infty_ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp_infty_subring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mem_ℓp.infty_pow {f : Π i, B i} (hf : mem_ℓp f ∞) (n : ℕ) : mem_ℓp (f ^ n) ∞ | (lp_infty_subring B).pow_mem hf n | lemma | mem_ℓp.infty_pow | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp_infty_subring",
"mem_ℓp",
"pow_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.nat_cast_mem_ℓp_infty (n : ℕ) : mem_ℓp (n : Π i, B i) ∞ | nat_cast_mem (lp_infty_subring B) n | lemma | nat_cast_mem_ℓp_infty | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp_infty_subring",
"mem_ℓp",
"nat_cast_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.int_cast_mem_ℓp_infty (z : ℤ) : mem_ℓp (z : Π i, B i) ∞ | coe_int_mem (lp_infty_subring B) z | lemma | int_cast_mem_ℓp_infty | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"coe_int_mem",
"lp_infty_subring",
"mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_coe_fn_one : ⇑(1 : lp B ∞) = 1 | rfl | lemma | lp.infty_coe_fn_one | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_coe_fn_pow (f : lp B ∞) (n : ℕ) : ⇑(f ^ n) = f ^ n | rfl | lemma | lp.infty_coe_fn_pow | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_coe_fn_nat_cast (n : ℕ) : ⇑(n : lp B ∞) = n | rfl | lemma | lp.infty_coe_fn_nat_cast | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_coe_fn_int_cast (z : ℤ) : ⇑(z : lp B ∞) = z | rfl | lemma | lp.infty_coe_fn_int_cast | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_normed_ring : normed_ring (lp B ∞) | { .. lp.infty_ring, .. lp.non_unital_normed_ring } | instance | lp.infty_normed_ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.infty_ring",
"normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_comm_ring : comm_ring (lp B ∞) | { mul_comm := λ f g, by { ext, simp only [lp.infty_coe_fn_mul, pi.mul_apply, mul_comm] },
.. lp.infty_ring } | instance | lp.infty_comm_ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"comm_ring",
"lp",
"lp.infty_coe_fn_mul",
"lp.infty_ring",
"mul_comm",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infty_normed_comm_ring : normed_comm_ring (lp B ∞) | { .. lp.infty_comm_ring, .. lp.infty_normed_ring } | instance | lp.infty_normed_comm_ring | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp",
"lp.infty_comm_ring",
"lp.infty_normed_ring",
"normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pi.algebra_of_normed_algebra : algebra 𝕜 (Π i, B i) | @pi.algebra I 𝕜 B _ _ $ λ i, normed_algebra.to_algebra | instance | pi.algebra_of_normed_algebra | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"algebra",
"pi.algebra"
] | A variant of `pi.algebra` that lean can't find otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.pre_lp.algebra : algebra 𝕜 (pre_lp B) | _root_.pi.algebra_of_normed_algebra | instance | pre_lp.algebra | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"algebra",
"pre_lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.algebra_map_mem_ℓp_infty (k : 𝕜) : mem_ℓp (algebra_map 𝕜 (Π i, B i) k) ∞ | begin
rw algebra.algebra_map_eq_smul_one,
exact (one_mem_ℓp_infty.const_smul k : mem_ℓp (k • 1 : Π i, B i) ∞)
end | lemma | algebra_map_mem_ℓp_infty | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lp_infty_subalgebra : subalgebra 𝕜 (pre_lp B) | { carrier := {f | mem_ℓp f ∞},
algebra_map_mem' := algebra_map_mem_ℓp_infty,
.. lp_infty_subring B } | def | lp_infty_subalgebra | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"algebra_map_mem_ℓp_infty",
"lp_infty_subring",
"mem_ℓp",
"pre_lp",
"subalgebra"
] | The `𝕜`-subalgebra of elements of `Π i : α, B i` whose `lp` norm is finite. This is `lp E ∞`,
with extra structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infty_normed_algebra : normed_algebra 𝕜 (lp B ∞) | { ..(lp_infty_subalgebra 𝕜 B).algebra,
..(lp.normed_space : normed_space 𝕜 (lp B ∞)) } | instance | lp.infty_normed_algebra | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"algebra",
"lp",
"lp_infty_subalgebra",
"normed_algebra",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single (p) (i : α) (a : E i) : lp E p | ⟨ λ j, if h : j = i then eq.rec a h.symm else 0,
begin
refine (mem_ℓp_zero _).of_exponent_ge (zero_le p),
refine (set.finite_singleton i).subset _,
intros j,
simp only [forall_exists_index, set.mem_singleton_iff, ne.def, dite_eq_right_iff,
set.mem_set_of_eq, not_forall],
rintros rfl,
sim... | def | lp.single | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"dite_eq_right_iff",
"forall_exists_index",
"lp",
"mem_ℓp_zero",
"not_forall",
"set.finite_singleton",
"set.mem_singleton_iff"
] | The element of `lp E p` which is `a : E i` at the index `i`, and zero elsewhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single_apply (p) (i : α) (a : E i) (j : α) :
lp.single p i a j = if h : j = i then eq.rec a h.symm else 0 | rfl | lemma | lp.single_apply | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_apply_self (p) (i : α) (a : E i) :
lp.single p i a i = a | by rw [lp.single_apply, dif_pos rfl] | lemma | lp.single_apply_self | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp.single",
"lp.single_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_apply_ne (p) (i : α) (a : E i) {j : α} (hij : j ≠ i) :
lp.single p i a j = 0 | by rw [lp.single_apply, dif_neg hij] | lemma | lp.single_apply_ne | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp.single",
"lp.single_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_neg (p) (i : α) (a : E i) :
lp.single p i (- a) = - lp.single p i a | begin
ext j,
by_cases hi : j = i,
{ subst hi,
simp [lp.single_apply_self] },
{ simp [lp.single_apply_ne p i _ hi] }
end | lemma | lp.single_neg | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp.single",
"lp.single_apply_ne",
"lp.single_apply_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_smul (p) (i : α) (a : E i) (c : 𝕜) :
lp.single p i (c • a) = c • lp.single p i a | begin
ext j,
by_cases hi : j = i,
{ subst hi,
simp [lp.single_apply_self] },
{ simp [lp.single_apply_ne p i _ hi] }
end | lemma | lp.single_smul | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp.single",
"lp.single_apply_ne",
"lp.single_apply_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sum_single (hp : 0 < p.to_real) (f : Π i, E i) (s : finset α) :
‖∑ i in s, lp.single p i (f i)‖ ^ p.to_real = ∑ i in s, ‖f i‖ ^ p.to_real | begin
refine (has_sum_norm hp (∑ i in s, lp.single p i (f i))).unique _,
simp only [lp.single_apply, coe_fn_sum, finset.sum_apply, finset.sum_dite_eq],
have h : ∀ i ∉ s, ‖ite (i ∈ s) (f i) 0‖ ^ p.to_real = 0,
{ intros i hi,
simp [if_neg hi, real.zero_rpow hp.ne'], },
have h' : ∀ i ∈ s, ‖f i‖ ^ p.to_real =... | lemma | lp.norm_sum_single | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"finset",
"has_sum_sum_of_ne_finset_zero",
"lp.single",
"lp.single_apply",
"real.zero_rpow",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_single (hp : 0 < p.to_real) (f : Π i, E i) (i : α) :
‖lp.single p i (f i)‖ = ‖f i‖ | begin
refine real.rpow_left_inj_on hp.ne' (norm_nonneg' _) (norm_nonneg _) _,
simpa using lp.norm_sum_single hp f {i},
end | lemma | lp.norm_single | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"lp.norm_sum_single",
"norm_nonneg'",
"real.rpow_left_inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sub_norm_compl_sub_single (hp : 0 < p.to_real) (f : lp E p) (s : finset α) :
‖f‖ ^ p.to_real - ‖f - ∑ i in s, lp.single p i (f i)‖ ^ p.to_real = ∑ i in s, ‖f i‖ ^ p.to_real | begin
refine ((has_sum_norm hp f).sub (has_sum_norm hp (f - ∑ i in s, lp.single p i (f i)))).unique _,
let F : α → ℝ := λ i, ‖f i‖ ^ p.to_real - ‖(f - ∑ i in s, lp.single p i (f i)) i‖ ^ p.to_real,
have hF : ∀ i ∉ s, F i = 0,
{ intros i hi,
suffices : ‖f i‖ ^ p.to_real - ‖f i - ite (i ∈ s) (f i) 0‖ ^ p.to_r... | lemma | lp.norm_sub_norm_compl_sub_single | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"finset",
"has_sum",
"has_sum_sum_of_ne_finset_zero",
"lp",
"lp.single",
"lp.single_apply",
"real.zero_rpow",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_compl_sum_single (hp : 0 < p.to_real) (f : lp E p) (s : finset α) :
‖f - ∑ i in s, lp.single p i (f i)‖ ^ p.to_real = ‖f‖ ^ p.to_real - ∑ i in s, ‖f i‖ ^ p.to_real | by linarith [lp.norm_sub_norm_compl_sub_single hp f s] | lemma | lp.norm_compl_sum_single | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"finset",
"lp",
"lp.norm_sub_norm_compl_sub_single",
"lp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_single [fact (1 ≤ p)] (hp : p ≠ ⊤) (f : lp E p) :
has_sum (λ i : α, lp.single p i (f i : E i)) f | begin
have hp₀ : 0 < p := zero_lt_one.trans_le (fact.out _),
have hp' : 0 < p.to_real := ennreal.to_real_pos hp₀.ne' hp,
have := lp.has_sum_norm hp' f,
rw [has_sum, metric.tendsto_nhds] at this ⊢,
intros ε hε,
refine (this _ (real.rpow_pos_of_pos hε p.to_real)).mono _,
intros s hs,
rw ← real.rpow_lt_rpo... | lemma | lp.has_sum_single | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"abs_norm",
"dist_comm",
"dist_nonneg",
"ennreal.to_real_pos",
"fact",
"has_sum",
"has_sum_single",
"lp",
"lp.has_sum_norm",
"lp.norm_compl_sum_single",
"lp.single",
"lp.single_neg",
"metric.tendsto_nhds",
"real.abs_rpow_of_nonneg",
"real.norm_eq_abs",
"real.rpow_lt_rpow_iff",
"real.... | The canonical finitely-supported approximations to an element `f` of `lp` converge to it, in the
`lp` topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_coe [_i : fact (1 ≤ p)] : uniform_continuous (coe : lp E p → Π i, E i) | begin
have hp : p ≠ 0 := (zero_lt_one.trans_le _i.elim).ne',
rw uniform_continuous_pi,
intros i,
rw normed_add_comm_group.uniformity_basis_dist.uniform_continuous_iff
normed_add_comm_group.uniformity_basis_dist,
intros ε hε,
refine ⟨ε, hε, _⟩,
rintros f g (hfg : ‖f - g‖ < ε),
have : ‖f i - g i‖ ≤ ‖f... | lemma | lp.uniform_continuous_coe | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"fact",
"lp",
"uniform_continuous",
"uniform_continuous_pi"
] | The coercion from `lp E p` to `Π i, E i` is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_apply_le_of_tendsto {C : ℝ} {F : ι → lp E ∞} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C)
{f : Π a, E a} (hf : tendsto (id (λ i, F i) : ι → Π a, E a) l (𝓝 f)) (a : α) :
‖f a‖ ≤ C | begin
have : tendsto (λ k, ‖F k a‖) l (𝓝 ‖f a‖) :=
(tendsto.comp (continuous_apply a).continuous_at hf).norm,
refine le_of_tendsto this (hCF.mono _),
intros k hCFk,
exact (norm_apply_le_norm ennreal.top_ne_zero (F k) a).trans hCFk,
end | lemma | lp.norm_apply_le_of_tendsto | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"continuous_apply",
"continuous_at",
"ennreal.top_ne_zero",
"le_of_tendsto",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_rpow_le_of_tendsto (hp : p ≠ ∞) {C : ℝ} {F : ι → lp E p} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C)
{f : Π a, E a} (hf : tendsto (id (λ i, F i) : ι → Π a, E a) l (𝓝 f)) (s : finset α) :
∑ (i : α) in s, ‖f i‖ ^ p.to_real ≤ C ^ p.to_real | begin
have hp' : p ≠ 0 := (zero_lt_one.trans_le _i.elim).ne',
have hp'' : 0 < p.to_real := ennreal.to_real_pos hp' hp,
let G : (Π a, E a) → ℝ := λ f, ∑ a in s, ‖f a‖ ^ p.to_real,
have hG : continuous G,
{ refine continuous_finset_sum s _,
intros a ha,
have : continuous (λ f : Π a, E a, f a):= continuo... | lemma | lp.sum_rpow_le_of_tendsto | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"continuous",
"continuous_apply",
"ennreal.to_real_pos",
"finset",
"le_of_tendsto",
"lp",
"lp.sum_rpow_le_norm_rpow",
"real.rpow_le_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_of_tendsto {C : ℝ} {F : ι → lp E p} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C) {f : lp E p}
(hf : tendsto (id (λ i, F i) : ι → Π a, E a) l (𝓝 f)) :
‖f‖ ≤ C | begin
obtain ⟨i, hi⟩ := hCF.exists,
have hC : 0 ≤ C := (norm_nonneg _).trans hi,
unfreezingI { rcases eq_top_or_lt_top p with rfl | hp },
{ apply norm_le_of_forall_le hC,
exact norm_apply_le_of_tendsto hCF hf, },
{ have : 0 < p := zero_lt_one.trans_le _i.elim,
have hp' : 0 < p.to_real := ennreal.to_re... | lemma | lp.norm_le_of_tendsto | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"ennreal.to_real_pos",
"eq_top_or_lt_top",
"lp"
] | "Semicontinuity of the `lp` norm": If all sufficiently large elements of a sequence in `lp E p`
have `lp` norm `≤ C`, then the pointwise limit, if it exists, also has `lp` norm `≤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ℓp_of_tendsto {F : ι → lp E p} (hF : metric.bounded (set.range F)) {f : Π a, E a}
(hf : tendsto (id (λ i, F i) : ι → Π a, E a) l (𝓝 f)) :
mem_ℓp f p | begin
obtain ⟨C, hC, hCF'⟩ := hF.exists_pos_norm_le,
have hCF : ∀ k, ‖F k‖ ≤ C := λ k, hCF' _ ⟨k, rfl⟩,
unfreezingI { rcases eq_top_or_lt_top p with rfl | hp },
{ apply mem_ℓp_infty,
use C,
rintros _ ⟨a, rfl⟩,
refine norm_apply_le_of_tendsto (eventually_of_forall hCF) hf a, },
{ apply mem_ℓp_gen',... | lemma | lp.mem_ℓp_of_tendsto | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"eq_top_or_lt_top",
"lp",
"mem_ℓp",
"mem_ℓp_gen'",
"mem_ℓp_infty",
"metric.bounded",
"set.range"
] | If `f` is the pointwise limit of a bounded sequence in `lp E p`, then `f` is in `lp E p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_lp_of_tendsto_pi {F : ℕ → lp E p} (hF : cauchy_seq F) {f : lp E p}
(hf : tendsto (id (λ i, F i) : ℕ → Π a, E a) at_top (𝓝 f)) :
tendsto F at_top (𝓝 f) | begin
rw metric.nhds_basis_closed_ball.tendsto_right_iff,
intros ε hε,
have hε' : {p : (lp E p) × (lp E p) | ‖p.1 - p.2‖ < ε} ∈ 𝓤 (lp E p),
{ exact normed_add_comm_group.uniformity_basis_dist.mem_of_mem hε },
refine (hF.eventually_eventually hε').mono _,
rintros n (hn : ∀ᶠ l in at_top, ‖(λ f, F n - f) (F l... | lemma | lp.tendsto_lp_of_tendsto_pi | analysis.normed_space | src/analysis/normed_space/lp_space.lean | [
"analysis.mean_inequalities",
"analysis.mean_inequalities_pow",
"analysis.special_functions.pow.continuity",
"topology.algebra.order.liminf_limsup"
] | [
"cauchy_seq",
"lp",
"tendsto_pi_nhds"
] | If a sequence is Cauchy in the `lp E p` topology and pointwise convergent to a element `f` of
`lp E p`, then it converges to `f` in the `lp E p` topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.topological_ring (I : Type*) (R : Type*)
[non_unital_ring R] [topological_space R] [topological_ring R] :
topological_ring (I → R) | pi.topological_ring | instance | function.topological_ring | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"non_unital_ring",
"topological_ring",
"topological_space"
] | A special case of `pi.topological_ring` for when `R` is not dependently typed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.algebra_ring (I : Type*) {R : Type*} (A : Type*) [comm_semiring R]
[ring A] [algebra R A] : algebra R (I → A) | pi.algebra _ _ | instance | function.algebra_ring | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"algebra",
"comm_semiring",
"pi.algebra",
"ring"
] | A special case of `function.algebra` for when A is a `ring` not a `semiring` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.matrix_algebra (I R A : Type*) (m : I → Type*)
[comm_semiring R] [semiring A] [algebra R A]
[Π i, fintype (m i)] [Π i, decidable_eq (m i)] :
algebra R (Π i, matrix (m i) (m i) A) | @pi.algebra I R (λ i, matrix (m i) (m i) A) _ _ (λ i, matrix.algebra) | instance | pi.matrix_algebra | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"algebra",
"comm_semiring",
"fintype",
"matrix",
"pi.algebra",
"semiring"
] | A special case of `pi.algebra` for when `f = λ i, matrix (m i) (m i) A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.matrix_topological_ring (I A : Type*) (m : I → Type*)
[ring A] [topological_space A] [topological_ring A]
[Π i, fintype (m i)] :
topological_ring (Π i, matrix (m i) (m i) A) | @pi.topological_ring _ (λ i, matrix (m i) (m i) A) _ _ (λ i, matrix.topological_ring) | instance | pi.matrix_topological_ring | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"fintype",
"matrix",
"ring",
"topological_ring",
"topological_space"
] | A special case of `pi.topological_ring` for when `f = λ i, matrix (m i) (m i) A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) | by simp_rw [exp_eq_tsum, diagonal_pow, ←diagonal_smul, ←diagonal_tsum] | lemma | matrix.exp_diagonal | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_eq_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_block_diagonal (v : m → matrix n n 𝔸) :
exp 𝕂 (block_diagonal v) = block_diagonal (exp 𝕂 v) | by simp_rw [exp_eq_tsum, ←block_diagonal_pow, ←block_diagonal_smul, ←block_diagonal_tsum] | lemma | matrix.exp_block_diagonal | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_eq_tsum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_block_diagonal' (v : Π i, matrix (n' i) (n' i) 𝔸) :
exp 𝕂 (block_diagonal' v) = block_diagonal' (exp 𝕂 v) | by simp_rw [exp_eq_tsum, ←block_diagonal'_pow, ←block_diagonal'_smul, ←block_diagonal'_tsum] | lemma | matrix.exp_block_diagonal' | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_eq_tsum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_conj_transpose [star_ring 𝔸] [has_continuous_star 𝔸] (A : matrix m m 𝔸) :
exp 𝕂 Aᴴ = (exp 𝕂 A)ᴴ | (star_exp A).symm | lemma | matrix.exp_conj_transpose | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"has_continuous_star",
"matrix",
"star_exp",
"star_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_hermitian.exp [star_ring 𝔸] [has_continuous_star 𝔸] {A : matrix m m 𝔸}
(h : A.is_hermitian) : (exp 𝕂 A).is_hermitian | (exp_conj_transpose _ _).symm.trans $ congr_arg _ h | lemma | matrix.is_hermitian.exp | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"has_continuous_star",
"matrix",
"star_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_transpose (A : matrix m m 𝔸) : exp 𝕂 Aᵀ = (exp 𝕂 A)ᵀ | by simp_rw [exp_eq_tsum, transpose_tsum, transpose_smul, transpose_pow] | lemma | matrix.exp_transpose | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_eq_tsum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symm.exp {A : matrix m m 𝔸} (h : A.is_symm) : (exp 𝕂 A).is_symm | (exp_transpose _ _).symm.trans $ congr_arg _ h | lemma | matrix.is_symm.exp | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_add_of_commute (A B : matrix m m 𝔸) (h : commute A B) :
exp 𝕂 (A + B) = exp 𝕂 A ⬝ exp 𝕂 B | begin
letI : semi_normed_ring (matrix m m 𝔸) := matrix.linfty_op_semi_normed_ring,
letI : normed_ring (matrix m m 𝔸) := matrix.linfty_op_normed_ring,
letI : normed_algebra 𝕂 (matrix m m 𝔸) := matrix.linfty_op_normed_algebra,
exact exp_add_of_commute h,
end | lemma | matrix.exp_add_of_commute | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"commute",
"exp",
"exp_add_of_commute",
"matrix",
"matrix.linfty_op_normed_algebra",
"matrix.linfty_op_normed_ring",
"matrix.linfty_op_semi_normed_ring",
"normed_algebra",
"normed_ring",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_sum_of_commute {ι} (s : finset ι) (f : ι → matrix m m 𝔸)
(h : (s : set ι).pairwise $ λ i j, commute (f i) (f j)) :
exp 𝕂 (∑ i in s, f i) = s.noncomm_prod (λ i, exp 𝕂 (f i))
(λ i hi j hj _, (h.of_refl hi hj).exp 𝕂) | begin
letI : semi_normed_ring (matrix m m 𝔸) := matrix.linfty_op_semi_normed_ring,
letI : normed_ring (matrix m m 𝔸) := matrix.linfty_op_normed_ring,
letI : normed_algebra 𝕂 (matrix m m 𝔸) := matrix.linfty_op_normed_algebra,
exact exp_sum_of_commute s f h,
end | lemma | matrix.exp_sum_of_commute | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"commute",
"exp",
"exp_sum_of_commute",
"finset",
"matrix",
"matrix.linfty_op_normed_algebra",
"matrix.linfty_op_normed_ring",
"matrix.linfty_op_semi_normed_ring",
"normed_algebra",
"normed_ring",
"pairwise",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_nsmul (n : ℕ) (A : matrix m m 𝔸) :
exp 𝕂 (n • A) = exp 𝕂 A ^ n | begin
letI : semi_normed_ring (matrix m m 𝔸) := matrix.linfty_op_semi_normed_ring,
letI : normed_ring (matrix m m 𝔸) := matrix.linfty_op_normed_ring,
letI : normed_algebra 𝕂 (matrix m m 𝔸) := matrix.linfty_op_normed_algebra,
exact exp_nsmul n A,
end | lemma | matrix.exp_nsmul | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_nsmul",
"matrix",
"matrix.linfty_op_normed_algebra",
"matrix.linfty_op_normed_ring",
"matrix.linfty_op_semi_normed_ring",
"normed_algebra",
"normed_ring",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_exp (A : matrix m m 𝔸) : is_unit (exp 𝕂 A) | begin
letI : semi_normed_ring (matrix m m 𝔸) := matrix.linfty_op_semi_normed_ring,
letI : normed_ring (matrix m m 𝔸) := matrix.linfty_op_normed_ring,
letI : normed_algebra 𝕂 (matrix m m 𝔸) := matrix.linfty_op_normed_algebra,
exact is_unit_exp _ A,
end | lemma | matrix.is_unit_exp | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"is_unit",
"is_unit_exp",
"matrix",
"matrix.linfty_op_normed_algebra",
"matrix.linfty_op_normed_ring",
"matrix.linfty_op_semi_normed_ring",
"normed_algebra",
"normed_ring",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_units_conj (U : (matrix m m 𝔸)ˣ) (A : matrix m m 𝔸) :
exp 𝕂 (↑U ⬝ A ⬝ ↑(U⁻¹) : matrix m m 𝔸) = ↑U ⬝ exp 𝕂 A ⬝ ↑(U⁻¹) | begin
letI : semi_normed_ring (matrix m m 𝔸) := matrix.linfty_op_semi_normed_ring,
letI : normed_ring (matrix m m 𝔸) := matrix.linfty_op_normed_ring,
letI : normed_algebra 𝕂 (matrix m m 𝔸) := matrix.linfty_op_normed_algebra,
exact exp_units_conj _ U A,
end | lemma | matrix.exp_units_conj | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_units_conj",
"matrix",
"matrix.linfty_op_normed_algebra",
"matrix.linfty_op_normed_ring",
"matrix.linfty_op_semi_normed_ring",
"normed_algebra",
"normed_ring",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_units_conj' (U : (matrix m m 𝔸)ˣ) (A : matrix m m 𝔸) :
exp 𝕂 (↑(U⁻¹) ⬝ A ⬝ U : matrix m m 𝔸) = ↑(U⁻¹) ⬝ exp 𝕂 A ⬝ U | exp_units_conj 𝕂 U⁻¹ A | lemma | matrix.exp_units_conj' | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_units_conj",
"exp_units_conj'",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_neg (A : matrix m m 𝔸) : exp 𝕂 (-A) = (exp 𝕂 A)⁻¹ | begin
rw nonsing_inv_eq_ring_inverse,
letI : semi_normed_ring (matrix m m 𝔸) := matrix.linfty_op_semi_normed_ring,
letI : normed_ring (matrix m m 𝔸) := matrix.linfty_op_normed_ring,
letI : normed_algebra 𝕂 (matrix m m 𝔸) := matrix.linfty_op_normed_algebra,
exact (ring.inverse_exp _ A).symm,
end | lemma | matrix.exp_neg | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_neg",
"matrix",
"matrix.linfty_op_normed_algebra",
"matrix.linfty_op_normed_ring",
"matrix.linfty_op_semi_normed_ring",
"normed_algebra",
"normed_ring",
"ring.inverse_exp",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_zsmul (z : ℤ) (A : matrix m m 𝔸) : exp 𝕂 (z • A) = exp 𝕂 A ^ z | begin
obtain ⟨n, rfl | rfl⟩ := z.eq_coe_or_neg,
{ rw [zpow_coe_nat, coe_nat_zsmul, exp_nsmul] },
{ have : is_unit (exp 𝕂 A).det := (matrix.is_unit_iff_is_unit_det _).mp (is_unit_exp _ _),
rw [matrix.zpow_neg this, zpow_coe_nat, neg_smul,
exp_neg, coe_nat_zsmul, exp_nsmul] },
end | lemma | matrix.exp_zsmul | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_neg",
"exp_nsmul",
"exp_zsmul",
"is_unit",
"is_unit_exp",
"matrix",
"matrix.is_unit_iff_is_unit_det",
"matrix.zpow_neg",
"neg_smul",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_conj (U : matrix m m 𝔸) (A : matrix m m 𝔸) (hy : is_unit U) :
exp 𝕂 (U ⬝ A ⬝ U⁻¹) = U ⬝ exp 𝕂 A ⬝ U⁻¹ | let ⟨u, hu⟩ := hy in hu ▸ by simpa only [matrix.coe_units_inv] using exp_units_conj 𝕂 u A | lemma | matrix.exp_conj | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_conj",
"exp_units_conj",
"is_unit",
"matrix",
"matrix.coe_units_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_conj' (U : matrix m m 𝔸) (A : matrix m m 𝔸) (hy : is_unit U) :
exp 𝕂 (U⁻¹ ⬝ A ⬝ U) = U⁻¹ ⬝ exp 𝕂 A ⬝ U | let ⟨u, hu⟩ := hy in hu ▸ by simpa only [matrix.coe_units_inv] using exp_units_conj' 𝕂 u A | lemma | matrix.exp_conj' | analysis.normed_space | src/analysis/normed_space/matrix_exponential.lean | [
"analysis.normed_space.exponential",
"analysis.matrix",
"linear_algebra.matrix.zpow",
"linear_algebra.matrix.hermitian",
"linear_algebra.matrix.symmetric",
"topology.uniform_space.matrix"
] | [
"exp",
"exp_conj'",
"exp_units_conj'",
"is_unit",
"matrix",
"matrix.coe_units_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
midpoint_fixed {x y : PE} :
∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y | begin
set z := midpoint ℝ x y,
-- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y },
haveI : nonempty s := ⟨⟨isometry_equiv.refl PE, rfl, rfl⟩⟩,
-- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
have h_bdd : bdd_above (ran... | lemma | isometry_equiv.midpoint_fixed | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [
"bdd_above",
"csupr_le",
"dist_comm",
"dist_triangle",
"le_csupr",
"le_div_iff'",
"midpoint",
"subtype.coe_mk",
"zero_lt_two'"
] | If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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