statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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inv_def (e : E ≃ₗᵢ[R] E) : (e⁻¹ : E ≃ₗᵢ[R] E) = e.symm | rfl | lemma | linear_isometry_equiv.inv_def | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_one : e.trans (1 : E₂ ≃ₗᵢ[R₂] E₂) = e | trans_refl _ | lemma | linear_isometry_equiv.trans_one | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_trans : (1 : E ≃ₗᵢ[R] E).trans e = e | refl_trans _ | lemma | linear_isometry_equiv.one_trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_mul (e : E ≃ₗᵢ[R] E) : (refl _ _) * e = e | trans_refl _ | lemma | linear_isometry_equiv.refl_mul | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_refl (e : E ≃ₗᵢ[R] E) : e * (refl _ _) = e | refl_trans _ | lemma | linear_isometry_equiv.mul_refl | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe : ⇑(e : E ≃SL[σ₁₂] E₂) = e | rfl | lemma | linear_isometry_equiv.coe_coe | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe' : ((e : E ≃SL[σ₁₂] E₂) : E →SL[σ₁₂] E₂) = e | rfl | lemma | linear_isometry_equiv.coe_coe' | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe'' : ⇑(e : E →SL[σ₁₂] E₂) = e | rfl | lemma | linear_isometry_equiv.coe_coe'' | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : e 0 = 0 | e.1.map_zero | lemma | linear_isometry_equiv.map_zero | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (x y : E) : e (x + y) = e x + e y | e.1.map_add x y | lemma | linear_isometry_equiv.map_add | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y : E) : e (x - y) = e x - e y | e.1.map_sub x y | lemma | linear_isometry_equiv.map_sub | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ (c : R) (x : E) : e (c • x) = σ₁₂ c • e x | e.1.map_smulₛₗ c x | lemma | linear_isometry_equiv.map_smulₛₗ | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul [module R E₂] {e : E ≃ₗᵢ[R] E₂} (c : R) (x : E) : e (c • x) = c • e x | e.1.map_smul c x | lemma | linear_isometry_equiv.map_smul | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_map (x : E) : ‖e x‖₊ = ‖x‖₊ | semilinear_isometry_class.nnnorm_map e x | lemma | linear_isometry_equiv.nnnorm_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"semilinear_isometry_class.nnnorm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_map (x y : E) : dist (e x) (e y) = dist x y | e.to_linear_isometry.dist_map x y | lemma | linear_isometry_equiv.dist_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_map (x y : E) : edist (e x) (e y) = edist x y | e.to_linear_isometry.edist_map x y | lemma | linear_isometry_equiv.edist_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_iff {x y : E} : e x = e y ↔ x = y | e.injective.eq_iff | lemma | linear_isometry_equiv.map_eq_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne {x y : E} (h : x ≠ y) : e x ≠ e y | e.injective.ne h | lemma | linear_isometry_equiv.map_ne | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_preimage (s : set E) : e '' s = e.symm ⁻¹' s | e.to_linear_equiv.image_eq_preimage s | lemma | linear_isometry_equiv.image_eq_preimage | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image (s : set E) : emetric.diam (e '' s) = emetric.diam s | e.isometry.ediam_image s | lemma | linear_isometry_equiv.ediam_image | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_image (s : set E) : metric.diam (e '' s) = metric.diam s | e.isometry.diam_image s | lemma | linear_isometry_equiv.diam_image | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ball (x : E₂) (r : ℝ) :
e ⁻¹' (metric.ball x r) = metric.ball (e.symm x) r | e.to_isometry_equiv.preimage_ball x r | lemma | linear_isometry_equiv.preimage_ball | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_sphere (x : E₂) (r : ℝ) :
e ⁻¹' (metric.sphere x r) = metric.sphere (e.symm x) r | e.to_isometry_equiv.preimage_sphere x r | lemma | linear_isometry_equiv.preimage_sphere | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closed_ball (x : E₂) (r : ℝ) :
e ⁻¹' (metric.closed_ball x r) = metric.closed_ball (e.symm x) r | e.to_isometry_equiv.preimage_closed_ball x r | lemma | linear_isometry_equiv.preimage_closed_ball | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_ball (x : E) (r : ℝ) :
e '' (metric.ball x r) = metric.ball (e x) r | e.to_isometry_equiv.image_ball x r | lemma | linear_isometry_equiv.image_ball | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_sphere (x : E) (r : ℝ) :
e '' (metric.sphere x r) = metric.sphere (e x) r | e.to_isometry_equiv.image_sphere x r | lemma | linear_isometry_equiv.image_sphere | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_closed_ball (x : E) (r : ℝ) :
e '' (metric.closed_ball x r) = metric.closed_ball (e x) r | e.to_isometry_equiv.image_closed_ball x r | lemma | linear_isometry_equiv.image_closed_ball | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_on_iff {f : α → E} {s : set α} :
continuous_on (e ∘ f) s ↔ continuous_on f s | e.isometry.comp_continuous_on_iff | lemma | linear_isometry_equiv.comp_continuous_on_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {f : α → E} :
continuous (e ∘ f) ↔ continuous f | e.isometry.comp_continuous_iff | lemma | linear_isometry_equiv.comp_continuous_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_map (p : submodule R E) [complete_space p] :
complete_space (p.map (e.to_linear_equiv : E →ₛₗ[σ₁₂] E₂)) | e.to_linear_isometry.complete_space_map' p | instance | linear_isometry_equiv.complete_space_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"complete_space",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : F →ₛₗᵢ[σ₁₂] E₂)
(hfr : function.surjective f) :
F ≃ₛₗᵢ[σ₁₂] E₂ | { norm_map' := f.norm_map,
.. linear_equiv.of_bijective f.to_linear_map ⟨f.injective, hfr⟩ } | def | linear_isometry_equiv.of_surjective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.of_bijective"
] | Construct a linear isometry equiv from a surjective linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_surjective (f : F →ₛₗᵢ[σ₁₂] E₂) (hfr : function.surjective f) :
⇑(linear_isometry_equiv.of_surjective f hfr) = f | by { ext, refl } | lemma | linear_isometry_equiv.coe_of_surjective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_isometry_equiv.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_linear_isometry (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E)
(h₁ : f.to_linear_map.comp g = linear_map.id) (h₂ : g.comp f.to_linear_map = linear_map.id) :
E ≃ₛₗᵢ[σ₁₂] E₂ | { norm_map' := λ x, f.norm_map x,
.. linear_equiv.of_linear f.to_linear_map g h₁ h₂ } | def | linear_isometry_equiv.of_linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.of_linear",
"linear_map.id"
] | If a linear isometry has an inverse, it is a linear isometric equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_linear_isometry (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E)
(h₁ : f.to_linear_map.comp g = linear_map.id) (h₂ : g.comp f.to_linear_map = linear_map.id) :
(of_linear_isometry f g h₁ h₂ : E → E₂) = (f : E → E₂) | rfl | lemma | linear_isometry_equiv.coe_of_linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_linear_isometry_symm (f : E →ₛₗᵢ[σ₁₂] E₂)
(g : E₂ →ₛₗ[σ₂₁] E) (h₁ : f.to_linear_map.comp g = linear_map.id)
(h₂ : g.comp f.to_linear_map = linear_map.id) :
((of_linear_isometry f g h₁ h₂).symm : E₂ → E) = (g : E₂ → E) | rfl | lemma | linear_isometry_equiv.coe_of_linear_isometry_symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg : E ≃ₗᵢ[R] E | { norm_map' := norm_neg,
.. linear_equiv.neg R } | def | linear_isometry_equiv.neg | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.neg"
] | The negation operation on a normed space `E`, considered as a linear isometry equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_neg : (neg R : E → E) = λ x, -x | rfl | lemma | linear_isometry_equiv.coe_neg | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_neg : (neg R : E ≃ₗᵢ[R] E).symm = neg R | rfl | lemma | linear_isometry_equiv.symm_neg | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_assoc [module R E₂] [module R E₃] : (E × E₂) × E₃ ≃ₗᵢ[R] E × E₂ × E₃ | { to_fun := equiv.prod_assoc E E₂ E₃,
inv_fun := (equiv.prod_assoc E E₂ E₃).symm,
map_add' := by simp,
map_smul' := by simp,
norm_map' :=
begin
rintros ⟨⟨e, f⟩, g⟩,
simp only [linear_equiv.coe_mk, equiv.prod_assoc_apply, prod.norm_def, max_assoc],
end,
.. equiv.prod_assoc E E₂ E₃, } | def | linear_isometry_equiv.prod_assoc | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"equiv.prod_assoc",
"inv_fun",
"linear_equiv.coe_mk",
"module",
"prod.norm_def"
] | The natural equivalence `(E × E₂) × E₃ ≃ E × (E₂ × E₃)` is a linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_assoc [module R E₂] [module R E₃] :
(prod_assoc R E E₂ E₃ : (E × E₂) × E₃ → E × E₂ × E₃) = equiv.prod_assoc E E₂ E₃ | rfl | lemma | linear_isometry_equiv.coe_prod_assoc | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"equiv.prod_assoc",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_assoc_symm [module R E₂] [module R E₃] :
((prod_assoc R E E₂ E₃).symm : E × E₂ × E₃ → (E × E₂) × E₃) = (equiv.prod_assoc E E₂ E₃).symm | rfl | lemma | linear_isometry_equiv.coe_prod_assoc_symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"equiv.prod_assoc",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_top {R : Type*} [ring R] [module R E] (p : submodule R E) (hp : p = ⊤) :
p ≃ₗᵢ[R] E | { to_linear_equiv := linear_equiv.of_top p hp, .. p.subtypeₗᵢ } | def | linear_isometry_equiv.of_top | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.of_top",
"module",
"ring",
"submodule"
] | If `p` is a submodule that is equal to `⊤`, then `linear_isometry_equiv.of_top p hp` is the
"identity" equivalence between `p` and `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_eq (hpq : p = q) :
p ≃ₗᵢ[R'] q | { norm_map' := λ x, rfl,
..linear_equiv.of_eq p q hpq } | def | linear_isometry_equiv.of_eq | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.of_eq",
"of_eq"
] | `linear_equiv.of_eq` as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_eq_apply (h : p = q) (x : p) : (of_eq p q h x : E) = x | rfl | lemma | linear_isometry_equiv.coe_of_eq_apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq_symm (h : p = q) : (of_eq p q h).symm = of_eq q p h.symm | rfl | lemma | linear_isometry_equiv.of_eq_symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq_rfl : of_eq p p rfl = linear_isometry_equiv.refl R' p | by ext; refl | lemma | linear_isometry_equiv.of_eq_rfl | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_isometry_equiv.refl",
"of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.ext_linear_isometry {ι : Type*} (b : basis ι R E) {f₁ f₂ : E →ₛₗᵢ[σ₁₂] E₂}
(h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ | linear_isometry.to_linear_map_injective $ b.ext h | lemma | basis.ext_linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"basis",
"linear_isometry.to_linear_map_injective"
] | Two linear isometries are equal if they are equal on basis vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.ext_linear_isometry_equiv {ι : Type*} (b : basis ι R E) {f₁ f₂ : E ≃ₛₗᵢ[σ₁₂] E₂}
(h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ | linear_isometry_equiv.to_linear_equiv_injective $ b.ext' h | lemma | basis.ext_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"basis",
"linear_isometry_equiv.to_linear_equiv_injective"
] | Two linear isometric equivalences are equal if they are equal on basis vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.equiv_range {R S : Type*} [semiring R] [ring S] [module S E]
[module R F] {σ₁₂ : R →+* S} {σ₂₁ : S →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁]
[ring_hom_inv_pair σ₂₁ σ₁₂] (f : F →ₛₗᵢ[σ₁₂] E) :
F ≃ₛₗᵢ[σ₁₂] f.to_linear_map.range | { to_linear_equiv := linear_equiv.of_injective f.to_linear_map f.injective, .. f } | def | linear_isometry.equiv_range | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.of_injective",
"module",
"ring",
"ring_hom_inv_pair",
"semiring"
] | Reinterpret a `linear_isometry` as a `linear_isometry_equiv` to the range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ℓp.all [finite α] (f : Π i, E i) : mem_ℓp f p | begin
rcases p.trichotomy with (rfl | rfl | h),
{ exact mem_ℓp_zero_iff.mpr {i : α | f i ≠ 0}.to_finite, },
{ exact mem_ℓp_infty_iff.mpr (set.finite.bdd_above (set.range (λ (i : α), ‖f i‖)).to_finite) },
{ casesI nonempty_fintype α, exact mem_ℓp_gen ⟨finset.univ.sum _, has_sum_fintype _⟩ }
end | lemma | mem_ℓp.all | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"finite",
"has_sum_fintype",
"mem_ℓp",
"mem_ℓp_gen",
"nonempty_fintype",
"set.finite.bdd_above",
"set.range"
] | When `α` is `finite`, every `f : pre_lp E p` satisfies `mem_ℓp f p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.lp_pi_Lp : lp E p ≃ pi_Lp p E | { to_fun := λ f, f,
inv_fun := λ f, ⟨f, mem_ℓp.all f⟩,
left_inv := λ f, lp.ext $ funext $ λ x, rfl,
right_inv := λ f, funext $ λ x, rfl } | def | equiv.lp_pi_Lp | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"inv_fun",
"lp",
"lp.ext",
"mem_ℓp.all",
"pi_Lp"
] | The canonical `equiv` between `lp E p ≃ pi_Lp p E` when `E : α → Type u` with `[fintype α]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_equiv_lp_pi_Lp (f : lp E p) : equiv.lp_pi_Lp f = f | rfl | lemma | coe_equiv_lp_pi_Lp | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"equiv.lp_pi_Lp",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_equiv_lp_pi_Lp_symm (f : pi_Lp p E) : (equiv.lp_pi_Lp.symm f : Π i, E i) = f | rfl | lemma | coe_equiv_lp_pi_Lp_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_lp_pi_Lp_norm (f : lp E p) : ‖equiv.lp_pi_Lp f‖ = ‖f‖ | begin
unfreezingI { rcases p.trichotomy with (rfl | rfl | h) },
{ rw [pi_Lp.norm_eq_card, lp.norm_eq_card_dsupport], refl },
{ rw [pi_Lp.norm_eq_csupr, lp.norm_eq_csupr], refl },
{ rw [pi_Lp.norm_eq_sum h, lp.norm_eq_tsum_rpow h, tsum_fintype], refl },
end | lemma | equiv_lp_pi_Lp_norm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"lp",
"lp.norm_eq_card_dsupport",
"lp.norm_eq_csupr",
"lp.norm_eq_tsum_rpow",
"pi_Lp.norm_eq_card",
"pi_Lp.norm_eq_csupr",
"pi_Lp.norm_eq_sum",
"tsum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv.lp_pi_Lp [fact (1 ≤ p)] : lp E p ≃+ pi_Lp p E | { map_add' := λ f g, rfl,
.. equiv.lp_pi_Lp } | def | add_equiv.lp_pi_Lp | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"equiv.lp_pi_Lp",
"fact",
"lp",
"pi_Lp"
] | The canonical `add_equiv` between `lp E p` and `pi_Lp p E` when `E : α → Type u` with
`[fintype α]` and `[fact (1 ≤ p)]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add_equiv_lp_pi_Lp [fact (1 ≤ p)] (f : lp E p) :
add_equiv.lp_pi_Lp f = f | rfl | lemma | coe_add_equiv_lp_pi_Lp | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"add_equiv.lp_pi_Lp",
"fact",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_equiv_lp_pi_Lp_symm [fact (1 ≤ p)] (f : pi_Lp p E) :
(add_equiv.lp_pi_Lp.symm f : Π i, E i) = f | rfl | lemma | coe_add_equiv_lp_pi_Lp_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"fact",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lp_pi_Lpₗᵢ [fact (1 ≤ p)] : lp E p ≃ₗᵢ[𝕜] pi_Lp p E | { map_smul' := λ k f, rfl,
norm_map' := equiv_lp_pi_Lp_norm,
.. add_equiv.lp_pi_Lp } | def | lp_pi_Lpₗᵢ | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"add_equiv.lp_pi_Lp",
"equiv_lp_pi_Lp_norm",
"fact",
"lp",
"pi_Lp"
] | The canonical `linear_isometry_equiv` between `lp E p` and `pi_Lp p E` when `E : α → Type u`
with `[fintype α]` and `[fact (1 ≤ p)]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lp_pi_Lpₗᵢ [fact (1 ≤ p)] (f : lp E p) :
lp_pi_Lpₗᵢ 𝕜 f = f | rfl | lemma | coe_lp_pi_Lpₗᵢ | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"fact",
"lp",
"lp_pi_Lpₗᵢ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lp_pi_Lpₗᵢ_symm [fact (1 ≤ p)] (f : pi_Lp p E) :
((lp_pi_Lpₗᵢ 𝕜).symm f : Π i, E i) = f | rfl | lemma | coe_lp_pi_Lpₗᵢ_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"fact",
"lp_pi_Lpₗᵢ",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv.lp_bcf :
lp (λ (_ : α), E) ∞ ≃+ (α →ᵇ E) | { to_fun := λ f, of_normed_add_comm_group_discrete f (‖f‖) $ le_csupr (mem_ℓp_infty_iff.mp f.prop),
inv_fun := λ f, ⟨f, f.bdd_above_range_norm_comp⟩,
left_inv := λ f, lp.ext rfl,
right_inv := λ f, ext $ λ x, rfl,
map_add' := λ f g, ext $ λ x, rfl } | def | add_equiv.lp_bcf | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"inv_fun",
"le_csupr",
"lp",
"lp.ext"
] | The canonical map between `lp (λ (_ : α), E) ∞` and `α →ᵇ E` as an `add_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add_equiv_lp_bcf (f : lp (λ (_ : α), E) ∞) :
(add_equiv.lp_bcf f : α → E) = f | rfl | lemma | coe_add_equiv_lp_bcf | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"add_equiv.lp_bcf",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_equiv_lp_bcf_symm (f : α →ᵇ E) : (add_equiv.lp_bcf.symm f : α → E) = f | rfl | lemma | coe_add_equiv_lp_bcf_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lp_bcfₗᵢ : lp (λ (_ : α), E) ∞ ≃ₗᵢ[𝕜] (α →ᵇ E) | { map_smul' := λ k f, rfl,
norm_map' := λ f, by { simp only [norm_eq_supr_norm, lp.norm_eq_csupr], refl },
.. add_equiv.lp_bcf } | def | lp_bcfₗᵢ | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"add_equiv.lp_bcf",
"lp",
"lp.norm_eq_csupr"
] | The canonical map between `lp (λ (_ : α), E) ∞` and `α →ᵇ E` as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lp_bcfₗᵢ (f : lp (λ (_ : α), E) ∞) : (lp_bcfₗᵢ 𝕜 f : α → E) = f | rfl | lemma | coe_lp_bcfₗᵢ | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"lp",
"lp_bcfₗᵢ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lp_bcfₗᵢ_symm (f : α →ᵇ E) : ((lp_bcfₗᵢ 𝕜).symm f : α → E) = f | rfl | lemma | coe_lp_bcfₗᵢ_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"lp_bcfₗᵢ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv.lp_bcf : lp (λ (_ : α), R) ∞ ≃+* (α →ᵇ R) | { map_mul' := λ f g, ext $ λ x, rfl, .. @add_equiv.lp_bcf _ R _ _ _ } | def | ring_equiv.lp_bcf | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"add_equiv.lp_bcf",
"lp"
] | The canonical map between `lp (λ (_ : α), R) ∞` and `α →ᵇ R` as a `ring_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_ring_equiv_lp_bcf (f : lp (λ (_ : α), R) ∞) :
(ring_equiv.lp_bcf R f : α → R) = f | rfl | lemma | coe_ring_equiv_lp_bcf | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"lp",
"ring_equiv.lp_bcf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_equiv_lp_bcf_symm (f : α →ᵇ R) :
((ring_equiv.lp_bcf R).symm f : α → R) = f | rfl | lemma | coe_ring_equiv_lp_bcf_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"ring_equiv.lp_bcf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv.lp_bcf : lp (λ (_ : α), A) ∞ ≃ₐ[𝕜] (α →ᵇ A) | { commutes' := λ k, rfl, .. ring_equiv.lp_bcf A } | def | alg_equiv.lp_bcf | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"lp",
"ring_equiv.lp_bcf"
] | The canonical map between `lp (λ (_ : α), A) ∞` and `α →ᵇ A` as an `alg_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_alg_equiv_lp_bcf (f : lp (λ (_ : α), A) ∞) :
(alg_equiv.lp_bcf α A 𝕜 f : α → A) = f | rfl | lemma | coe_alg_equiv_lp_bcf | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"alg_equiv.lp_bcf",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_alg_equiv_lp_bcf_symm (f : α →ᵇ A) :
((alg_equiv.lp_bcf α A 𝕜).symm f : α → A) = f | rfl | lemma | coe_alg_equiv_lp_bcf_symm | analysis.normed_space | src/analysis/normed_space/lp_equiv.lean | [
"analysis.normed_space.lp_space",
"analysis.normed_space.pi_Lp",
"topology.continuous_function.bounded"
] | [
"alg_equiv.lp_bcf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp (f : Π i, E i) (p : ℝ≥0∞) : Prop | if p = 0 then (set.finite {i | f i ≠ 0}) else
(if p = ∞ then bdd_above (set.range (λ i, ‖f i‖)) else summable (λ i, ‖f i‖ ^ p.to_real)) | def | 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"
] | [
"bdd_above",
"set.finite",
"set.range",
"summable"
] | The property that `f : Π i : α, E i`
* is finitely supported, if `p = 0`, or
* admits an upper bound for `set.range (λ i, ‖f i‖)`, if `p = ∞`, or
* has the series `∑' i, ‖f i‖ ^ p` be summable, if `0 < p < ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ℓp_zero_iff {f : Π i, E i} : mem_ℓp f 0 ↔ set.finite {i | f i ≠ 0} | by dsimp [mem_ℓp]; rw [if_pos rfl] | lemma | mem_ℓp_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"
] | [
"mem_ℓp",
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp_zero {f : Π i, E i} (hf : set.finite {i | f i ≠ 0}) : mem_ℓp f 0 | mem_ℓp_zero_iff.2 hf | lemma | mem_ℓp_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"
] | [
"mem_ℓp",
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp_infty_iff {f : Π i, E i} : mem_ℓp f ∞ ↔ bdd_above (set.range (λ i, ‖f i‖)) | by dsimp [mem_ℓp]; rw [if_neg ennreal.top_ne_zero, if_pos rfl] | lemma | mem_ℓp_infty_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"
] | [
"bdd_above",
"ennreal.top_ne_zero",
"mem_ℓp",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp_infty {f : Π i, E i} (hf : bdd_above (set.range (λ i, ‖f i‖))) : mem_ℓp f ∞ | mem_ℓp_infty_iff.2 hf | lemma | 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"
] | [
"bdd_above",
"mem_ℓp",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp_gen_iff (hp : 0 < p.to_real) {f : Π i, E i} :
mem_ℓp f p ↔ summable (λ i, ‖f i‖ ^ p.to_real) | begin
rw ennreal.to_real_pos_iff at hp,
dsimp [mem_ℓp],
rw [if_neg hp.1.ne', if_neg hp.2.ne],
end | lemma | mem_ℓp_gen_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"
] | [
"ennreal.to_real_pos_iff",
"mem_ℓp",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp_gen {f : Π i, E i} (hf : summable (λ i, ‖f i‖ ^ p.to_real)) :
mem_ℓp f p | begin
rcases p.trichotomy with rfl | rfl | hp,
{ apply mem_ℓp_zero,
have H : summable (λ i : α, (1:ℝ)) := by simpa using hf,
exact (finite_of_summable_const (by norm_num) H).subset (set.subset_univ _) },
{ apply mem_ℓp_infty,
have H : summable (λ i : α, (1:ℝ)) := by simpa using hf,
simpa using ((f... | lemma | mem_ℓp_gen | 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"
] | [
"bdd_above",
"finite_of_summable_const",
"mem_ℓp",
"mem_ℓp_gen_iff",
"mem_ℓp_infty",
"mem_ℓp_zero",
"set.subset_univ",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℓp_gen' {C : ℝ} {f : Π i, E i} (hf : ∀ s : finset α, ∑ i in s, ‖f i‖ ^ p.to_real ≤ C) :
mem_ℓp f p | begin
apply mem_ℓp_gen,
use ⨆ s : finset α, ∑ i in s, ‖f i‖ ^ p.to_real,
apply has_sum_of_is_lub_of_nonneg,
{ intros b,
exact real.rpow_nonneg_of_nonneg (norm_nonneg _) _ },
apply is_lub_csupr,
use C,
rintros - ⟨s, rfl⟩,
exact hf s
end | lemma | mem_ℓp_gen' | 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_of_is_lub_of_nonneg",
"is_lub_csupr",
"mem_ℓp",
"mem_ℓp_gen",
"real.rpow_nonneg_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem_ℓp : mem_ℓp (0 : Π i, E i) p | begin
rcases p.trichotomy with rfl | rfl | hp,
{ apply mem_ℓp_zero,
simp },
{ apply mem_ℓp_infty,
simp only [norm_zero, pi.zero_apply],
exact bdd_above_singleton.mono set.range_const_subset, },
{ apply mem_ℓp_gen,
simp [real.zero_rpow hp.ne', summable_zero], }
end | lemma | zero_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"
] | [
"mem_ℓp",
"mem_ℓp_gen",
"mem_ℓp_infty",
"mem_ℓp_zero",
"real.zero_rpow",
"set.range_const_subset",
"summable_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem_ℓp' : mem_ℓp (λ i : α, (0 : E i)) p | zero_mem_ℓp | lemma | zero_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"
] | [
"mem_ℓp",
"zero_mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_dsupport {f : Π i, E i} (hf : mem_ℓp f 0) : set.finite {i | f i ≠ 0} | mem_ℓp_zero_iff.1 hf | lemma | mem_ℓp.finite_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"
] | [
"mem_ℓp",
"set.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above {f : Π i, E i} (hf : mem_ℓp f ∞) : bdd_above (set.range (λ i, ‖f i‖)) | mem_ℓp_infty_iff.1 hf | lemma | mem_ℓp.bdd_above | 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"
] | [
"bdd_above",
"mem_ℓp",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable (hp : 0 < p.to_real) {f : Π i, E i} (hf : mem_ℓp f p) :
summable (λ i, ‖f i‖ ^ p.to_real) | (mem_ℓp_gen_iff hp).1 hf | lemma | mem_ℓp.summable | 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_iff",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg {f : Π i, E i} (hf : mem_ℓp f p) : mem_ℓp (-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.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"
] | [
"mem_ℓp",
"mem_ℓp_gen",
"mem_ℓp_infty",
"mem_ℓp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_iff {f : Π i, E i} : mem_ℓp (-f) p ↔ mem_ℓp f p | ⟨λ h, neg_neg f ▸ h.neg, mem_ℓp.neg⟩ | lemma | mem_ℓp.neg_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_exponent_ge {p q : ℝ≥0∞} {f : Π i, E i}
(hfq : mem_ℓp f q) (hpq : q ≤ p) :
mem_ℓp f p | begin
rcases ennreal.trichotomy₂ hpq with ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | ⟨rfl, hp⟩ | ⟨rfl, rfl⟩ | ⟨hq, rfl⟩
| ⟨hq, hp, hpq'⟩,
{ exact hfq },
{ apply mem_ℓp_infty,
obtain ⟨C, hC⟩ := (hfq.finite_dsupport.image (λ i, ‖f i‖)).bdd_above,
use max 0 C,
rintros x ⟨i, rfl⟩,
by_cases hi : f i = 0,
{ simp... | lemma | mem_ℓp.of_exponent_ge | 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_of_nonneg",
"bdd_above",
"ennreal.trichotomy₂",
"eventually_lt_of_tendsto_lt",
"finite",
"mem_ℓp",
"mem_ℓp_gen",
"mem_ℓp_infty",
"mul_inv_cancel",
"real.one_le_rpow",
"real.rpow_le_rpow",
"real.rpow_le_rpow_of_exponent_ge'",
"real.rpow_mul",
"real.rpow_nonneg_of_nonneg",
"real.zero_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add {f g : Π i, E i} (hf : mem_ℓp f p) (hg : mem_ℓp g p) : mem_ℓp (f + g) p | begin
rcases p.trichotomy with rfl | rfl | hp,
{ apply mem_ℓp_zero,
refine (hf.finite_dsupport.union hg.finite_dsupport).subset (λ i, _),
simp only [pi.add_apply, ne.def, set.mem_union, set.mem_set_of_eq],
contrapose!,
rintros ⟨hf', hg'⟩,
simp [hf', hg'] },
{ apply mem_ℓp_infty,
obtain ⟨A,... | lemma | mem_ℓp.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"
] | [
"finset",
"finset.univ",
"mem_ℓp",
"mem_ℓp_gen",
"mem_ℓp_infty",
"mem_ℓp_zero",
"nnreal.rpow_add_le_add_rpow",
"real.rpow_le_rpow",
"real.rpow_nonneg_of_nonneg",
"real.rpow_sum_le_const_mul_sum_rpow_of_nonneg",
"set.mem_union",
"summable_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub {f g : Π i, E i} (hf : mem_ℓp f p) (hg : mem_ℓp g p) : mem_ℓp (f - g) p | by { rw sub_eq_add_neg, exact hf.add hg.neg } | lemma | mem_ℓp.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"
] | [
"mem_ℓp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sum {ι} (s : finset ι) {f : ι → Π i, E i} (hf : ∀ i ∈ s, mem_ℓp (f i) p) :
mem_ℓp (λ a, ∑ i in s, f i a) p | begin
haveI : decidable_eq ι := classical.dec_eq _,
revert hf,
refine finset.induction_on s _ _,
{ simp only [zero_mem_ℓp', finset.sum_empty, implies_true_iff], },
{ intros i s his ih hf,
simp only [his, finset.sum_insert, not_false_iff],
exact (hf i (s.mem_insert_self i)).add (ih (λ j hj, hf j (finse... | lemma | mem_ℓp.finset_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"
] | [
"classical.dec_eq",
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"ih",
"mem_ℓp",
"zero_mem_ℓp'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_smul {f : Π i, E i} (hf : mem_ℓp f p) (c : 𝕜) : mem_ℓp (c • f) p | begin
rcases p.trichotomy with rfl | rfl | hp,
{ apply mem_ℓp_zero,
refine hf.finite_dsupport.subset (λ i, (_ : ¬c • f i = 0 → ¬f i = 0)),
exact not_imp_not.mpr (λ hf', hf'.symm ▸ (smul_zero c)) },
{ obtain ⟨A, hA⟩ := hf.bdd_above,
refine mem_ℓp_infty ⟨‖c‖ * A, _⟩,
rintros a ⟨i, rfl⟩,
refine (... | lemma | 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"
] | [
"ennreal.to_real_nonneg",
"mem_ℓp",
"mem_ℓp_gen",
"mem_ℓp_infty",
"mem_ℓp_zero",
"mul_le_mul_of_nonneg_left",
"nnnorm_smul_le",
"nnreal.rpow_le_rpow",
"nnreal.summable_coe",
"nnreal.summable_of_le",
"norm_smul_le",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_mul {f : α → 𝕜} (hf : mem_ℓp f p) (c : 𝕜) : mem_ℓp (λ x, c * f x) p | @mem_ℓp.const_smul α (λ i, 𝕜) _ _ 𝕜 _ _ (λ i, by apply_instance) _ hf c | lemma | mem_ℓp.const_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.const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_lp (E : α → Type*) [Π i, normed_add_comm_group (E i)] : Type* | Π i, E i | def | pre_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"
] | [
"normed_add_comm_group"
] | We define `pre_lp E` to be a type synonym for `Π i, E i` which, importantly, does not inherit
the `pi` topology on `Π i, E i` (otherwise this topology would descend to `lp E p` and conflict
with the normed group topology we will later equip it with.)
We choose to deal with this issue by making a type synonym for `Π i,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_lp.unique [is_empty α] : unique (pre_lp E) | pi.unique_of_is_empty E | instance | pre_lp.unique | 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",
"pi.unique_of_is_empty",
"pre_lp",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lp (E : α → Type*) [Π i, normed_add_comm_group (E i)]
(p : ℝ≥0∞) : add_subgroup (pre_lp E) | { carrier := {f | mem_ℓp f p},
zero_mem' := zero_mem_ℓp,
add_mem' := λ f g, mem_ℓp.add,
neg_mem' := λ f, mem_ℓp.neg } | def | 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"
] | [
"add_subgroup",
"mem_ℓp",
"mem_ℓp.add",
"mem_ℓp.neg",
"normed_add_comm_group",
"pre_lp",
"zero_mem_ℓp"
] | lp space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : lp E p} (h : (f : Π i, E i) = g) : f = g | subtype.ext h | lemma | lp.ext | 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",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : lp E p} : f = g ↔ (f : Π i, E i) = g | subtype.ext_iff | lemma | lp.ext_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"
] | [
"lp",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero' [is_empty α] (f : lp E p) : f = 0 | subsingleton.elim f 0 | lemma | lp.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"
] | [
"is_empty",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone {p q : ℝ≥0∞} (hpq : q ≤ p) : lp E q ≤ lp E p | λ f hf, mem_ℓp.of_exponent_ge hf hpq | lemma | lp.monotone | 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.of_exponent_ge",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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