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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orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | begin
refine ⟨_, by { rintro rfl, exact bot_orthogonal_eq_top }⟩,
intro h,
have : K ⊓ Kᗮ = ⊥ := K.orthogonal_disjoint.eq_bot,
rwa [h, inf_comm, top_inf_eq] at this
end | lemma | submodule.orthogonal_eq_top_iff | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inf_comm",
"top_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family_self :
orthogonal_family 𝕜 (λ b, ↥(cond b K Kᗮ)) (λ b, (cond b K Kᗮ).subtypeₗᵢ) | | tt tt := absurd rfl
| tt ff := λ _ x y, inner_right_of_mem_orthogonal x.prop y.prop
| ff tt := λ _ x y, inner_left_of_mem_orthogonal y.prop x.prop
| ff ff := absurd rfl | lemma | submodule.orthogonal_family_self | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"orthogonal_family"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bilin_form_of_real_inner_orthogonal {E} [normed_add_comm_group E] [inner_product_space ℝ E]
(K : submodule ℝ E) :
bilin_form_of_real_inner.orthogonal K = Kᗮ | rfl | lemma | bilin_form_of_real_inner_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_product_space",
"normed_add_comm_group",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho (U V : submodule 𝕜 E) : Prop | U ≤ Vᗮ | def | submodule.is_ortho | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | The proposition that two submodules are orthogonal. Has notation `U ⟂ V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_ortho_iff_le {U V : submodule 𝕜 E} : U ⟂ V ↔ U ≤ Vᗮ | iff.rfl | lemma | submodule.is_ortho_iff_le | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.symm {U V : submodule 𝕜 E} (h : U ⟂ V) : V ⟂ U | (le_orthogonal_orthogonal _).trans (orthogonal_le h) | lemma | submodule.is_ortho.symm | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_comm {U V : submodule 𝕜 E} : U ⟂ V ↔ V ⟂ U | ⟨is_ortho.symm, is_ortho.symm⟩ | lemma | submodule.is_ortho_comm | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetric_is_ortho : symmetric (is_ortho : submodule 𝕜 E → submodule 𝕜 E → Prop) | λ _ _, is_ortho.symm | lemma | submodule.symmetric_is_ortho | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.inner_eq {U V : submodule 𝕜 E} (h : U ⟂ V) {u v : E} (hu : u ∈ U) (hv : v ∈ V) :
⟪u, v⟫ = 0 | h.symm hv _ hu | lemma | submodule.is_ortho.inner_eq | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_iff_inner_eq {U V : submodule 𝕜 E} : U ⟂ V ↔ ∀ (u ∈ U) (v ∈ V), ⟪u, v⟫ = 0 | forall₄_congr $ λ u hu v hv, inner_eq_zero_symm | lemma | submodule.is_ortho_iff_inner_eq | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"forall₄_congr",
"inner_eq_zero_symm",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_bot_left {V : submodule 𝕜 E} : ⊥ ⟂ V | bot_le | lemma | submodule.is_ortho_bot_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"bot_le",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_bot_right {U : submodule 𝕜 E} : U ⟂ ⊥ | is_ortho_bot_left.symm | lemma | submodule.is_ortho_bot_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.mono_left {U₁ U₂ V : submodule 𝕜 E} (hU : U₂ ≤ U₁) (h : U₁ ⟂ V) : U₂ ⟂ V | hU.trans h | lemma | submodule.is_ortho.mono_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.mono_right {U V₁ V₂ : submodule 𝕜 E} (hV : V₂ ≤ V₁) (h : U ⟂ V₁) : U ⟂ V₂ | (h.symm.mono_left hV).symm | lemma | submodule.is_ortho.mono_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.mono {U₁ V₁ U₂ V₂ : submodule 𝕜 E} (hU : U₂ ≤ U₁) (hV : V₂ ≤ V₁) (h : U₁ ⟂ V₁) :
U₂ ⟂ V₂ | (h.mono_right hV).mono_left hU | lemma | submodule.is_ortho.mono | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_self {U : submodule 𝕜 E} : U ⟂ U ↔ U = ⊥ | ⟨λ h, eq_bot_iff.mpr $ λ x hx, inner_self_eq_zero.mp (h hx x hx), λ h, h.symm ▸ is_ortho_bot_left⟩ | lemma | submodule.is_ortho_self | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_orthogonal_right (U : submodule 𝕜 E) : U ⟂ Uᗮ | le_orthogonal_orthogonal _ | lemma | submodule.is_ortho_orthogonal_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_orthogonal_left (U : submodule 𝕜 E) : Uᗮ ⟂ U | (is_ortho_orthogonal_right U).symm | lemma | submodule.is_ortho_orthogonal_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.le {U V : submodule 𝕜 E} (h : U ⟂ V) : U ≤ Vᗮ | h | lemma | submodule.is_ortho.le | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.ge {U V : submodule 𝕜 E} (h : U ⟂ V) : V ≤ Uᗮ | h.symm | lemma | submodule.is_ortho.ge | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_top_right {U : submodule 𝕜 E} : U ⟂ ⊤ ↔ U = ⊥ | ⟨λ h, eq_bot_iff.mpr $ λ x hx, inner_self_eq_zero.mp (h hx _ mem_top),
λ h, h.symm ▸ is_ortho_bot_left⟩ | lemma | submodule.is_ortho_top_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_top_left {V : submodule 𝕜 E} : ⊤ ⟂ V ↔ V = ⊥ | is_ortho_comm.trans is_ortho_top_right | lemma | submodule.is_ortho_top_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.disjoint {U V : submodule 𝕜 E} (h : U ⟂ V) : disjoint U V | (submodule.orthogonal_disjoint _).mono_right h.symm | lemma | submodule.is_ortho.disjoint | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"disjoint",
"submodule",
"submodule.orthogonal_disjoint"
] | Orthogonal submodules are disjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_ortho_sup_left {U₁ U₂ V : submodule 𝕜 E} : U₁ ⊔ U₂ ⟂ V ↔ U₁ ⟂ V ∧ U₂ ⟂ V | sup_le_iff | lemma | submodule.is_ortho_sup_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_sup_right {U V₁ V₂ : submodule 𝕜 E} : U ⟂ V₁ ⊔ V₂ ↔ U ⟂ V₁ ∧ U ⟂ V₂ | is_ortho_comm.trans $ is_ortho_sup_left.trans $ is_ortho_comm.and is_ortho_comm | lemma | submodule.is_ortho_sup_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_Sup_left {U : set (submodule 𝕜 E)} {V : submodule 𝕜 E} :
Sup U ⟂ V ↔ ∀ Uᵢ ∈ U, Uᵢ ⟂ V | Sup_le_iff | lemma | submodule.is_ortho_Sup_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"Sup_le_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_Sup_right {U : submodule 𝕜 E} {V : set (submodule 𝕜 E)} :
U ⟂ Sup V ↔ ∀ Vᵢ ∈ V, U ⟂ Vᵢ | is_ortho_comm.trans $ is_ortho_Sup_left.trans $ by simp_rw is_ortho_comm | lemma | submodule.is_ortho_Sup_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_supr_left {ι : Sort*} {U : ι → submodule 𝕜 E} {V : submodule 𝕜 E} :
supr U ⟂ V ↔ ∀ i, U i ⟂ V | supr_le_iff | lemma | submodule.is_ortho_supr_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule",
"supr",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_supr_right {ι : Sort*} {U : submodule 𝕜 E} {V : ι → submodule 𝕜 E} :
U ⟂ supr V ↔ ∀ i, U ⟂ V i | is_ortho_comm.trans $ is_ortho_supr_left.trans $ by simp_rw is_ortho_comm | lemma | submodule.is_ortho_supr_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho_span {s t : set E} :
span 𝕜 s ⟂ span 𝕜 t ↔ ∀ ⦃u⦄, u ∈ s → ∀ ⦃v⦄, v ∈ t → ⟪u, v⟫ = 0 | begin
simp_rw [span_eq_supr_of_singleton_spans s, span_eq_supr_of_singleton_spans t,
is_ortho_supr_left, is_ortho_supr_right, is_ortho_iff_le, span_le, set.subset_def,
set_like.mem_coe, mem_orthogonal_singleton_iff_inner_left, set.mem_singleton_iff, forall_eq],
end | lemma | submodule.is_ortho_span | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"forall_eq",
"set.mem_singleton_iff",
"set.subset_def",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.map (f : E →ₗᵢ[𝕜] F) {U V : submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f | begin
rw is_ortho_iff_inner_eq at *,
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
linear_isometry.inner_map_map],
exact h,
end | lemma | submodule.is_ortho.map | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"and_imp",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"linear_isometry.inner_map_map",
"mem_map",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.comap (f : E →ₗᵢ[𝕜] F) {U V : submodule 𝕜 F} (h : U ⟂ V) : U.comap f ⟂ V.comap f | begin
rw is_ortho_iff_inner_eq at *,
simp_rw [mem_comap, ←f.inner_map_map],
intros u hu v hv,
exact h _ hu _ hv,
end | lemma | submodule.is_ortho.comap | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.map_iff (f : E ≃ₗᵢ[𝕜] F) {U V : submodule 𝕜 E} : U.map f ⟂ V.map f ↔ U ⟂ V | ⟨λ h, begin
have hf : ∀ p : submodule 𝕜 E, (p.map f).comap f.to_linear_isometry = p :=
comap_map_eq_of_injective f.injective,
simpa only [hf] using h.comap f.to_linear_isometry,
end, is_ortho.map f.to_linear_isometry⟩ | lemma | submodule.is_ortho.map_iff | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ortho.comap_iff (f : E ≃ₗᵢ[𝕜] F) {U V : submodule 𝕜 F} :
U.comap f ⟂ V.comap f ↔ U ⟂ V | ⟨λ h, begin
have hf : ∀ p : submodule 𝕜 F, (p.comap f).map f.to_linear_isometry = p :=
map_comap_eq_of_surjective f.surjective,
simpa only [hf] using h.map f.to_linear_isometry,
end, is_ortho.comap f.to_linear_isometry⟩ | lemma | submodule.is_ortho.comap_iff | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family_iff_pairwise {ι} {V : ι → submodule 𝕜 E} :
orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ) ↔ pairwise ((⟂) on V) | forall₃_congr $ λ i j hij,
subtype.forall.trans $ forall₂_congr $ λ x hx, subtype.forall.trans $ forall₂_congr $ λ y hy,
inner_eq_zero_symm | lemma | orthogonal_family_iff_pairwise | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"forall₂_congr",
"forall₃_congr",
"inner_eq_zero_symm",
"orthogonal_family",
"pairwise",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.is_ortho {ι} {V : ι → submodule 𝕜 E}
(hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) {i j : ι} (hij : i ≠ j) :
V i ⟂ V j | hV.pairwise hij | lemma | orthogonal_family.is_ortho | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"orthogonal_family",
"submodule"
] | Two submodules in an orthogonal family with different indices are orthogonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_Lp.inner_product_space {ι : Type*} [fintype ι] (f : ι → Type*)
[Π i, normed_add_comm_group (f i)] [Π i, inner_product_space 𝕜 (f i)] :
inner_product_space 𝕜 (pi_Lp 2 f) | { inner := λ x y, ∑ i, inner (x i) (y i),
norm_sq_eq_inner := λ x,
by simp only [pi_Lp.norm_sq_eq_of_L2, add_monoid_hom.map_sum, ← norm_sq_eq_inner, one_div],
conj_symm :=
begin
intros x y,
unfold inner,
rw ring_hom.map_sum,
apply finset.sum_congr rfl,
rintros z -,
apply inner_conj_sym... | instance | pi_Lp.inner_product_space | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"finset.mul_sum",
"fintype",
"inner_add_left",
"inner_conj_symm",
"inner_product_space",
"inner_smul_left",
"normed_add_comm_group",
"one_div",
"pi_Lp",
"pi_Lp.norm_sq_eq_of_L2",
"ring_hom.map_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Lp.inner_apply {ι : Type*} [fintype ι] {f : ι → Type*}
[Π i, normed_add_comm_group (f i)] [Π i, inner_product_space 𝕜 (f i)] (x y : pi_Lp 2 f) :
⟪x, y⟫ = ∑ i, ⟪x i, y i⟫ | rfl | lemma | pi_Lp.inner_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"fintype",
"inner_product_space",
"normed_add_comm_group",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space (𝕜 : Type*) [is_R_or_C 𝕜]
(n : Type*) [fintype n] : Type* | pi_Lp 2 (λ (i : n), 𝕜) | def | euclidean_space | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"fintype",
"is_R_or_C",
"pi_Lp"
] | The standard real/complex Euclidean space, functions on a finite type. For an `n`-dimensional
space use `euclidean_space 𝕜 (fin n)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
euclidean_space.nnnorm_eq {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
(x : euclidean_space 𝕜 n) : ‖x‖₊ = nnreal.sqrt (∑ i, ‖x i‖₊ ^ 2) | pi_Lp.nnnorm_eq_of_L2 x | lemma | euclidean_space.nnnorm_eq | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"fintype",
"is_R_or_C",
"nnreal.sqrt",
"pi_Lp.nnnorm_eq_of_L2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.norm_eq {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
(x : euclidean_space 𝕜 n) : ‖x‖ = real.sqrt (∑ i, ‖x i‖ ^ 2) | by simpa only [real.coe_sqrt, nnreal.coe_sum] using congr_arg (coe : ℝ≥0 → ℝ) x.nnnorm_eq | lemma | euclidean_space.norm_eq | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"fintype",
"is_R_or_C",
"nnreal.coe_sum",
"real.coe_sqrt",
"real.sqrt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.dist_eq {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
(x y : euclidean_space 𝕜 n) : dist x y = (∑ i, dist (x i) (y i) ^ 2).sqrt | (pi_Lp.dist_eq_of_L2 x y : _) | lemma | euclidean_space.dist_eq | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"fintype",
"is_R_or_C",
"pi_Lp.dist_eq_of_L2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.nndist_eq {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
(x y : euclidean_space 𝕜 n) : nndist x y = (∑ i, nndist (x i) (y i) ^ 2).sqrt | (pi_Lp.nndist_eq_of_L2 x y : _) | lemma | euclidean_space.nndist_eq | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"fintype",
"is_R_or_C",
"pi_Lp.nndist_eq_of_L2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.edist_eq {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
(x y : euclidean_space 𝕜 n) : edist x y = (∑ i, edist (x i) (y i) ^ 2) ^ (1 / 2 : ℝ) | (pi_Lp.edist_eq_of_L2 x y : _) | lemma | euclidean_space.edist_eq | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"fintype",
"is_R_or_C",
"pi_Lp.edist_eq_of_L2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finrank_euclidean_space :
finite_dimensional.finrank 𝕜 (euclidean_space 𝕜 ι) = fintype.card ι | by simp | lemma | finrank_euclidean_space | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"finite_dimensional.finrank",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finrank_euclidean_space_fin {n : ℕ} :
finite_dimensional.finrank 𝕜 (euclidean_space 𝕜 (fin n)) = n | by simp | lemma | finrank_euclidean_space_fin | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"finite_dimensional.finrank"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.inner_eq_star_dot_product (x y : euclidean_space 𝕜 ι) :
⟪x, y⟫ = matrix.dot_product (star $ pi_Lp.equiv _ _ x) (pi_Lp.equiv _ _ y) | rfl | lemma | euclidean_space.inner_eq_star_dot_product | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"matrix.dot_product",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.inner_pi_Lp_equiv_symm (x y : ι → 𝕜) :
⟪(pi_Lp.equiv 2 _).symm x, (pi_Lp.equiv 2 _).symm y⟫ = matrix.dot_product (star x) y | rfl | lemma | euclidean_space.inner_pi_Lp_equiv_symm | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"matrix.dot_product",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
direct_sum.is_internal.isometry_L2_of_orthogonal_family
[decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V)
(hV' : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) :
E ≃ₗᵢ[𝕜] pi_Lp 2 (λ i, V i) | begin
let e₁ := direct_sum.linear_equiv_fun_on_fintype 𝕜 ι (λ i, V i),
let e₂ := linear_equiv.of_bijective (direct_sum.coe_linear_map V) hV,
refine linear_equiv.isometry_of_inner (e₂.symm.trans e₁) _,
suffices : ∀ v w, ⟪v, w⟫ = ⟪e₂ (e₁.symm v), e₂ (e₁.symm w)⟫,
{ intros v₀ w₀,
convert this (e₁ (e₂.symm v... | def | direct_sum.is_internal.isometry_L2_of_orthogonal_family | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"direct_sum.coe_linear_map",
"direct_sum.is_internal",
"direct_sum.linear_equiv_fun_on_fintype",
"linear_equiv.apply_symm_apply",
"linear_equiv.isometry_of_inner",
"linear_equiv.of_bijective",
"linear_equiv.symm_apply_apply",
"orthogonal_family",
"pi_Lp",
"pi_Lp.inner_apply",
"submodule",
"sum... | A finite, mutually orthogonal family of subspaces of `E`, which span `E`, induce an isometry
from `E` to `pi_Lp 2` of the subspaces equipped with the `L2` inner product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.isometry_L2_of_orthogonal_family_symm_apply
[decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V)
(hV' : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ))
(w : pi_Lp 2 (λ i, V i)) :
(hV.isometry_L2_of_orthogonal_family hV').symm w = ∑ i, (w i : E) | begin
classical,
let e₁ := direct_sum.linear_equiv_fun_on_fintype 𝕜 ι (λ i, V i),
let e₂ := linear_equiv.of_bijective (direct_sum.coe_linear_map V) hV,
suffices : ∀ v : ⨁ i, V i, e₂ v = ∑ i, e₁ v i,
{ exact this (e₁.symm w) },
intros v,
simp [e₂, direct_sum.coe_linear_map, direct_sum.to_module, dfinsupp.... | lemma | direct_sum.is_internal.isometry_L2_of_orthogonal_family_symm_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"dfinsupp.sum_add_hom_apply",
"direct_sum.coe_linear_map",
"direct_sum.is_internal",
"direct_sum.linear_equiv_fun_on_fintype",
"direct_sum.to_module",
"linear_equiv.of_bijective",
"orthogonal_family",
"pi_Lp",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.equiv :
euclidean_space 𝕜 ι ≃L[𝕜] (ι → 𝕜) | (pi_Lp.linear_equiv 2 𝕜 (λ i : ι, 𝕜)).to_continuous_linear_equiv | def | euclidean_space.equiv | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"pi_Lp.linear_equiv"
] | `pi_Lp.linear_equiv` upgraded to a continuous linear map between `euclidean_space 𝕜 ι`
and `ι → 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
euclidean_space.projₗ (i : ι) :
euclidean_space 𝕜 ι →ₗ[𝕜] 𝕜 | (linear_map.proj i).comp (pi_Lp.linear_equiv 2 𝕜 (λ i : ι, 𝕜) : euclidean_space 𝕜 ι →ₗ[𝕜] ι → 𝕜) | def | euclidean_space.projₗ | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"linear_map.proj",
"pi_Lp.linear_equiv"
] | The projection on the `i`-th coordinate of `euclidean_space 𝕜 ι`, as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
euclidean_space.proj (i : ι) :
euclidean_space 𝕜 ι →L[𝕜] 𝕜 | ⟨euclidean_space.projₗ i, continuous_apply i⟩ | def | euclidean_space.proj | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"continuous_apply",
"euclidean_space"
] | The projection on the `i`-th coordinate of `euclidean_space 𝕜 ι`,
as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
euclidean_space.single [decidable_eq ι] (i : ι) (a : 𝕜) :
euclidean_space 𝕜 ι | (pi_Lp.equiv _ _).symm (pi.single i a) | def | euclidean_space.single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"pi_Lp.equiv"
] | The vector given in euclidean space by being `1 : 𝕜` at coordinate `i : ι` and `0 : 𝕜` at
all other coordinates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_Lp.equiv_single [decidable_eq ι] (i : ι) (a : 𝕜) :
pi_Lp.equiv _ _ (euclidean_space.single i a) = pi.single i a | rfl | lemma | pi_Lp.equiv_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Lp.equiv_symm_single [decidable_eq ι] (i : ι) (a : 𝕜) :
(pi_Lp.equiv _ _).symm (pi.single i a) = euclidean_space.single i a | rfl | lemma | pi_Lp.equiv_symm_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.single_apply [decidable_eq ι] (i : ι) (a : 𝕜) (j : ι) :
(euclidean_space.single i a) j = ite (j = i) a 0 | by { rw [euclidean_space.single, pi_Lp.equiv_symm_apply, ← pi.single_apply i a j] } | theorem | euclidean_space.single_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"pi_Lp.equiv_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.inner_single_left [decidable_eq ι] (i : ι) (a : 𝕜) (v : euclidean_space 𝕜 ι) :
⟪euclidean_space.single i (a : 𝕜), v⟫ = conj a * (v i) | by simp [apply_ite conj] | lemma | euclidean_space.inner_single_left | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"apply_ite",
"euclidean_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.inner_single_right [decidable_eq ι] (i : ι) (a : 𝕜)
(v : euclidean_space 𝕜 ι) :
⟪v, euclidean_space.single i (a : 𝕜)⟫ = a * conj (v i) | by simp [apply_ite conj, mul_comm] | lemma | euclidean_space.inner_single_right | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"apply_ite",
"euclidean_space",
"euclidean_space.single",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.norm_single [decidable_eq ι] (i : ι) (a : 𝕜) :
‖euclidean_space.single i (a : 𝕜)‖ = ‖a‖ | (pi_Lp.norm_equiv_symm_single 2 (λ i, 𝕜) i a : _) | lemma | euclidean_space.norm_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"pi_Lp.norm_equiv_symm_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.nnnorm_single [decidable_eq ι] (i : ι) (a : 𝕜) :
‖euclidean_space.single i (a : 𝕜)‖₊ = ‖a‖₊ | (pi_Lp.nnnorm_equiv_symm_single 2 (λ i, 𝕜) i a : _) | lemma | euclidean_space.nnnorm_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"pi_Lp.nnnorm_equiv_symm_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.dist_single_same [decidable_eq ι] (i : ι) (a b : 𝕜) :
dist (euclidean_space.single i (a : 𝕜)) (euclidean_space.single i (b : 𝕜)) = dist a b | (pi_Lp.dist_equiv_symm_single_same 2 (λ i, 𝕜) i a b : _) | lemma | euclidean_space.dist_single_same | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"pi_Lp.dist_equiv_symm_single_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.nndist_single_same [decidable_eq ι] (i : ι) (a b : 𝕜) :
nndist (euclidean_space.single i (a : 𝕜)) (euclidean_space.single i (b : 𝕜)) = nndist a b | (pi_Lp.nndist_equiv_symm_single_same 2 (λ i, 𝕜) i a b : _) | lemma | euclidean_space.nndist_single_same | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"pi_Lp.nndist_equiv_symm_single_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.edist_single_same [decidable_eq ι] (i : ι) (a b : 𝕜) :
edist (euclidean_space.single i (a : 𝕜)) (euclidean_space.single i (b : 𝕜)) = edist a b | (pi_Lp.edist_equiv_symm_single_same 2 (λ i, 𝕜) i a b : _) | lemma | euclidean_space.edist_single_same | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"pi_Lp.edist_equiv_symm_single_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_space.orthonormal_single [decidable_eq ι] :
orthonormal 𝕜 (λ i : ι, euclidean_space.single i (1 : 𝕜)) | begin
simp_rw [orthonormal_iff_ite, euclidean_space.inner_single_left, map_one, one_mul,
euclidean_space.single_apply],
intros i j,
refl,
end | lemma | euclidean_space.orthonormal_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.inner_single_left",
"euclidean_space.single",
"euclidean_space.single_apply",
"map_one",
"one_mul",
"orthonormal",
"orthonormal_iff_ite"
] | `euclidean_space.single` forms an orthonormal family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
euclidean_space.pi_Lp_congr_left_single [decidable_eq ι] {ι' : Type*} [fintype ι']
[decidable_eq ι'] (e : ι' ≃ ι) (i' : ι') (v : 𝕜):
linear_isometry_equiv.pi_Lp_congr_left 2 𝕜 𝕜 e (euclidean_space.single i' v) =
euclidean_space.single (e i') v | linear_isometry_equiv.pi_Lp_congr_left_single e i' _ | lemma | euclidean_space.pi_Lp_congr_left_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"fintype",
"linear_isometry_equiv.pi_Lp_congr_left",
"linear_isometry_equiv.pi_Lp_congr_left_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal_basis | of_repr :: (repr : E ≃ₗᵢ[𝕜] euclidean_space 𝕜 ι) | structure | orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space"
] | An orthonormal basis on E is an identification of `E` with its dimensional-matching
`euclidean_space 𝕜 ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_repr [decidable_eq ι] (e : E ≃ₗᵢ[𝕜] euclidean_space 𝕜 ι) :
⇑(orthonormal_basis.of_repr e) = λ i, e.symm (euclidean_space.single i (1 : 𝕜)) | begin
rw coe_fn,
unfold has_coe_to_fun.coe,
funext,
congr,
simp only [eq_iff_true_of_subsingleton],
end | lemma | orthonormal_basis.coe_of_repr | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"eq_iff_true_of_subsingleton",
"euclidean_space",
"euclidean_space.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_symm_single [decidable_eq ι] (b : orthonormal_basis ι 𝕜 E) (i : ι) :
b.repr.symm (euclidean_space.single i (1:𝕜)) = b i | by { classical, congr, simp, } | lemma | orthonormal_basis.repr_symm_single | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_self [decidable_eq ι] (b : orthonormal_basis ι 𝕜 E) (i : ι) :
b.repr (b i) = euclidean_space.single i (1:𝕜) | by rw [← b.repr_symm_single i, linear_isometry_equiv.apply_symm_apply] | lemma | orthonormal_basis.repr_self | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.single",
"linear_isometry_equiv.apply_symm_apply",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_apply_apply (b : orthonormal_basis ι 𝕜 E) (v : E) (i : ι) :
b.repr v i = ⟪b i, v⟫ | begin
classical,
rw [← b.repr.inner_map_map (b i) v, b.repr_self i, euclidean_space.inner_single_left],
simp only [one_mul, eq_self_iff_true, map_one],
end | lemma | orthonormal_basis.repr_apply_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.inner_single_left",
"map_one",
"one_mul",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal (b : orthonormal_basis ι 𝕜 E) : orthonormal 𝕜 b | begin
classical,
rw orthonormal_iff_ite,
intros i j,
rw [← b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j,
euclidean_space.inner_single_left, euclidean_space.single_apply, map_one, one_mul],
end | lemma | orthonormal_basis.orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.inner_single_left",
"euclidean_space.single_apply",
"map_one",
"one_mul",
"orthonormal",
"orthonormal_basis",
"orthonormal_iff_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_basis (b : orthonormal_basis ι 𝕜 E) : basis ι 𝕜 E | basis.of_equiv_fun b.repr.to_linear_equiv | def | orthonormal_basis.to_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"basis.of_equiv_fun",
"orthonormal_basis"
] | The `basis ι 𝕜 E` underlying the `orthonormal_basis` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_basis (b : orthonormal_basis ι 𝕜 E) :
(⇑b.to_basis : ι → E) = ⇑b | begin
change ⇑(basis.of_equiv_fun b.repr.to_linear_equiv) = b,
ext j,
classical,
rw basis.coe_of_equiv_fun,
congr,
end | lemma | orthonormal_basis.coe_to_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.coe_of_equiv_fun",
"basis.of_equiv_fun",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_basis_repr (b : orthonormal_basis ι 𝕜 E) :
b.to_basis.equiv_fun = b.repr.to_linear_equiv | basis.equiv_fun_of_equiv_fun _ | lemma | orthonormal_basis.coe_to_basis_repr | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.equiv_fun_of_equiv_fun",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_basis_repr_apply (b : orthonormal_basis ι 𝕜 E) (x : E) (i : ι) :
b.to_basis.repr x i = b.repr x i | by {rw [← basis.equiv_fun_apply, orthonormal_basis.coe_to_basis_repr,
linear_isometry_equiv.coe_to_linear_equiv]} | lemma | orthonormal_basis.coe_to_basis_repr_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.equiv_fun_apply",
"linear_isometry_equiv.coe_to_linear_equiv",
"orthonormal_basis",
"orthonormal_basis.coe_to_basis_repr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_repr (b : orthonormal_basis ι 𝕜 E) (x : E) :
∑ i, b.repr x i • b i = x | by { simp_rw [← b.coe_to_basis_repr_apply, ← b.coe_to_basis], exact b.to_basis.sum_repr x } | lemma | orthonormal_basis.sum_repr | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_repr_symm (b : orthonormal_basis ι 𝕜 E) (v : euclidean_space 𝕜 ι) :
∑ i , v i • b i = (b.repr.symm v) | by { simpa using (b.to_basis.equiv_fun_symm_apply v).symm } | lemma | orthonormal_basis.sum_repr_symm | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_inner_mul_inner (b : orthonormal_basis ι 𝕜 E) (x y : E) :
∑ i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫ | begin
have := congr_arg (innerSL 𝕜 x) (b.sum_repr y),
rw map_sum at this,
convert this,
ext i,
rw [smul_hom_class.map_smul, b.repr_apply_apply, mul_comm],
refl,
end | lemma | orthonormal_basis.sum_inner_mul_inner | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"innerSL",
"mul_comm",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_projection_eq_sum {U : submodule 𝕜 E} [complete_space U]
(b : orthonormal_basis ι 𝕜 U) (x : E) :
orthogonal_projection U x = ∑ i, ⟪(b i : E), x⟫ • b i | by simpa only [b.repr_apply_apply, inner_orthogonal_projection_eq_of_mem_left]
using (b.sum_repr (orthogonal_projection U x)).symm | lemma | orthonormal_basis.orthogonal_projection_eq_sum | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complete_space",
"inner_orthogonal_projection_eq_of_mem_left",
"orthogonal_projection",
"orthonormal_basis",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {G : Type*}
[normed_add_comm_group G] [inner_product_space 𝕜 G] (b : orthonormal_basis ι 𝕜 E)
(L : E ≃ₗᵢ[𝕜] G) :
orthonormal_basis ι 𝕜 G | { repr := L.symm.trans b.repr } | def | orthonormal_basis.map | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"inner_product_space",
"normed_add_comm_group",
"orthonormal_basis"
] | Mapping an orthonormal basis along a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_apply {G : Type*} [normed_add_comm_group G] [inner_product_space 𝕜 G]
(b : orthonormal_basis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) (i : ι) :
b.map L i = L (b i) | rfl | lemma | orthonormal_basis.map_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"inner_product_space",
"normed_add_comm_group",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_basis_map {G : Type*} [normed_add_comm_group G] [inner_product_space 𝕜 G]
(b : orthonormal_basis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) :
(b.map L).to_basis = b.to_basis.map L.to_linear_equiv | rfl | lemma | orthonormal_basis.to_basis_map | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"inner_product_space",
"normed_add_comm_group",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.basis.to_orthonormal_basis (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
orthonormal_basis ι 𝕜 E | orthonormal_basis.of_repr $
linear_equiv.isometry_of_inner v.equiv_fun
begin
intros x y,
let p : euclidean_space 𝕜 ι := v.equiv_fun x,
let q : euclidean_space 𝕜 ι := v.equiv_fun y,
have key : ⟪p, q⟫ = ⟪∑ i, p i • v i, ∑ i, q i • v i⟫,
{ simp [sum_inner, inner_smul_left, hv.inner_right_fintype] },
convert ... | def | basis.to_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"euclidean_space",
"inner_smul_left",
"linear_equiv.isometry_of_inner",
"orthonormal",
"orthonormal_basis",
"sum_inner"
] | A basis that is orthonormal is an orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.basis.coe_to_orthonormal_basis_repr (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
((v.to_orthonormal_basis hv).repr : E → euclidean_space 𝕜 ι) = v.equiv_fun | rfl | lemma | basis.coe_to_orthonormal_basis_repr | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"euclidean_space",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.basis.coe_to_orthonormal_basis_repr_symm
(v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
((v.to_orthonormal_basis hv).repr.symm : euclidean_space 𝕜 ι → E) = v.equiv_fun.symm | rfl | lemma | basis.coe_to_orthonormal_basis_repr_symm | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"euclidean_space",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.basis.to_basis_to_orthonormal_basis (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
(v.to_orthonormal_basis hv).to_basis = v | by simp [basis.to_orthonormal_basis, orthonormal_basis.to_basis] | lemma | basis.to_basis_to_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"basis.to_orthonormal_basis",
"orthonormal",
"orthonormal_basis.to_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.basis.coe_to_orthonormal_basis (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
(v.to_orthonormal_basis hv : ι → E) = (v : ι → E) | calc (v.to_orthonormal_basis hv : ι → E) = ((v.to_orthonormal_basis hv).to_basis : ι → E) :
by { classical, rw orthonormal_basis.coe_to_basis }
... = (v : ι → E) : by simp | lemma | basis.coe_to_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"orthonormal",
"orthonormal_basis.coe_to_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (hon : orthonormal 𝕜 v) (hsp: ⊤ ≤ submodule.span 𝕜 (set.range v)):
orthonormal_basis ι 𝕜 E | (basis.mk (orthonormal.linear_independent hon) hsp).to_orthonormal_basis (by rwa basis.coe_mk) | def | orthonormal_basis.mk | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.coe_mk",
"basis.mk",
"orthonormal",
"orthonormal.linear_independent",
"orthonormal_basis",
"set.range",
"submodule.span"
] | A finite orthonormal set that spans is an orthonormal basis | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk (hon : orthonormal 𝕜 v) (hsp: ⊤ ≤ submodule.span 𝕜 (set.range v)) :
⇑(orthonormal_basis.mk hon hsp) = v | by classical; rw [orthonormal_basis.mk, _root_.basis.coe_to_orthonormal_basis, basis.coe_mk] | lemma | orthonormal_basis.coe_mk | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.coe_mk",
"orthonormal",
"orthonormal_basis.mk",
"set.range",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span [decidable_eq E] {v' : ι' → E} (h : orthonormal 𝕜 v') (s : finset ι') :
orthonormal_basis s 𝕜 (span 𝕜 (s.image v' : set E)) | let
e₀' : basis s 𝕜 _ := basis.span (h.linear_independent.comp (coe : s → ι') subtype.coe_injective),
e₀ : orthonormal_basis s 𝕜 _ := orthonormal_basis.mk
begin
convert orthonormal_span (h.comp (coe : s → ι') subtype.coe_injective),
ext,
simp [e₀', basis.span_apply], | def | orthonormal_basis.span | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis",
"basis.span",
"basis.span_apply",
"finset",
"orthonormal",
"orthonormal_basis",
"orthonormal_basis.mk",
"orthonormal_span",
"subtype.coe_injective"
] | Any finite subset of a orthonormal family is an `orthonormal_basis` for its span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_apply [decidable_eq E]
{v' : ι' → E} (h : orthonormal 𝕜 v') (s : finset ι') (i : s) :
(orthonormal_basis.span h s i : E) = v' i | by simp only [orthonormal_basis.span, basis.span_apply, linear_isometry_equiv.of_eq_symm,
orthonormal_basis.map_apply, orthonormal_basis.coe_mk,
linear_isometry_equiv.coe_of_eq_apply] | lemma | span_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.span_apply",
"finset",
"linear_isometry_equiv.coe_of_eq_apply",
"linear_isometry_equiv.of_eq_symm",
"orthonormal",
"orthonormal_basis.coe_mk",
"orthonormal_basis.map_apply",
"orthonormal_basis.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_orthogonal_eq_bot (hon : orthonormal 𝕜 v) (hsp : (span 𝕜 (set.range v))ᗮ = ⊥) :
orthonormal_basis ι 𝕜 E | orthonormal_basis.mk hon
begin
refine eq.ge _,
haveI : finite_dimensional 𝕜 (span 𝕜 (range v)) :=
finite_dimensional.span_of_finite 𝕜 (finite_range v),
haveI : complete_space (span 𝕜 (range v)) := finite_dimensional.complete 𝕜 _,
rwa orthogonal_eq_bot_iff at hsp,
end | def | mk_of_orthogonal_eq_bot | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complete_space",
"eq.ge",
"finite_dimensional",
"finite_dimensional.complete",
"finite_dimensional.span_of_finite",
"orthonormal",
"orthonormal_basis",
"orthonormal_basis.mk",
"set.range"
] | A finite orthonormal family of vectors whose span has trivial orthogonal complement is an
orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_orthogonal_eq_bot_mk (hon : orthonormal 𝕜 v)
(hsp : (span 𝕜 (set.range v))ᗮ = ⊥) :
⇑(orthonormal_basis.mk_of_orthogonal_eq_bot hon hsp) = v | orthonormal_basis.coe_mk hon _ | lemma | coe_of_orthogonal_eq_bot_mk | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthonormal",
"orthonormal_basis.coe_mk",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reindex (b : orthonormal_basis ι 𝕜 E) (e : ι ≃ ι') : orthonormal_basis ι' 𝕜 E | orthonormal_basis.of_repr (b.repr.trans (linear_isometry_equiv.pi_Lp_congr_left 2 𝕜 𝕜 e)) | def | reindex | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"linear_isometry_equiv.pi_Lp_congr_left",
"orthonormal_basis"
] | `b.reindex (e : ι ≃ ι')` is an `orthonormal_basis` indexed by `ι'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reindex_apply (b : orthonormal_basis ι 𝕜 E) (e : ι ≃ ι') (i' : ι') :
(b.reindex e) i' = b (e.symm i') | begin
classical,
dsimp [reindex, orthonormal_basis.has_coe_to_fun],
rw coe_of_repr,
dsimp,
rw [← b.repr_symm_single, linear_isometry_equiv.pi_Lp_congr_left_symm,
euclidean_space.pi_Lp_congr_left_single],
end | lemma | reindex_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.pi_Lp_congr_left_single",
"linear_isometry_equiv.pi_Lp_congr_left_symm",
"orthonormal_basis",
"reindex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_reindex (b : orthonormal_basis ι 𝕜 E) (e : ι ≃ ι') :
⇑(b.reindex e) = ⇑b ∘ ⇑(e.symm) | funext (b.reindex_apply e) | lemma | coe_reindex | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_reindex
(b : orthonormal_basis ι 𝕜 E) (e : ι ≃ ι') (x : E) (i' : ι') :
(b.reindex e).repr x i' = b.repr x (e.symm i') | by { classical,
rw [orthonormal_basis.repr_apply_apply, b.repr_apply_apply, orthonormal_basis.coe_reindex] } | lemma | repr_reindex | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthonormal_basis",
"orthonormal_basis.repr_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.orthonormal_basis_one_I : orthonormal_basis (fin 2) ℝ ℂ | (complex.basis_one_I.to_orthonormal_basis
begin
rw orthonormal_iff_ite,
intros i, fin_cases i;
intros j; fin_cases j;
simp [real_inner_eq_re_inner]
end) | def | complex.orthonormal_basis_one_I | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthonormal_basis",
"orthonormal_iff_ite",
"real_inner_eq_re_inner"
] | `![1, I]` is an orthonormal basis for `ℂ` considered as a real inner product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.orthonormal_basis_one_I_repr_apply (z : ℂ) :
complex.orthonormal_basis_one_I.repr z = ![z.re, z.im] | rfl | lemma | complex.orthonormal_basis_one_I_repr_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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