statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
continuous_linear_equiv_of_bilin (coercive : is_coercive B) : V ≃L[ℝ] V | continuous_linear_equiv.of_bijective
B♯
coercive.ker_eq_bot
coercive.range_eq_top | def | is_coercive.continuous_linear_equiv_of_bilin | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"continuous_linear_equiv.of_bijective",
"is_coercive"
] | The Lax-Milgram equivalence of a coercive bounded bilinear operator:
for all `v : V`, `continuous_linear_equiv_of_bilin B v` is the unique element `V`
such that `⟪continuous_linear_equiv_of_bilin B v, w⟫ = B v w`.
The Lax-Milgram theorem states that this is a continuous equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv_of_bilin_apply (coercive : is_coercive B) (v w : V) :
⟪coercive.continuous_linear_equiv_of_bilin v, w⟫_ℝ = B v w | continuous_linear_map_of_bilin_apply ℝ B v w | lemma | is_coercive.continuous_linear_equiv_of_bilin_apply | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"is_coercive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_continuous_linear_equiv_of_bilin (coercive : is_coercive B) {v f : V}
(is_lax_milgram : (∀ w, ⟪f, w⟫_ℝ = B v w)) :
f = coercive.continuous_linear_equiv_of_bilin v | unique_continuous_linear_map_of_bilin ℝ B is_lax_milgram | lemma | is_coercive.unique_continuous_linear_equiv_of_bilin | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"is_coercive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_formal_adjoint (T : E →ₗ.[𝕜] F) (S : F →ₗ.[𝕜] E) : Prop | ∀ (x : T.domain) (y : S.domain), ⟪T x, y⟫ = ⟪(x : E), S y⟫ | def | linear_pmap.is_formal_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | An operator `T` is a formal adjoint of `S` if for all `x` in the domain of `T` and `y` in the
domain of `S`, we have that `⟪T x, y⟫ = ⟪x, S y⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_formal_adjoint.symm (h : T.is_formal_adjoint S) : S.is_formal_adjoint T | λ y _, by rw [←inner_conj_symm, ←inner_conj_symm (y : F), h] | lemma | linear_pmap.is_formal_adjoint.symm | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_domain : submodule 𝕜 F | { carrier := {y | continuous ((innerₛₗ 𝕜 y).comp T.to_fun)},
zero_mem' := by { rw [set.mem_set_of_eq, linear_map.map_zero, linear_map.zero_comp],
exact continuous_zero },
add_mem' := λ x y hx hy, by { rw [set.mem_set_of_eq, linear_map.map_add] at *, exact hx.add hy },
smul_mem' := λ a x hx, by { rw [set.me... | def | linear_pmap.adjoint_domain | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"continuous",
"innerₛₗ",
"linear_map.map_add",
"linear_map.map_smulₛₗ",
"linear_map.map_zero",
"linear_map.zero_comp",
"submodule"
] | The domain of the adjoint operator.
This definition is needed to construct the adjoint operator and the preferred version to use is
`T.adjoint.domain` instead of `T.adjoint_domain`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_domain_mk_clm (y : T.adjoint_domain) : T.domain →L[𝕜] 𝕜 | ⟨(innerₛₗ 𝕜 (y : F)).comp T.to_fun, y.prop⟩ | def | linear_pmap.adjoint_domain_mk_clm | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"innerₛₗ"
] | The operator `λ x, ⟪y, T x⟫` considered as a continuous linear operator from `T.adjoint_domain`
to `𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_domain_mk_clm_apply (y : T.adjoint_domain) (x : T.domain) :
adjoint_domain_mk_clm T y x = ⟪(y : F), T x⟫ | rfl | lemma | linear_pmap.adjoint_domain_mk_clm_apply | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_domain_mk_clm_extend (y : T.adjoint_domain) :
E →L[𝕜] 𝕜 | (T.adjoint_domain_mk_clm y).extend (submodule.subtypeL T.domain)
hT.dense_range_coe uniform_embedding_subtype_coe.to_uniform_inducing | def | linear_pmap.adjoint_domain_mk_clm_extend | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"extend",
"submodule.subtypeL"
] | The unique continuous extension of the operator `adjoint_domain_mk_clm` to `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_domain_mk_clm_extend_apply (y : T.adjoint_domain) (x : T.domain) :
adjoint_domain_mk_clm_extend hT y (x : E) = ⟪(y : F), T x⟫ | continuous_linear_map.extend_eq _ _ _ _ _ | lemma | linear_pmap.adjoint_domain_mk_clm_extend_apply | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"continuous_linear_map.extend_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_aux : T.adjoint_domain →ₗ[𝕜] E | { to_fun := λ y, (inner_product_space.to_dual 𝕜 E).symm (adjoint_domain_mk_clm_extend hT y),
map_add' := λ x y, hT.eq_of_inner_left $ λ _,
by simp only [inner_add_left, submodule.coe_add, inner_product_space.to_dual_symm_apply,
adjoint_domain_mk_clm_extend_apply],
map_smul' := λ _ _, hT.eq_of_inner_left ... | def | linear_pmap.adjoint_aux | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"inner_add_left",
"inner_product_space.to_dual",
"inner_product_space.to_dual_symm_apply",
"inner_smul_left",
"ring_hom.id_apply",
"submodule.coe_add",
"submodule.coe_smul_of_tower"
] | The adjoint as a linear map from its domain to `E`.
This is an auxiliary definition needed to define the adjoint operator as a `linear_pmap` without
the assumption that `T.domain` is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_aux_inner (y : T.adjoint_domain) (x : T.domain) :
⟪adjoint_aux hT y, x⟫ = ⟪(y : F), T x⟫ | by simp only [adjoint_aux, linear_map.coe_mk, inner_product_space.to_dual_symm_apply,
adjoint_domain_mk_clm_extend_apply] | lemma | linear_pmap.adjoint_aux_inner | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"inner_product_space.to_dual_symm_apply",
"linear_map.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_aux_unique (y : T.adjoint_domain) {x₀ : E}
(hx₀ : ∀ x : T.domain, ⟪x₀, x⟫ = ⟪(y : F), T x⟫) : adjoint_aux hT y = x₀ | hT.eq_of_inner_left (λ v, (adjoint_aux_inner hT _ _).trans (hx₀ v).symm) | lemma | linear_pmap.adjoint_aux_unique | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint : F →ₗ.[𝕜] E | { domain := T.adjoint_domain,
to_fun := if hT : dense (T.domain : set E) then adjoint_aux hT else 0 } | def | linear_pmap.adjoint | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"dense"
] | The adjoint operator as a partially defined linear operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_adjoint_domain_iff (y : F) :
y ∈ T†.domain ↔ continuous ((innerₛₗ 𝕜 y).comp T.to_fun) | iff.rfl | lemma | linear_pmap.mem_adjoint_domain_iff | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"continuous",
"innerₛₗ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_adjoint_domain_of_exists (y : F) (h : ∃ w : E, ∀ (x : T.domain), ⟪w, x⟫ = ⟪y, T x⟫) :
y ∈ T†.domain | begin
cases h with w hw,
rw T.mem_adjoint_domain_iff,
have : continuous ((innerSL 𝕜 w).comp T.domain.subtypeL) := by continuity,
convert this using 1,
exact funext (λ x, (hw x).symm),
end | lemma | linear_pmap.mem_adjoint_domain_of_exists | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"continuity",
"continuous",
"innerSL"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_apply_of_not_dense (hT : ¬ dense (T.domain : set E)) (y : T†.domain) : T† y = 0 | begin
change (if hT : dense (T.domain : set E) then adjoint_aux hT else 0) y = _,
simp only [hT, not_false_iff, dif_neg, linear_map.zero_apply],
end | lemma | linear_pmap.adjoint_apply_of_not_dense | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"dense",
"linear_map.zero_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_apply_of_dense (y : T†.domain) : T† y = adjoint_aux hT y | begin
change (if hT : dense (T.domain : set E) then adjoint_aux hT else 0) y = _,
simp only [hT, dif_pos, linear_map.coe_mk],
end | lemma | linear_pmap.adjoint_apply_of_dense | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"dense",
"linear_map.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_apply_eq (y : T†.domain) {x₀ : E}
(hx₀ : ∀ x : T.domain, ⟪x₀, x⟫ = ⟪(y : F), T x⟫) : T† y = x₀ | (adjoint_apply_of_dense hT y).symm ▸ adjoint_aux_unique hT _ hx₀ | lemma | linear_pmap.adjoint_apply_eq | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoint_is_formal_adjoint : T†.is_formal_adjoint T | λ x, (adjoint_apply_of_dense hT x).symm ▸ adjoint_aux_inner hT x | lemma | linear_pmap.adjoint_is_formal_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | The fundamental property of the adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_formal_adjoint.le_adjoint (h : T.is_formal_adjoint S) : S ≤ T† | -- Trivially, every `x : S.domain` is in `T.adjoint.domain`
⟨λ x hx, mem_adjoint_domain_of_exists _ ⟨S ⟨x, hx⟩, h.symm ⟨x, hx⟩⟩,
-- Equality on `S.domain` follows from equality
-- `⟪v, S x⟫ = ⟪v, T.adjoint y⟫` for all `v : T.domain`:
λ _ _ hxy, (adjoint_apply_eq hT _ (λ _, by rw [h.symm, hxy])).symm⟩ | lemma | linear_pmap.is_formal_adjoint.le_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [] | The adjoint is maximal in the sense that it contains every formal adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pmap_adjoint_eq_adjoint_to_pmap_of_dense (hp : dense (p : set E)) :
(A.to_pmap p).adjoint = A.adjoint.to_pmap ⊤ | begin
ext,
{ simp only [to_linear_map_eq_coe, linear_map.to_pmap_domain, submodule.mem_top, iff_true,
linear_pmap.mem_adjoint_domain_iff, linear_map.coe_comp, innerₛₗ_apply_coe],
exact ((innerSL 𝕜 x).comp $ A.comp $ submodule.subtypeL _).cont },
intros x y hxy,
refine linear_pmap.adjoint_apply_eq hp ... | lemma | continuous_linear_map.to_pmap_adjoint_eq_adjoint_to_pmap_of_dense | analysis.inner_product_space | src/analysis/inner_product_space/linear_pmap.lean | [
"analysis.inner_product_space.adjoint",
"topology.algebra.module.linear_pmap",
"topology.algebra.module.basic"
] | [
"coe_coe",
"cont",
"dense",
"innerSL",
"innerₛₗ_apply_coe",
"linear_map.coe_comp",
"linear_map.to_pmap_apply",
"linear_map.to_pmap_domain",
"linear_pmap.adjoint_apply_eq",
"linear_pmap.mem_adjoint_domain_iff",
"submodule.mem_top",
"submodule.subtypeL"
] | Restricting `A` to a dense submodule and taking the `linear_pmap.adjoint` is the same
as taking the `continuous_linear_map.adjoint` interpreted as a `linear_pmap`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_spaceable : Prop | (parallelogram_identity :
∀ x y : E, ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) | class | inner_product_spaceable | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [] | Predicate for the parallelogram identity to hold in a normed group. This is a scalar-less
version of `inner_product_space`. If you have an `inner_product_spaceable` assumption, you can
locally upgrade that to `inner_product_space 𝕜 E` using `casesI nonempty_inner_product_space 𝕜 E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space.to_inner_product_spaceable [inner_product_space 𝕜 E] :
inner_product_spaceable E | ⟨parallelogram_law_with_norm 𝕜⟩ | lemma | inner_product_space.to_inner_product_spaceable | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_product_space",
"inner_product_spaceable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_product_space.to_inner_product_spaceable_of_real [inner_product_space ℝ E] :
inner_product_spaceable E | ⟨parallelogram_law_with_norm ℝ⟩ | instance | inner_product_space.to_inner_product_spaceable_of_real | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_product_space",
"inner_product_spaceable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_ (x y : E) : 𝕜 | 4⁻¹ * ((𝓚 ‖x + y‖) * (𝓚 ‖x + y‖) - (𝓚 ‖x - y‖) * (𝓚 ‖x - y‖)
+ (I:𝕜) * (𝓚 ‖(I:𝕜) • x + y‖) * (𝓚 ‖(I:𝕜) • x + y‖)
- (I:𝕜) * (𝓚 ‖(I:𝕜) • x - y‖) * (𝓚 ‖(I:𝕜) • x - y‖)) | def | inner_ | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [] | Auxiliary definition of the inner product derived from the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_prop (r : 𝕜) : Prop | ∀ x y : E, inner_ 𝕜 (r • x) y = conj r * inner_ 𝕜 x y | def | inner_product_spaceable.inner_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_"
] | Auxiliary definition for the `add_left` property | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_prop_neg_one : inner_prop E ((-1 : ℤ) : 𝕜) | begin
intros x y,
simp only [inner_, neg_mul_eq_neg_mul, one_mul, int.cast_one, one_smul, ring_hom.map_one,
map_neg, int.cast_neg, neg_smul, neg_one_mul],
rw neg_mul_comm,
congr' 1,
have h₁ : ‖-x - y‖ = ‖x + y‖,
{ rw [←neg_add', norm_neg], },
have h₂ : ‖-x + y‖ = ‖x - y‖,
{ rw [←neg_sub, norm_neg, s... | lemma | inner_product_spaceable.inner_prop_neg_one | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_",
"int.cast_neg",
"int.cast_one",
"neg_mul_comm",
"neg_mul_eq_neg_mul",
"neg_one_mul",
"neg_smul",
"one_mul",
"one_smul",
"ring",
"ring_hom.map_one",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.inner_ {f g : ℝ → E} (hf : continuous f) (hg : continuous g) :
continuous (λ x, inner_ 𝕜 (f x) (g x)) | by { unfold inner_, continuity } | lemma | inner_product_spaceable.continuous.inner_ | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"continuity",
"continuous",
"inner_"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) | begin
simp only [inner_],
have h₁ : norm_sq (4 : 𝕜) = 16,
{ have : ((4 : ℝ) : 𝕜) = (4 : 𝕜),
{ simp only [of_real_one, of_real_bit0] },
rw [←this, norm_sq_eq_def',
is_R_or_C.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)],
norm_num },
have h₂ : ‖x + x‖ = 2 * ‖x‖,
{ rw [←two_smul 𝕜, norm_smul,... | lemma | inner_product_spaceable.inner_.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"bit0_zero",
"inner_",
"is_R_or_C.norm_of_nonneg",
"is_R_or_C.norm_two",
"mul_zero",
"norm_smul",
"ring",
"zero_div",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y | begin
simp only [inner_],
have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹,
{ rw [is_R_or_C.conj_inv, ←of_real_one, ←of_real_bit0, ←of_real_bit0, conj_of_real] },
rw [map_mul, h4],
congr' 1,
simp only [map_sub, map_add, algebra_map_eq_of_real, ←of_real_mul, conj_of_real, map_mul, conj_I],
rw [add_comm y x, norm_sub_rev],
... | lemma | inner_product_spaceable.inner_.conj_symm | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_",
"is_R_or_C.conj_inv",
"map_mul",
"neg_mul",
"neg_one_smul",
"norm_smul",
"one_mul",
"smul_add",
"smul_smul",
"smul_sub",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux1 (x y z : E) :
‖x + y + z‖ * ‖x + y + z‖ =
(‖2 • x + y‖ * ‖2 • x + y‖ + ‖2 • z + y‖ * ‖2 • z + y‖) / 2 - ‖x - z‖ * ‖x - z‖ | begin
rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm],
convert parallelogram_identity (x + y + z) (x - z) using 4; { rw two_smul, abel }
end | lemma | inner_product_spaceable.add_left_aux1 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"eq_div_iff",
"mul_comm",
"two_ne_zero'",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux2 (x y z : E) :
‖x + y - z‖ * ‖x + y - z‖ =
(‖2 • x + y‖ * ‖2 • x + y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 - ‖x + z‖ * ‖x + z‖ | begin
rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm],
have h₀ := parallelogram_identity (x + y - z) (x + z),
convert h₀ using 4; { rw two_smul, abel }
end | lemma | inner_product_spaceable.add_left_aux2 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"eq_div_iff",
"mul_comm",
"two_ne_zero'",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux2' (x y z : E) :
‖x + y + z‖ * ‖x + y + z‖ - ‖x + y - z‖ * ‖x + y - z‖ =
‖x + z‖ * ‖x + z‖ - ‖x - z‖ * ‖x - z‖ +
(‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 | by { rw [add_left_aux1 , add_left_aux2], ring } | lemma | inner_product_spaceable.add_left_aux2' | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux3 (y z : E) :
‖2 • z + y‖ * ‖2 • z + y‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖ | begin
apply eq_sub_of_add_eq,
convert parallelogram_identity (y + z) z using 4; { try { rw two_smul }, abel }
end | lemma | inner_product_spaceable.add_left_aux3 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux4 (y z : E) :
‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖ | begin
apply eq_sub_of_add_eq',
have h₀ := parallelogram_identity (y - z) z,
convert h₀ using 4; { try { rw two_smul }, abel }
end | lemma | inner_product_spaceable.add_left_aux4 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux4' (y z : E) :
(‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 =
(‖y + z‖ * ‖y + z‖) - (‖y - z‖ * ‖y - z‖) | by { rw [add_left_aux3 , add_left_aux4], ring } | lemma | inner_product_spaceable.add_left_aux4' | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux5 (x y z : E) :
‖(I : 𝕜) • (x + y) + z‖ * ‖(I : 𝕜) • (x + y) + z‖ =
(‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖ +
‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖) / 2 - ‖(I : 𝕜) • x - z‖ * ‖(I : 𝕜) • x - z‖ | begin
rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm],
have h₀ := parallelogram_identity ((I : 𝕜) • (x + y) + z) ((I : 𝕜) • x - z),
convert h₀ using 4; { try { simp only [two_smul, smul_add] }, abel }
end | lemma | inner_product_spaceable.add_left_aux5 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"eq_div_iff",
"mul_comm",
"smul_add",
"two_ne_zero'",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux6 (x y z : E) :
‖(I : 𝕜) • (x + y) - z‖ * ‖(I : 𝕜) • (x + y) - z‖ =
(‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖ +
‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖) / 2 -
‖(I : 𝕜) • x + z‖ * ‖(I : 𝕜) • x + z‖ | begin
rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm],
have h₀ := parallelogram_identity ((I : 𝕜) • (x + y) - z) ((I : 𝕜) • x + z),
convert h₀ using 4; { try { simp only [two_smul, smul_add] }, abel }
end | lemma | inner_product_spaceable.add_left_aux6 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"eq_div_iff",
"mul_comm",
"smul_add",
"two_ne_zero'",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux7 (y z : E) :
‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖ =
2 * (‖(I : 𝕜) • y + z‖ * ‖(I : 𝕜) • y + z‖ + ‖z‖ * ‖z‖) - ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ | begin
apply eq_sub_of_add_eq,
have h₀ := parallelogram_identity ((I : 𝕜) • y + z) z,
convert h₀ using 4; { try { simp only [two_smul, smul_add] }, abel }
end | lemma | inner_product_spaceable.add_left_aux7 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"smul_add",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_aux8 (y z : E) :
‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖ =
2 * (‖(I : 𝕜) • y - z‖ * ‖(I : 𝕜) • y - z‖ + ‖z‖ * ‖z‖) - ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ | begin
apply eq_sub_of_add_eq',
have h₀ := parallelogram_identity ((I : 𝕜) • y - z) z,
convert h₀ using 4; { try { simp only [two_smul, smul_add] }, abel }
end | lemma | inner_product_spaceable.add_left_aux8 | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"smul_add",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left (x y z : E) : inner_ 𝕜 (x + y) z = inner_ 𝕜 x z + inner_ 𝕜 y z | begin
simp only [inner_, ←mul_add],
congr,
simp only [mul_assoc, ←map_mul, add_sub_assoc, ←mul_sub, ←map_sub],
rw add_add_add_comm,
simp only [←map_add, ←mul_add],
congr,
{ rw [←add_sub_assoc, add_left_aux2', add_left_aux4'] },
{ rw [add_left_aux5, add_left_aux6, add_left_aux7, add_left_aux8],
simp ... | lemma | inner_product_spaceable.add_left | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"div_eq_mul_inv",
"inner_",
"map_mul",
"mul_assoc",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat (n : ℕ) (x y : E) : inner_ 𝕜 ((n : 𝕜) • x) y = (n : 𝕜) * inner_ 𝕜 x y | begin
induction n with n ih,
{ simp only [inner_, nat.nat_zero_eq_zero, zero_sub, nat.cast_zero, zero_mul, eq_self_iff_true,
zero_smul, zero_add, mul_zero, sub_self, norm_neg, smul_zero], },
{ simp only [nat.cast_succ, add_smul, one_smul],
rw [add_left, ih, add_mul, one_mul] }
end | lemma | inner_product_spaceable.nat | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"add_smul",
"ih",
"inner_",
"mul_zero",
"nat.cast_succ",
"nat.cast_zero",
"one_mul",
"one_smul",
"smul_zero",
"zero_mul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_prop (r : ℕ) : inner_prop E (r : 𝕜) | λ x y, by { simp only [map_nat_cast], exact nat r x y } | lemma | inner_product_spaceable.nat_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"map_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_prop (n : ℤ) : inner_prop E (n : 𝕜) | begin
intros x y,
rw ←n.sign_mul_nat_abs,
simp only [int.cast_coe_nat, map_nat_cast, map_int_cast, int.cast_mul, map_mul, mul_smul],
obtain hn | rfl | hn := lt_trichotomy n 0,
{ rw [int.sign_eq_neg_one_of_neg hn, inner_prop_neg_one ((n.nat_abs : 𝕜) • x), nat],
simp only [map_neg, neg_mul, one_mul, mul_eq... | lemma | inner_product_spaceable.int_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_",
"int.cast_coe_nat",
"int.cast_mul",
"int.cast_neg",
"int.cast_one",
"int.cast_zero",
"int.nat_abs_eq_zero",
"map_int_cast",
"map_mul",
"map_nat_cast",
"map_one",
"mul_eq_mul_left_iff",
"mul_zero",
"nat.cast_eq_zero",
"nat.cast_zero",
"neg_mul",
"one_mul",
"one_smul",
"s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_prop (r : ℚ) : inner_prop E (r : 𝕜) | begin
intros x y,
have : (r.denom : 𝕜) ≠ 0,
{ haveI : char_zero 𝕜 := is_R_or_C.char_zero_R_or_C,
exact_mod_cast r.pos.ne' },
rw [←r.num_div_denom, ←mul_right_inj' this, ←nat r.denom _ y, smul_smul, rat.cast_div],
simp only [map_nat_cast, rat.cast_coe_nat, map_int_cast, rat.cast_coe_int, map_div₀],
rw ... | lemma | inner_product_spaceable.rat_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"char_zero",
"is_R_or_C.char_zero_R_or_C",
"map_div₀",
"map_int_cast",
"map_nat_cast",
"mul_div_cancel'",
"rat.cast_coe_int",
"rat.cast_coe_nat",
"rat.cast_div",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_prop (r : ℝ) : inner_prop E (r : 𝕜) | begin
intros x y,
revert r,
rw ←function.funext_iff,
refine rat.dense_embedding_coe_real.dense.equalizer _ _ (funext $ λ X, _),
{ exact (continuous_of_real.smul continuous_const).inner_ continuous_const },
{ exact (continuous_conj.comp continuous_of_real).mul continuous_const },
{ simp only [function.comp... | lemma | inner_product_spaceable.real_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"continuous_const",
"inner_",
"is_R_or_C.of_real_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
I_prop : inner_prop E (I : 𝕜) | begin
by_cases hI : (I : 𝕜) = 0,
{ rw [hI, ←nat.cast_zero], exact nat_prop _ },
intros x y,
have hI' : (-I : 𝕜) * I = 1,
{ rw [←inv_I, inv_mul_cancel hI], },
rw [conj_I, inner_, inner_, mul_left_comm],
congr' 1,
rw [smul_smul, I_mul_I_of_nonzero hI, neg_one_smul],
rw [mul_sub, mul_add, mul_sub,
... | lemma | inner_product_spaceable.I_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_",
"inv_mul_cancel",
"mul_assoc",
"mul_left_comm",
"neg_one_smul",
"one_mul",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_prop (r : 𝕜) : inner_prop E r | begin
intros x y,
rw [←re_add_im r, add_smul, add_left, real_prop _ x, ←smul_smul, real_prop _ _ y, I_prop,
map_add, map_mul, conj_of_real, conj_of_real, conj_I],
ring,
end | lemma | inner_product_spaceable.inner_prop | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"add_smul",
"map_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_product_space.of_norm
(h : ∀ x y : E, ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) :
inner_product_space 𝕜 E | begin
haveI : inner_product_spaceable E := ⟨h⟩,
exact
{ inner := inner_ 𝕜,
norm_sq_eq_inner := inner_.norm_sq,
conj_symm := inner_.conj_symm,
add_left := add_left,
smul_left := λ _ _ _, inner_prop _ _ _ }
end | def | inner_product_space.of_norm | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_",
"inner_product_space",
"inner_product_spaceable"
] | **Fréchet–von Neumann–Jordan Theorem**. A normed space `E` whose norm satisfies the
parallelogram identity can be given a compatible inner product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_inner_product_space : nonempty (inner_product_space 𝕜 E) | ⟨{ inner := inner_ 𝕜,
norm_sq_eq_inner := inner_.norm_sq,
conj_symm := inner_.conj_symm,
add_left := add_left,
smul_left := λ _ _ _, inner_prop _ _ _ }⟩ | lemma | nonempty_inner_product_space | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"inner_",
"inner_product_space"
] | **Fréchet–von Neumann–Jordan Theorem**. A normed space `E` whose norm satisfies the
parallelogram identity can be given a compatible inner product. Do
`casesI nonempty_inner_product_space 𝕜 E` to locally upgrade `inner_product_spaceable E` to
`inner_product_space 𝕜 E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_spaceable.to_uniform_convex_space : uniform_convex_space E | by { casesI nonempty_inner_product_space ℝ E, apply_instance } | instance | inner_product_spaceable.to_uniform_convex_space | analysis.inner_product_space | src/analysis/inner_product_space/of_norm.lean | [
"topology.algebra.algebra",
"analysis.inner_product_space.basic"
] | [
"nonempty_inner_product_space",
"uniform_convex_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
det_to_matrix_orthonormal_basis_of_same_orientation
(h : e.to_basis.orientation = f.to_basis.orientation) :
e.to_basis.det f = 1 | begin
apply (e.det_to_matrix_orthonormal_basis_real f).resolve_right,
have : 0 < e.to_basis.det f,
{ rw e.to_basis.orientation_eq_iff_det_pos at h,
simpa using h },
linarith,
end | lemma | orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | The change-of-basis matrix between two orthonormal bases with the same orientation has
determinant 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det_to_matrix_orthonormal_basis_of_opposite_orientation
(h : e.to_basis.orientation ≠ f.to_basis.orientation) :
e.to_basis.det f = -1 | begin
contrapose! h,
simp [e.to_basis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormal_basis_real f).resolve_right h],
end | lemma | orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | The change-of-basis matrix between two orthonormal bases with the opposite orientations has
determinant -1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
same_orientation_iff_det_eq_det :
e.to_basis.det = f.to_basis.det ↔ e.to_basis.orientation = f.to_basis.orientation | begin
split,
{ intros h,
dsimp [basis.orientation],
congr' },
{ intros h,
rw e.to_basis.det.eq_smul_basis_det f.to_basis,
simp [e.det_to_matrix_orthonormal_basis_of_same_orientation f h], },
end | lemma | orthonormal_basis.same_orientation_iff_det_eq_det | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"basis.orientation"
] | Two orthonormal bases with the same orientation determine the same "determinant" top-dimensional
form on `E`, and conversely. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det_eq_neg_det_of_opposite_orientation
(h : e.to_basis.orientation ≠ f.to_basis.orientation) :
e.to_basis.det = -f.to_basis.det | begin
rw e.to_basis.det.eq_smul_basis_det f.to_basis,
simp [e.det_to_matrix_orthonormal_basis_of_opposite_orientation f h],
end | lemma | orthonormal_basis.det_eq_neg_det_of_opposite_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | Two orthonormal bases with opposite orientations determine opposite "determinant"
top-dimensional forms on `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_adjust_to_orientation : orthonormal ℝ (e.to_basis.adjust_to_orientation x) | begin
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg,
simpa using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x
end | lemma | orthonormal_basis.orthonormal_adjust_to_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal"
] | `orthonormal_basis.adjust_to_orientation`, applied to an orthonormal basis, preserves the
property of orthonormality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjust_to_orientation : orthonormal_basis ι ℝ E | (e.to_basis.adjust_to_orientation x).to_orthonormal_basis (e.orthonormal_adjust_to_orientation x) | def | orthonormal_basis.adjust_to_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal_basis"
] | Given an orthonormal basis and an orientation, return an orthonormal basis giving that
orientation: either the original basis, or one constructed by negating a single (arbitrary) basis
vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_basis_adjust_to_orientation :
(e.adjust_to_orientation x).to_basis = e.to_basis.adjust_to_orientation x | (e.to_basis.adjust_to_orientation x).to_basis_to_orthonormal_basis _ | lemma | orthonormal_basis.to_basis_adjust_to_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orientation_adjust_to_orientation :
(e.adjust_to_orientation x).to_basis.orientation = x | begin
rw e.to_basis_adjust_to_orientation,
exact e.to_basis.orientation_adjust_to_orientation x,
end | lemma | orthonormal_basis.orientation_adjust_to_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | `adjust_to_orientation` gives an orthonormal basis with the required orientation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjust_to_orientation_apply_eq_or_eq_neg (i : ι) :
e.adjust_to_orientation x i = e i ∨ e.adjust_to_orientation x i = -(e i) | by simpa [← e.to_basis_adjust_to_orientation]
using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x i | lemma | orthonormal_basis.adjust_to_orientation_apply_eq_or_eq_neg | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | Every basis vector from `adjust_to_orientation` is either that from the original basis or its
negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det_adjust_to_orientation :
(e.adjust_to_orientation x).to_basis.det = e.to_basis.det
∨ (e.adjust_to_orientation x).to_basis.det = -e.to_basis.det | by simpa using e.to_basis.det_adjust_to_orientation x | lemma | orthonormal_basis.det_adjust_to_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_det_adjust_to_orientation (v : ι → E) : | |(e.adjust_to_orientation x).to_basis.det v| = |e.to_basis.det v| :=
by simp [to_basis_adjust_to_orientation] | lemma | orthonormal_basis.abs_det_adjust_to_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_orthonormal_basis (hn : 0 < n) (h : finrank ℝ E = n)
(x : orientation ℝ E (fin n)) : orthonormal_basis (fin n) ℝ E | begin
haveI := fin.pos_iff_nonempty.1 hn,
haveI := finite_dimensional_of_finrank (h.symm ▸ hn : 0 < finrank ℝ E),
exact ((std_orthonormal_basis _ _).reindex $ fin_congr h).adjust_to_orientation x
end | def | orientation.fin_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"fin_congr",
"orientation",
"orthonormal_basis",
"reindex",
"std_orthonormal_basis"
] | An orthonormal basis, indexed by `fin n`, with the given orientation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_orthonormal_basis_orientation (hn : 0 < n)
(h : finrank ℝ E = n) (x : orientation ℝ E (fin n)) :
(x.fin_orthonormal_basis hn h).to_basis.orientation = x | begin
haveI := fin.pos_iff_nonempty.1 hn,
haveI := finite_dimensional_of_finrank (h.symm ▸ hn : 0 < finrank ℝ E),
exact ((std_orthonormal_basis _ _).reindex $ fin_congr h).orientation_adjust_to_orientation x
end | lemma | orientation.fin_orthonormal_basis_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"fin_congr",
"orientation",
"reindex",
"std_orthonormal_basis"
] | `orientation.fin_orthonormal_basis` gives a basis with the required orientation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_form : alternating_map ℝ E ℝ (fin n) | begin
classical,
unfreezingI { cases n },
{ let opos : alternating_map ℝ E ℝ (fin 0) := alternating_map.const_of_is_empty ℝ E (fin 0) (1:ℝ),
exact o.eq_or_eq_neg_of_is_empty.by_cases (λ _, opos) (λ _, -opos) },
{ exact (o.fin_orthonormal_basis n.succ_pos _i.out).to_basis.det }
end | def | orientation.volume_form | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"alternating_map",
"alternating_map.const_of_is_empty"
] | The volume form on an oriented real inner product space, a nonvanishing top-dimensional
alternating form uniquely defined by compatibility with the orientation and inner product structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_form_zero_pos [_i : fact (finrank ℝ E = 0)] :
orientation.volume_form (positive_orientation : orientation ℝ E (fin 0))
= alternating_map.const_linear_equiv_of_is_empty 1 | by simp [volume_form, or.by_cases, if_pos] | lemma | orientation.volume_form_zero_pos | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"alternating_map.const_linear_equiv_of_is_empty",
"fact",
"orientation",
"orientation.volume_form"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
volume_form_zero_neg [_i : fact (finrank ℝ E = 0)] :
orientation.volume_form (-positive_orientation : orientation ℝ E (fin 0))
= - alternating_map.const_linear_equiv_of_is_empty 1 | begin
dsimp [volume_form, or.by_cases, positive_orientation],
apply if_neg,
rw [ray_eq_iff, same_ray_comm],
intros h,
simpa using
congr_arg alternating_map.const_linear_equiv_of_is_empty.symm (eq_zero_of_same_ray_self_neg h),
end | lemma | orientation.volume_form_zero_neg | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"alternating_map.const_linear_equiv_of_is_empty",
"eq_zero_of_same_ray_self_neg",
"fact",
"orientation",
"orientation.volume_form",
"ray_eq_iff",
"same_ray_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
volume_form_robust (b : orthonormal_basis (fin n) ℝ E) (hb : b.to_basis.orientation = o) :
o.volume_form = b.to_basis.det | begin
unfreezingI { cases n },
{ classical,
have : o = positive_orientation := hb.symm.trans b.to_basis.orientation_is_empty,
simp [volume_form, or.by_cases, dif_pos this] },
{ dsimp [volume_form],
rw [same_orientation_iff_det_eq_det, hb],
exact o.fin_orthonormal_basis_orientation _ _ },
end | lemma | orientation.volume_form_robust | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal_basis"
] | The volume form on an oriented real inner product space can be evaluated as the determinant with
respect to any orthonormal basis of the space compatible with the orientation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_form_robust_neg (b : orthonormal_basis (fin n) ℝ E)
(hb : b.to_basis.orientation ≠ o) :
o.volume_form = - b.to_basis.det | begin
unfreezingI { cases n },
{ classical,
have : positive_orientation ≠ o := by rwa b.to_basis.orientation_is_empty at hb,
simp [volume_form, or.by_cases, dif_neg this.symm] },
let e : orthonormal_basis (fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (fact.out _),
dsimp [volume_form],
apply e... | lemma | orientation.volume_form_robust_neg | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal_basis"
] | The volume form on an oriented real inner product space can be evaluated as the determinant with
respect to any orthonormal basis of the space compatible with the orientation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_form_neg_orientation : (-o).volume_form = - o.volume_form | begin
unfreezingI { cases n },
{ refine o.eq_or_eq_neg_of_is_empty.elim _ _; rintros rfl; simp [volume_form_zero_neg] },
let e : orthonormal_basis (fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (fact.out _),
have h₁ : e.to_basis.orientation = o := o.fin_orthonormal_basis_orientation _ _,
have h₂ : e.t... | lemma | orientation.volume_form_neg_orientation | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
volume_form_robust' (b : orthonormal_basis (fin n) ℝ E) (v : fin n → E) : | |o.volume_form v| = |b.to_basis.det v| :=
begin
unfreezingI { cases n },
{ refine o.eq_or_eq_neg_of_is_empty.elim _ _; rintros rfl; simp },
{ rw [o.volume_form_robust (b.adjust_to_orientation o) (b.orientation_adjust_to_orientation o),
b.abs_det_adjust_to_orientation] },
end | lemma | orientation.volume_form_robust' | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_volume_form_apply_le (v : fin n → E) : |o.volume_form v| ≤ ∏ i : fin n, ‖v i‖ | begin
unfreezingI { cases n },
{ refine o.eq_or_eq_neg_of_is_empty.elim _ _; rintros rfl; simp },
haveI : finite_dimensional ℝ E := fact_finite_dimensional_of_finrank_eq_succ n,
have : finrank ℝ E = fintype.card (fin n.succ) := by simpa using _i.out,
let b : orthonormal_basis (fin n.succ) ℝ E := gram_schmidt_... | lemma | orientation.abs_volume_form_apply_le | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"abs_real_inner_le_norm",
"finite_dimensional",
"finset.abs_prod",
"finset.prod_le_prod",
"fintype.card",
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_det",
"orthonormal_basis"
] | Let `v` be an indexed family of `n` vectors in an oriented `n`-dimensional real inner
product space `E`. The output of the volume form of `E` when evaluated on `v` is bounded in absolute
value by the product of the norms of the vectors `v i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_form_apply_le (v : fin n → E) : o.volume_form v ≤ ∏ i : fin n, ‖v i‖ | (le_abs_self _).trans (o.abs_volume_form_apply_le v) | lemma | orientation.volume_form_apply_le | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_volume_form_apply_of_pairwise_orthogonal
{v : fin n → E} (hv : pairwise (λ i j, ⟪v i, v j⟫ = 0)) : | |o.volume_form v| = ∏ i : fin n, ‖v i‖ :=
begin
unfreezingI { cases n },
{ refine o.eq_or_eq_neg_of_is_empty.elim _ _; rintros rfl; simp },
haveI : finite_dimensional ℝ E := fact_finite_dimensional_of_finrank_eq_succ n,
have hdim : finrank ℝ E = fintype.card (fin n.succ) := by simpa using _i.out,
let b : orth... | lemma | orientation.abs_volume_form_apply_of_pairwise_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"abs_of_nonneg",
"finite_dimensional",
"finset.abs_prod",
"finset.mem_univ",
"finset.prod_eq_zero",
"fintype.card",
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_apply_of_orthogonal",
"gram_schmidt_orthonormal_basis_det",
"inner_smul_left",
"is_R_or_C.conj_to_real",
"orthon... | Let `v` be an indexed family of `n` orthogonal vectors in an oriented `n`-dimensional
real inner product space `E`. The output of the volume form of `E` when evaluated on `v` is, up to
sign, the product of the norms of the vectors `v i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_volume_form_apply_of_orthonormal (v : orthonormal_basis (fin n) ℝ E) : | |o.volume_form v| = 1 :=
by simpa [o.volume_form_robust' v v] using congr_arg abs v.to_basis.det_self | lemma | orientation.abs_volume_form_apply_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"orthonormal_basis"
] | The output of the volume form of an oriented real inner product space `E` when evaluated on an
orthonormal basis is ±1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_form_map {F : Type*}
[normed_add_comm_group F] [inner_product_space ℝ F] [fact (finrank ℝ F = n)]
(φ : E ≃ₗᵢ[ℝ] F) (x : fin n → F) :
(orientation.map (fin n) φ.to_linear_equiv o).volume_form x = o.volume_form (φ.symm ∘ x) | begin
unfreezingI { cases n },
{ refine o.eq_or_eq_neg_of_is_empty.elim _ _; rintros rfl; simp },
let e : orthonormal_basis (fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (fact.out _),
have he : e.to_basis.orientation = o :=
(o.fin_orthonormal_basis_orientation n.succ_pos (fact.out _)),
have heφ :... | lemma | orientation.volume_form_map | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"fact",
"inner_product_space",
"normed_add_comm_group",
"orientation.map",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
volume_form_comp_linear_isometry_equiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < (φ.to_linear_equiv : E →ₗ[ℝ] E).det) (x : fin n → E) :
o.volume_form (φ ∘ x) = o.volume_form x | begin
convert o.volume_form_map φ (φ ∘ x),
{ symmetry,
rwa ← o.map_eq_iff_det_pos φ.to_linear_equiv at hφ,
rw [_i.out, fintype.card_fin] },
{ ext,
simp }
end | lemma | orientation.volume_form_comp_linear_isometry_equiv | analysis.inner_product_space | src/analysis/inner_product_space/orientation.lean | [
"analysis.inner_product_space.gram_schmidt_ortho",
"linear_algebra.orientation"
] | [
"fintype.card_fin"
] | The volume form is invariant under pullback by a positively-oriented isometric automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal : submodule 𝕜 E | { carrier := {v | ∀ u ∈ K, ⟪u, v⟫ = 0},
zero_mem' := λ _ _, inner_zero_right _,
add_mem' := λ x y hx hy u hu, by rw [inner_add_right, hx u hu, hy u hu, add_zero],
smul_mem' := λ c x hx u hu, by rw [inner_smul_right, hx u hu, mul_zero] } | def | submodule.orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_add_right",
"inner_smul_right",
"inner_zero_right",
"mul_zero",
"submodule"
] | The subspace of vectors orthogonal to a given subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_orthogonal (v : E) : v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪u, v⟫ = 0 | iff.rfl | lemma | submodule.mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [] | When a vector is in `Kᗮ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_orthogonal' (v : E) : v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪v, u⟫ = 0 | by simp_rw [mem_orthogonal, inner_eq_zero_symm] | lemma | submodule.mem_orthogonal' | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_eq_zero_symm"
] | When a vector is in `Kᗮ`, with the inner product the
other way round. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_right_of_mem_orthogonal {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) : ⟪u, v⟫ = 0 | (K.mem_orthogonal v).1 hv u hu | lemma | submodule.inner_right_of_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [] | A vector in `K` is orthogonal to one in `Kᗮ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_left_of_mem_orthogonal {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) : ⟪v, u⟫ = 0 | by rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv | lemma | submodule.inner_left_of_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_eq_zero_symm"
] | A vector in `Kᗮ` is orthogonal to one in `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_orthogonal_singleton_iff_inner_right {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪u, v⟫ = 0 | begin
refine ⟨inner_right_of_mem_orthogonal (mem_span_singleton_self u), _⟩,
intros hv w hw,
rw mem_span_singleton at hw,
obtain ⟨c, rfl⟩ := hw,
simp [inner_smul_left, hv],
end | lemma | submodule.mem_orthogonal_singleton_iff_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_smul_left"
] | A vector is in `(𝕜 ∙ u)ᗮ` iff it is orthogonal to `u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_orthogonal_singleton_iff_inner_left {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪v, u⟫ = 0 | by rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm] | lemma | submodule.mem_orthogonal_singleton_iff_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_eq_zero_symm"
] | A vector in `(𝕜 ∙ u)ᗮ` is orthogonal to `u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_mem_orthogonal_of_inner_left {x y : E}
(h : ∀ (v : K), ⟪x, v⟫ = ⟪y, v⟫) : x - y ∈ Kᗮ | begin
rw mem_orthogonal',
intros u hu,
rw [inner_sub_left, sub_eq_zero],
exact h ⟨u, hu⟩,
end | lemma | submodule.sub_mem_orthogonal_of_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_sub_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem_orthogonal_of_inner_right {x y : E}
(h : ∀ (v : K), ⟪(v : E), x⟫ = ⟪(v : E), y⟫) : x - y ∈ Kᗮ | begin
intros u hu,
rw [inner_sub_right, sub_eq_zero],
exact h ⟨u, hu⟩,
end | lemma | submodule.sub_mem_orthogonal_of_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"inner_sub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_orthogonal_eq_bot : K ⊓ Kᗮ = ⊥ | begin
rw eq_bot_iff,
intros x,
rw mem_inf,
exact λ ⟨hx, ho⟩, inner_self_eq_zero.1 (ho x hx)
end | lemma | submodule.inf_orthogonal_eq_bot | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"eq_bot_iff"
] | `K` and `Kᗮ` have trivial intersection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_disjoint : disjoint K Kᗮ | by simp [disjoint_iff, K.inf_orthogonal_eq_bot] | lemma | submodule.orthogonal_disjoint | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"disjoint",
"disjoint_iff"
] | `K` and `Kᗮ` have trivial intersection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_eq_inter : Kᗮ = ⨅ v : K, linear_map.ker (innerSL 𝕜 (v : E)) | begin
apply le_antisymm,
{ rw le_infi_iff,
rintros ⟨v, hv⟩ w hw,
simpa using hw _ hv },
{ intros v hv w hw,
simp only [mem_infi] at hv,
exact hv ⟨w, hw⟩ }
end | lemma | submodule.orthogonal_eq_inter | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"innerSL",
"le_infi_iff",
"linear_map.ker"
] | `Kᗮ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_orthogonal : is_closed (Kᗮ : set E) | begin
rw orthogonal_eq_inter K,
have := λ v : K, continuous_linear_map.is_closed_ker (innerSL 𝕜 (v : E)),
convert is_closed_Inter this,
simp only [infi_coe],
end | lemma | submodule.is_closed_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"continuous_linear_map.is_closed_ker",
"innerSL",
"is_closed",
"is_closed_Inter"
] | The orthogonal complement of any submodule `K` is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_gc :
@galois_connection (submodule 𝕜 E) (submodule 𝕜 E)ᵒᵈ _ _
orthogonal orthogonal | λ K₁ K₂, ⟨λ h v hv u hu, inner_left_of_mem_orthogonal hv (h hu),
λ h v hv u hu, inner_left_of_mem_orthogonal hv (h hu)⟩ | lemma | submodule.orthogonal_gc | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"galois_connection",
"submodule"
] | `orthogonal` gives a `galois_connection` between
`submodule 𝕜 E` and its `order_dual`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_le {K₁ K₂ : submodule 𝕜 E} (h : K₁ ≤ K₂) : K₂ᗮ ≤ K₁ᗮ | (orthogonal_gc 𝕜 E).monotone_l h | lemma | submodule.orthogonal_le | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | `orthogonal` reverses the `≤` ordering of two
subspaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_orthogonal_monotone {K₁ K₂ : submodule 𝕜 E} (h : K₁ ≤ K₂) :
K₁ᗮᗮ ≤ K₂ᗮᗮ | orthogonal_le (orthogonal_le h) | lemma | submodule.orthogonal_orthogonal_monotone | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | `orthogonal.orthogonal` preserves the `≤` ordering of two
subspaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_orthogonal_orthogonal : K ≤ Kᗮᗮ | (orthogonal_gc 𝕜 E).le_u_l _ | lemma | submodule.le_orthogonal_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [] | `K` is contained in `Kᗮᗮ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_orthogonal (K₁ K₂ : submodule 𝕜 E) : K₁ᗮ ⊓ K₂ᗮ = (K₁ ⊔ K₂)ᗮ | (orthogonal_gc 𝕜 E).l_sup.symm | lemma | submodule.inf_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_orthogonal {ι : Type*} (K : ι → submodule 𝕜 E) : (⨅ i, (K i)ᗮ) = (supr K)ᗮ | (orthogonal_gc 𝕜 E).l_supr.symm | lemma | submodule.infi_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule",
"supr"
] | The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Inf_orthogonal (s : set $ submodule 𝕜 E) : (⨅ K ∈ s, Kᗮ) = (Sup s)ᗮ | (orthogonal_gc 𝕜 E).l_Sup.symm | lemma | submodule.Inf_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"submodule"
] | The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
top_orthogonal_eq_bot : (⊤ : submodule 𝕜 E)ᗮ = ⊥ | begin
ext,
rw [mem_bot, mem_orthogonal],
exact ⟨λ h, inner_self_eq_zero.mp (h x mem_top), by { rintro rfl, simp }⟩
end | lemma | submodule.top_orthogonal_eq_bot | 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 | |
bot_orthogonal_eq_top : (⊥ : submodule 𝕜 E)ᗮ = ⊤ | begin
rw [← top_orthogonal_eq_bot, eq_top_iff],
exact le_orthogonal_orthogonal ⊤
end | lemma | submodule.bot_orthogonal_eq_top | analysis.inner_product_space | src/analysis/inner_product_space/orthogonal.lean | [
"linear_algebra.bilinear_form",
"analysis.inner_product_space.basic"
] | [
"eq_top_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.