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 |
|---|---|---|---|---|---|---|---|---|---|---|
re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : E) :
re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2 | by { rw @norm_add_mul_self 𝕜, ring } | lemma | re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_add_mul_self",
"ring"
] | Polarization identity: The real part of the inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : E) :
re ⟪x, y⟫ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 | by { rw [@norm_sub_mul_self 𝕜], ring } | lemma | re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_sub_mul_self",
"ring"
] | Polarization identity: The real part of the inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four (x y : E) :
re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4 | by { rw [@norm_add_mul_self 𝕜, @norm_sub_mul_self 𝕜], ring } | lemma | re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_add_mul_self",
"norm_sub_mul_self",
"ring"
] | Polarization identity: The real part of the inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_inner_eq_norm_sub_I_smul_mul_self_sub_norm_add_I_smul_mul_self_div_four (x y : E) :
im ⟪x, y⟫ = (‖x - IK • y‖ * ‖x - IK • y‖ - ‖x + IK • y‖ * ‖x + IK • y‖) / 4 | by { simp only [@norm_add_mul_self 𝕜, @norm_sub_mul_self 𝕜, inner_smul_right, I_mul_re], ring } | lemma | im_inner_eq_norm_sub_I_smul_mul_self_sub_norm_add_I_smul_mul_self_div_four | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_smul_right",
"norm_add_mul_self",
"norm_sub_mul_self",
"ring"
] | Polarization identity: The imaginary part of the inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_eq_sum_norm_sq_div_four (x y : E) :
⟪x, y⟫ = (‖x + y‖ ^ 2 - ‖x - y‖ ^ 2 + (‖x - IK • y‖ ^ 2 - ‖x + IK • y‖ ^ 2) * IK) / 4 | begin
rw [← re_add_im ⟪x, y⟫, re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four,
im_inner_eq_norm_sub_I_smul_mul_self_sub_norm_add_I_smul_mul_self_div_four],
push_cast,
simp only [sq, ← mul_div_right_comm, ← add_div]
end | lemma | inner_eq_sum_norm_sq_div_four | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"add_div",
"im_inner_eq_norm_sub_I_smul_mul_self_sub_norm_add_I_smul_mul_self_div_four",
"mul_div_right_comm",
"re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four"
] | Polarization identity: The inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_div_norm_sq_smul {x y : F} (hx : x ≠ 0) (hy : y ≠ 0) (R : ℝ) :
dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = (R ^ 2 / (‖x‖ * ‖y‖)) * dist x y | have hx' : ‖x‖ ≠ 0, from norm_ne_zero_iff.2 hx,
have hy' : ‖y‖ ≠ 0, from norm_ne_zero_iff.2 hy,
calc dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y)
= sqrt (‖(R / ‖x‖) ^ 2 • x - (R / ‖y‖) ^ 2 • y‖^2) :
by rw [dist_eq_norm, sqrt_sq (norm_nonneg _)]
... = sqrt ((R ^ 2 / (‖x‖ * ‖y‖)) ^ 2 * ‖x - y‖ ^ 2) :
congr_arg sq... | lemma | dist_div_norm_sq_smul | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"div_nonneg",
"inner_smul_right",
"mul_self_nonneg",
"norm_smul",
"norm_sub_mul_self_real",
"real.norm_of_nonneg",
"real_inner_smul_left",
"ring",
"sq_nonneg"
] | Formula for the distance between the images of two nonzero points under an inversion with center
zero. See also `euclidean_geometry.dist_inversion_inversion` for inversions around a general
point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space.to_uniform_convex_space : uniform_convex_space F | ⟨λ ε hε, begin
refine ⟨2 - sqrt (4 - ε^2), sub_pos_of_lt $ (sqrt_lt' zero_lt_two).2 _, λ x hx y hy hxy, _⟩,
{ norm_num,
exact pow_pos hε _ },
rw sub_sub_cancel,
refine le_sqrt_of_sq_le _,
rw [sq, eq_sub_iff_add_eq.2 (parallelogram_law_with_norm ℝ x y), ←sq (‖x - y‖), hx, hy],
norm_num,
exact pow_le_po... | instance | inner_product_space.to_uniform_convex_space | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"parallelogram_law_with_norm",
"pow_le_pow_of_le_left",
"pow_pos",
"uniform_convex_space",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_map_polarization (T : V →ₗ[ℂ] V) (x y : V):
⟪ T y, x ⟫_ℂ = (⟪T (x + y) , x + y⟫_ℂ - ⟪T (x - y) , x - y⟫_ℂ +
complex.I * ⟪T (x + complex.I • y) , x + complex.I • y⟫_ℂ -
complex.I * ⟪T (x - complex.I • y), x - complex.I • y ⟫_ℂ) / 4 | begin
simp only [map_add, map_sub, inner_add_left, inner_add_right, linear_map.map_smul,
inner_smul_left, inner_smul_right, complex.conj_I, ←pow_two, complex.I_sq,
inner_sub_left, inner_sub_right, mul_add, ←mul_assoc, mul_neg, neg_neg,
sub_neg_eq_add, one_mul, neg_one_mul, mul_s... | lemma | inner_map_polarization | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"complex.I",
"complex.I_sq",
"complex.conj_I",
"inner_add_left",
"inner_add_right",
"inner_smul_left",
"inner_smul_right",
"inner_sub_left",
"inner_sub_right",
"linear_map.map_smul",
"mul_neg",
"neg_one_mul",
"one_mul",
"ring"
] | A complex polarization identity, with a linear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_map_polarization' (T : V →ₗ[ℂ] V) (x y : V):
⟪ T x, y ⟫_ℂ = (⟪T (x + y) , x + y⟫_ℂ - ⟪T (x - y) , x - y⟫_ℂ -
complex.I * ⟪T (x + complex.I • y) , x + complex.I • y⟫_ℂ +
complex.I * ⟪T (x - complex.I • y), x - complex.I • y ⟫_ℂ) / 4 | begin
simp only [map_add, map_sub, inner_add_left, inner_add_right, linear_map.map_smul,
inner_smul_left, inner_smul_right, complex.conj_I, ←pow_two, complex.I_sq,
inner_sub_left, inner_sub_right, mul_add, ←mul_assoc, mul_neg, neg_neg,
sub_neg_eq_add, one_mul, neg_one_mul, mul_s... | lemma | inner_map_polarization' | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"complex.I",
"complex.I_sq",
"complex.conj_I",
"inner_add_left",
"inner_add_right",
"inner_smul_left",
"inner_smul_right",
"inner_sub_left",
"inner_sub_right",
"linear_map.map_smul",
"mul_neg",
"neg_one_mul",
"one_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_map_self_eq_zero (T : V →ₗ[ℂ] V) :
(∀ (x : V), ⟪T x, x⟫_ℂ = 0) ↔ T = 0 | begin
split,
{ intro hT,
ext x,
simp only [linear_map.zero_apply, ← @inner_self_eq_zero ℂ, inner_map_polarization, hT],
norm_num },
{ rintro rfl x,
simp only [linear_map.zero_apply, inner_zero_left] }
end | lemma | inner_map_self_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_map_polarization",
"inner_self_eq_zero",
"inner_zero_left",
"linear_map.zero_apply"
] | A linear map `T` is zero, if and only if the identity `⟪T x, x⟫_ℂ = 0` holds for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_inner_map (S T : V →ₗ[ℂ] V) :
(∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T | begin
rw [←sub_eq_zero, ←inner_map_self_eq_zero],
refine forall_congr (λ x, _),
rw [linear_map.sub_apply, inner_sub_left, sub_eq_zero],
end | lemma | ext_inner_map | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_sub_left",
"linear_map.sub_apply"
] | Two linear maps `S` and `T` are equal, if and only if the identity `⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ` holds
for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.inner_map_map (f : E →ₗᵢ[𝕜] E') (x y : E) : ⟪f x, f y⟫ = ⟪x, y⟫ | by simp [inner_eq_sum_norm_sq_div_four, ← f.norm_map] | lemma | linear_isometry.inner_map_map | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_eq_sum_norm_sq_div_four"
] | A linear isometry preserves the inner product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.inner_map_map (f : E ≃ₗᵢ[𝕜] E') (x y : E) :
⟪f x, f y⟫ = ⟪x, y⟫ | f.to_linear_isometry.inner_map_map x y | lemma | linear_isometry_equiv.inner_map_map | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | A linear isometric equivalence preserves the inner product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.isometry_of_inner (f : E →ₗ[𝕜] E') (h : ∀ x y, ⟪f x, f y⟫ = ⟪x, y⟫) : E →ₗᵢ[𝕜] E' | ⟨f, λ x, by simp only [@norm_eq_sqrt_inner 𝕜, h]⟩ | def | linear_map.isometry_of_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_eq_sqrt_inner"
] | A linear map that preserves the inner product is a linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.coe_isometry_of_inner (f : E →ₗ[𝕜] E') (h) :
⇑(f.isometry_of_inner h) = f | rfl | lemma | linear_map.coe_isometry_of_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.isometry_of_inner_to_linear_map (f : E →ₗ[𝕜] E') (h) :
(f.isometry_of_inner h).to_linear_map = f | rfl | lemma | linear_map.isometry_of_inner_to_linear_map | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.isometry_of_inner (f : E ≃ₗ[𝕜] E') (h : ∀ x y, ⟪f x, f y⟫ = ⟪x, y⟫) :
E ≃ₗᵢ[𝕜] E' | ⟨f, ((f : E →ₗ[𝕜] E').isometry_of_inner h).norm_map⟩ | def | linear_equiv.isometry_of_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | A linear equivalence that preserves the inner product is a linear isometric equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv.coe_isometry_of_inner (f : E ≃ₗ[𝕜] E') (h) :
⇑(f.isometry_of_inner h) = f | rfl | lemma | linear_equiv.coe_isometry_of_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.isometry_of_inner_to_linear_equiv (f : E ≃ₗ[𝕜] E') (h) :
(f.isometry_of_inner h).to_linear_equiv = f | rfl | lemma | linear_equiv.isometry_of_inner_to_linear_equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.orthonormal_comp_iff {v : ι → E} (f : E →ₗᵢ[𝕜] E') :
orthonormal 𝕜 (f ∘ v) ↔ orthonormal 𝕜 v | begin
classical,
simp_rw [orthonormal_iff_ite, linear_isometry.inner_map_map]
end | lemma | linear_isometry.orthonormal_comp_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"linear_isometry.inner_map_map",
"orthonormal",
"orthonormal_iff_ite"
] | A linear isometry preserves the property of being orthonormal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.comp_linear_isometry {v : ι → E} (hv : orthonormal 𝕜 v) (f : E →ₗᵢ[𝕜] E') :
orthonormal 𝕜 (f ∘ v) | by rwa f.orthonormal_comp_iff | lemma | orthonormal.comp_linear_isometry | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"orthonormal"
] | A linear isometry preserves the property of being orthonormal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.comp_linear_isometry_equiv {v : ι → E} (hv : orthonormal 𝕜 v) (f : E ≃ₗᵢ[𝕜] E') :
orthonormal 𝕜 (f ∘ v) | hv.comp_linear_isometry f.to_linear_isometry | lemma | orthonormal.comp_linear_isometry_equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"orthonormal"
] | A linear isometric equivalence preserves the property of being orthonormal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.map_linear_isometry_equiv {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v)
(f : E ≃ₗᵢ[𝕜] E') : orthonormal 𝕜 (v.map f.to_linear_equiv) | hv.comp_linear_isometry_equiv f | lemma | orthonormal.map_linear_isometry_equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | A linear isometric equivalence, applied with `basis.map`, preserves the property of being
orthonormal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.isometry_of_orthonormal (f : E →ₗ[𝕜] E') {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v)
(hf : orthonormal 𝕜 (f ∘ v)) : E →ₗᵢ[𝕜] E' | f.isometry_of_inner $ λ x y, by rw [←v.total_repr x, ←v.total_repr y, finsupp.apply_total,
finsupp.apply_total, hv.inner_finsupp_eq_sum_left,
hf.inner_finsupp_eq_sum_left] | def | linear_map.isometry_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"finsupp.apply_total",
"orthonormal"
] | A linear map that sends an orthonormal basis to orthonormal vectors is a linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.coe_isometry_of_orthonormal (f : E →ₗ[𝕜] E') {v : basis ι 𝕜 E}
(hv : orthonormal 𝕜 v) (hf : orthonormal 𝕜 (f ∘ v)) :
⇑(f.isometry_of_orthonormal hv hf) = f | rfl | lemma | linear_map.coe_isometry_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.isometry_of_orthonormal_to_linear_map (f : E →ₗ[𝕜] E') {v : basis ι 𝕜 E}
(hv : orthonormal 𝕜 v) (hf : orthonormal 𝕜 (f ∘ v)) :
(f.isometry_of_orthonormal hv hf).to_linear_map = f | rfl | lemma | linear_map.isometry_of_orthonormal_to_linear_map | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.isometry_of_orthonormal (f : E ≃ₗ[𝕜] E') {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v)
(hf : orthonormal 𝕜 (f ∘ v)) : E ≃ₗᵢ[𝕜] E' | f.isometry_of_inner $ λ x y, begin
rw ←linear_equiv.coe_coe at hf,
rw [←v.total_repr x, ←v.total_repr y, ←linear_equiv.coe_coe, finsupp.apply_total,
finsupp.apply_total, hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]
end | def | linear_equiv.isometry_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"finsupp.apply_total",
"orthonormal"
] | A linear equivalence that sends an orthonormal basis to orthonormal vectors is a linear
isometric equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv.coe_isometry_of_orthonormal (f : E ≃ₗ[𝕜] E') {v : basis ι 𝕜 E}
(hv : orthonormal 𝕜 v) (hf : orthonormal 𝕜 (f ∘ v)) :
⇑(f.isometry_of_orthonormal hv hf) = f | rfl | lemma | linear_equiv.coe_isometry_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.isometry_of_orthonormal_to_linear_equiv (f : E ≃ₗ[𝕜] E')
{v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v) (hf : orthonormal 𝕜 (f ∘ v)) :
(f.isometry_of_orthonormal hv hf).to_linear_equiv = f | rfl | lemma | linear_equiv.isometry_of_orthonormal_to_linear_equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal.equiv {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v) {v' : basis ι' 𝕜 E'}
(hv' : orthonormal 𝕜 v') (e : ι ≃ ι') : E ≃ₗᵢ[𝕜] E' | (v.equiv v' e).isometry_of_orthonormal hv begin
have h : (v.equiv v' e) ∘ v = v' ∘ e,
{ ext i,
simp },
rw h,
exact hv'.comp _ e.injective
end | def | orthonormal.equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | A linear isometric equivalence that sends an orthonormal basis to a given orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.equiv_to_linear_equiv {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v)
{v' : basis ι' 𝕜 E'} (hv' : orthonormal 𝕜 v') (e : ι ≃ ι') :
(hv.equiv hv' e).to_linear_equiv = v.equiv v' e | rfl | lemma | orthonormal.equiv_to_linear_equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal.equiv_apply {ι' : Type*} {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v)
{v' : basis ι' 𝕜 E'} (hv' : orthonormal 𝕜 v') (e : ι ≃ ι') (i : ι) :
hv.equiv hv' e (v i) = v' (e i) | basis.equiv_apply _ _ _ _ | lemma | orthonormal.equiv_apply | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"basis.equiv_apply",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal.equiv_refl {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v) :
hv.equiv hv (equiv.refl ι) = linear_isometry_equiv.refl 𝕜 E | v.ext_linear_isometry_equiv $ λ i,
by simp only [orthonormal.equiv_apply, equiv.coe_refl, id.def, linear_isometry_equiv.coe_refl] | lemma | orthonormal.equiv_refl | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"equiv.coe_refl",
"equiv.refl",
"linear_isometry_equiv.coe_refl",
"linear_isometry_equiv.refl",
"orthonormal",
"orthonormal.equiv_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal.equiv_symm {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v) {v' : basis ι' 𝕜 E'}
(hv' : orthonormal 𝕜 v') (e : ι ≃ ι') : (hv.equiv hv' e).symm = hv'.equiv hv e.symm | v'.ext_linear_isometry_equiv $ λ i, (hv.equiv hv' e).injective $
by simp only [linear_isometry_equiv.apply_symm_apply, orthonormal.equiv_apply, e.apply_symm_apply] | lemma | orthonormal.equiv_symm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"linear_isometry_equiv.apply_symm_apply",
"orthonormal",
"orthonormal.equiv_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal.equiv_trans {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v) {v' : basis ι' 𝕜 E'}
(hv' : orthonormal 𝕜 v') (e : ι ≃ ι') {v'' : basis ι'' 𝕜 E''} (hv'' : orthonormal 𝕜 v'')
(e' : ι' ≃ ι'') : (hv.equiv hv' e).trans (hv'.equiv hv'' e') = hv.equiv hv'' (e.trans e') | v.ext_linear_isometry_equiv $ λ i,
by simp only [linear_isometry_equiv.trans_apply, orthonormal.equiv_apply, e.coe_trans] | lemma | orthonormal.equiv_trans | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"linear_isometry_equiv.trans_apply",
"orthonormal",
"orthonormal.equiv_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal.map_equiv {v : basis ι 𝕜 E} (hv : orthonormal 𝕜 v) {v' : basis ι' 𝕜 E'}
(hv' : orthonormal 𝕜 v') (e : ι ≃ ι') :
v.map ((hv.equiv hv' e).to_linear_equiv) = v'.reindex e.symm | v.map_equiv _ _ | lemma | orthonormal.map_equiv | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : F) :
⟪x, y⟫_ℝ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2 | re_to_real.symm.trans $
re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two x y | lemma | real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two"
] | Polarization identity: The real inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : F) :
⟪x, y⟫_ℝ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 | re_to_real.symm.trans $
re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two x y | lemma | real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two"
] | Polarization identity: The real inner product, in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero (x y : F) :
‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ ⟪x, y⟫_ℝ = 0 | begin
rw [@norm_add_mul_self ℝ, add_right_cancel_iff, add_right_eq_self, mul_eq_zero],
norm_num
end | lemma | norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"mul_eq_zero",
"norm_add_mul_self"
] | Pythagorean theorem, if-and-only-if vector inner product form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_eq_sqrt_iff_real_inner_eq_zero {x y : F} :
‖x + y‖ = sqrt (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0 | by rw [←norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero, eq_comm,
sqrt_eq_iff_mul_self_eq (add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)) (norm_nonneg _)] | lemma | norm_add_eq_sqrt_iff_real_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"mul_self_nonneg"
] | Pythagorean theorem, if-and-if vector inner product form using square roots. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (x y : E) (h : ⟪x, y⟫ = 0) :
‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ | begin
rw [@norm_add_mul_self 𝕜, add_right_cancel_iff, add_right_eq_self, mul_eq_zero],
apply or.inr,
simp only [h, zero_re'],
end | lemma | norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"mul_eq_zero",
"norm_add_mul_self"
] | Pythagorean theorem, vector inner product form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_sq_eq_norm_sq_add_norm_sq_real {x y : F} (h : ⟪x, y⟫_ℝ = 0) :
‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ | (norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero x y).2 h | lemma | norm_add_sq_eq_norm_sq_add_norm_sq_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero"
] | Pythagorean theorem, vector inner product form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero (x y : F) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ ⟪x, y⟫_ℝ = 0 | begin
rw [@norm_sub_mul_self ℝ, add_right_cancel_iff, sub_eq_add_neg, add_right_eq_self, neg_eq_zero,
mul_eq_zero],
norm_num
end | lemma | norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"mul_eq_zero",
"norm_sub_mul_self"
] | Pythagorean theorem, subtracting vectors, if-and-only-if vector
inner product form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_eq_sqrt_iff_real_inner_eq_zero {x y : F} :
‖x - y‖ = sqrt (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0 | by rw [←norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero, eq_comm,
sqrt_eq_iff_mul_self_eq (add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)) (norm_nonneg _)] | lemma | norm_sub_eq_sqrt_iff_real_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"mul_self_nonneg"
] | Pythagorean theorem, subtracting vectors, if-and-if vector inner product form using square
roots. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_sq_eq_norm_sq_add_norm_sq_real {x y : F} (h : ⟪x, y⟫_ℝ = 0) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ | (norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero x y).2 h | lemma | norm_sub_sq_eq_norm_sq_add_norm_sq_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero"
] | Pythagorean theorem, subtracting vectors, vector inner product
form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_add_sub_eq_zero_iff (x y : F) : ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ = ‖y‖ | begin
conv_rhs { rw ←mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _) },
simp only [←@inner_self_eq_norm_mul_norm ℝ, inner_add_left, inner_sub_right,
real_inner_comm y x, sub_eq_zero, re_to_real],
split,
{ intro h,
rw [add_comm] at h,
linarith },
{ intro h,
linarith }
end | lemma | real_inner_add_sub_eq_zero_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_left",
"inner_self_eq_norm_mul_norm",
"inner_sub_right",
"real_inner_comm"
] | The sum and difference of two vectors are orthogonal if and only
if they have the same norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_eq_norm_add {v w : E} (h : ⟪v, w⟫ = 0) : ‖w - v‖ = ‖w + v‖ | begin
rw ←mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _),
simp only [h, ←@inner_self_eq_norm_mul_norm 𝕜, sub_neg_eq_add, sub_zero, map_sub, zero_re',
zero_sub,
add_zero, map_add, inner_add_right, inner_sub_left, inner_sub_right, inner_re_symm, zero_add]
end | lemma | norm_sub_eq_norm_add | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_right",
"inner_re_symm",
"inner_self_eq_norm_mul_norm",
"inner_sub_left",
"inner_sub_right"
] | Given two orthogonal vectors, their sum and difference have equal norms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_real_inner_div_norm_mul_norm_le_one (x y : F) : |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| ≤ 1 | begin
rw [abs_div, abs_mul, abs_norm, abs_norm],
exact div_le_one_of_le (abs_real_inner_le_norm x y) (by positivity)
end | lemma | abs_real_inner_div_norm_mul_norm_le_one | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"abs_div",
"abs_mul",
"abs_norm",
"abs_real_inner_le_norm",
"div_le_one_of_le"
] | The real inner product of two vectors, divided by the product of their
norms, has absolute value at most 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_smul_self_left (x : F) (r : ℝ) : ⟪r • x, x⟫_ℝ = r * (‖x‖ * ‖x‖) | by rw [real_inner_smul_left, ←real_inner_self_eq_norm_mul_norm] | lemma | real_inner_smul_self_left | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"real_inner_smul_left"
] | The inner product of a vector with a multiple of itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_smul_self_right (x : F) (r : ℝ) : ⟪x, r • x⟫_ℝ = r * (‖x‖ * ‖x‖) | by rw [inner_smul_right, ←real_inner_self_eq_norm_mul_norm] | lemma | real_inner_smul_self_right | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_smul_right"
] | The inner product of a vector with a multiple of itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul
{x : E} {r : 𝕜} (hx : x ≠ 0) (hr : r ≠ 0) : ‖⟪x, r • x⟫‖ / (‖x‖ * ‖r • x‖) = 1 | begin
have hx' : ‖x‖ ≠ 0 := by simp [hx],
have hr' : ‖r‖ ≠ 0 := by simp [hr],
rw [inner_smul_right, norm_mul, ← inner_self_re_eq_norm, inner_self_eq_norm_mul_norm, norm_smul],
rw [← mul_assoc, ← div_div, mul_div_cancel _ hx',
← div_div, mul_comm, mul_div_cancel _ hr', div_self hx'],
end | lemma | norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"div_div",
"div_self",
"inner_self_eq_norm_mul_norm",
"inner_self_re_eq_norm",
"inner_smul_right",
"mul_assoc",
"mul_comm",
"mul_div_cancel",
"norm_mul",
"norm_smul"
] | The inner product of a nonzero vector with a nonzero multiple of
itself, divided by the product of their norms, has absolute value
1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul
{x : F} {r : ℝ} (hx : x ≠ 0) (hr : r ≠ 0) : |⟪x, r • x⟫_ℝ| / (‖x‖ * ‖r • x‖) = 1 | norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr | lemma | abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul"
] | The inner product of a nonzero vector with a nonzero multiple of
itself, divided by the product of their norms, has absolute value
1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul
{x : F} {r : ℝ} (hx : x ≠ 0) (hr : 0 < r) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = 1 | begin
rw [real_inner_smul_self_right, norm_smul, real.norm_eq_abs, ←mul_assoc ‖x‖, mul_comm _ (|r|),
mul_assoc, abs_of_nonneg hr.le, div_self],
exact mul_ne_zero hr.ne' (mul_self_ne_zero.2 (norm_ne_zero_iff.2 hx))
end | lemma | real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"abs_of_nonneg",
"div_self",
"mul_assoc",
"mul_comm",
"mul_ne_zero",
"norm_smul",
"real.norm_eq_abs",
"real_inner_smul_self_right"
] | The inner product of a nonzero vector with a positive multiple of
itself, divided by the product of their norms, has value 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul
{x : F} {r : ℝ} (hx : x ≠ 0) (hr : r < 0) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = -1 | begin
rw [real_inner_smul_self_right, norm_smul, real.norm_eq_abs, ←mul_assoc ‖x‖, mul_comm _ (|r|),
mul_assoc, abs_of_neg hr, neg_mul, div_neg_eq_neg_div, div_self],
exact mul_ne_zero hr.ne (mul_self_ne_zero.2 (norm_ne_zero_iff.2 hx))
end | lemma | real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"abs_of_neg",
"div_neg_eq_neg_div",
"div_self",
"mul_assoc",
"mul_comm",
"mul_ne_zero",
"neg_mul",
"norm_smul",
"real.norm_eq_abs",
"real_inner_smul_self_right"
] | The inner product of a nonzero vector with a negative multiple of
itself, divided by the product of their norms, has value -1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_inner_eq_norm_tfae (x y : E) :
tfae [‖⟪x, y⟫‖ = ‖x‖ * ‖y‖,
x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x,
x = 0 ∨ ∃ r : 𝕜, y = r • x,
x = 0 ∨ y ∈ 𝕜 ∙ x] | begin
tfae_have : 1 → 2,
{ refine λ h, or_iff_not_imp_left.2 (λ hx₀, _),
have : ‖x‖ ^ 2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.2 hx₀),
rw [← sq_eq_sq (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)),
mul_pow, ← mul_right_inj' this, eq_comm, ← sub_eq_zero, ← mul_sub] at h,
simp only [@n... | lemma | norm_inner_eq_norm_tfae | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"div_eq_inv_mul",
"inner_product_space.core",
"inner_product_space.core.cauchy_schwarz_aux",
"inner_product_space.core.norm_sq_eq_zero",
"inner_product_space.to_core",
"inner_self_eq_norm_mul_norm",
"inner_self_eq_norm_sq_to_K",
"inner_self_ne_zero",
"inner_smul_right",
"inv_smul_smul₀",
"mul_le... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inner_eq_norm_iff {x y : E} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) :
‖⟪x, y⟫‖ = ‖x‖ * ‖y‖ ↔ ∃ (r : 𝕜), r ≠ 0 ∧ y = r • x | calc ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖ ↔ x = 0 ∨ ∃ r : 𝕜, y = r • x :
(@norm_inner_eq_norm_tfae 𝕜 _ _ _ _ x y).out 0 2
... ↔ ∃ r : 𝕜, y = r • x : or_iff_right hx₀
... ↔ ∃ r : 𝕜, r ≠ 0 ∧ y = r • x :
⟨λ ⟨r, h⟩, ⟨r, λ hr₀, hy₀ $ h.symm ▸ smul_eq_zero.2 $ or.inl hr₀, h⟩, λ ⟨r, hr₀, h⟩, ⟨r, h⟩⟩ | lemma | norm_inner_eq_norm_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_inner_eq_norm_tfae",
"or_iff_right"
] | If the inner product of two vectors is equal to the product of their norms, then the two vectors
are multiples of each other. One form of the equality case for Cauchy-Schwarz.
Compare `inner_eq_norm_mul_iff`, which takes the stronger hypothesis `⟪x, y⟫ = ‖x‖ * ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_inner_div_norm_mul_norm_eq_one_iff (x y : E) :
‖(⟪x, y⟫ / (‖x‖ * ‖y‖))‖ = 1 ↔ (x ≠ 0 ∧ ∃ (r : 𝕜), r ≠ 0 ∧ y = r • x) | begin
split,
{ intro h,
have hx₀ : x ≠ 0 := λ h₀, by simpa [h₀] using h,
have hy₀ : y ≠ 0 := λ h₀, by simpa [h₀] using h,
refine ⟨hx₀, (norm_inner_eq_norm_iff hx₀ hy₀).1 $ eq_of_div_eq_one _⟩,
simpa using h },
{ rintro ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩,
simp only [norm_div, norm_mul, norm_of_real, abs_norm... | lemma | norm_inner_div_norm_mul_norm_eq_one_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"abs_norm",
"eq_of_div_eq_one",
"norm_div",
"norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul",
"norm_inner_eq_norm_iff",
"norm_mul"
] | The inner product of two vectors, divided by the product of their
norms, has absolute value 1 if and only if they are nonzero and one is
a multiple of the other. One form of equality case for Cauchy-Schwarz. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_real_inner_div_norm_mul_norm_eq_one_iff (x y : F) : | |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| = 1 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), r ≠ 0 ∧ y = r • x) :=
@norm_inner_div_norm_mul_norm_eq_one_iff ℝ F _ _ _ x y | lemma | abs_real_inner_div_norm_mul_norm_eq_one_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"norm_inner_div_norm_mul_norm_eq_one_iff"
] | The inner product of two vectors, divided by the product of their
norms, has absolute value 1 if and only if they are nonzero and one is
a multiple of the other. One form of equality case for Cauchy-Schwarz. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_eq_norm_mul_iff_div {x y : E} (h₀ : x ≠ 0) :
⟪x, y⟫ = (‖x‖ : 𝕜) * ‖y‖ ↔ (‖y‖ / ‖x‖ : 𝕜) • x = y | begin
have h₀' := h₀,
rw [← norm_ne_zero_iff, ne.def, ← @of_real_eq_zero 𝕜] at h₀',
split; intro h,
{ have : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫ : 𝕜) • x :=
((@norm_inner_eq_norm_tfae 𝕜 _ _ _ _ x y).out 0 1).1 (by simp [h]),
rw [this.resolve_left h₀, h],
simp [norm_smul, inner_self_norm_to_K, h₀'] },
... | lemma | inner_eq_norm_mul_iff_div | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_self_eq_norm_sq_to_K",
"inner_self_norm_to_K",
"inner_smul_right",
"mul_left_comm",
"norm_inner_eq_norm_tfae",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_eq_norm_mul_iff {x y : E} :
⟪x, y⟫ = (‖x‖ : 𝕜) * ‖y‖ ↔ (‖y‖ : 𝕜) • x = (‖x‖ : 𝕜) • y | begin
rcases eq_or_ne x 0 with (rfl | h₀),
{ simp },
{ rw [inner_eq_norm_mul_iff_div h₀, div_eq_inv_mul, mul_smul, inv_smul_eq_iff₀],
rwa [ne.def, of_real_eq_zero, norm_eq_zero] },
end | lemma | inner_eq_norm_mul_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"div_eq_inv_mul",
"eq_or_ne",
"inner_eq_norm_mul_iff_div",
"inv_smul_eq_iff₀",
"norm_eq_zero"
] | If the inner product of two vectors is equal to the product of their norms (i.e.,
`⟪x, y⟫ = ‖x‖ * ‖y‖`), then the two vectors are nonnegative real multiples of each other. One form
of the equality case for Cauchy-Schwarz.
Compare `norm_inner_eq_norm_iff`, which takes the weaker hypothesis `abs ⟪x, y⟫ = ‖x‖ * ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_eq_norm_mul_iff_real {x y : F} : ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y | inner_eq_norm_mul_iff | lemma | inner_eq_norm_mul_iff_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_eq_norm_mul_iff"
] | If the inner product of two vectors is equal to the product of their norms (i.e.,
`⟪x, y⟫ = ‖x‖ * ‖y‖`), then the two vectors are nonnegative real multiples of each other. One form
of the equality case for Cauchy-Schwarz.
Compare `norm_inner_eq_norm_iff`, which takes the weaker hypothesis `abs ⟪x, y⟫ = ‖x‖ * ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_div_norm_mul_norm_eq_one_iff (x y : F) :
⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), 0 < r ∧ y = r • x) | begin
split,
{ intro h,
have hx₀ : x ≠ 0 := λ h₀, by simpa [h₀] using h,
have hy₀ : y ≠ 0 := λ h₀, by simpa [h₀] using h,
refine ⟨hx₀, ‖y‖ / ‖x‖, div_pos (norm_pos_iff.2 hy₀) (norm_pos_iff.2 hx₀), _⟩,
exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm },
{ rintro ⟨hx, ⟨r, ⟨hr, rfl... | lemma | real_inner_div_norm_mul_norm_eq_one_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"div_pos",
"eq_of_div_eq_one",
"inner_eq_norm_mul_iff_div",
"real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul"
] | The inner product of two vectors, divided by the product of their
norms, has value 1 if and only if they are nonzero and one is
a positive multiple of the other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_div_norm_mul_norm_eq_neg_one_iff (x y : F) :
⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = -1 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), r < 0 ∧ y = r • x) | begin
rw [← neg_eq_iff_eq_neg, ← neg_div, ← inner_neg_right, ← norm_neg y,
real_inner_div_norm_mul_norm_eq_one_iff, (@neg_surjective ℝ _).exists],
refine iff.rfl.and (exists_congr $ λ r, _),
rw [neg_pos, neg_smul, neg_inj]
end | lemma | real_inner_div_norm_mul_norm_eq_neg_one_iff | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_neg_right",
"neg_div",
"neg_smul",
"real_inner_div_norm_mul_norm_eq_one_iff"
] | The inner product of two vectors, divided by the product of their
norms, has value -1 if and only if they are nonzero and one is
a negative multiple of the other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_eq_one_iff_of_norm_one {x y : E} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
⟪x, y⟫ = 1 ↔ x = y | by { convert inner_eq_norm_mul_iff using 2; simp [hx, hy] } | lemma | inner_eq_one_iff_of_norm_one | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_eq_norm_mul_iff"
] | If the inner product of two unit vectors is `1`, then the two vectors are equal. One form of
the equality case for Cauchy-Schwarz. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_lt_norm_mul_iff_real {x y : F} :
⟪x, y⟫_ℝ < ‖x‖ * ‖y‖ ↔ ‖y‖ • x ≠ ‖x‖ • y | calc ⟪x, y⟫_ℝ < ‖x‖ * ‖y‖
↔ ⟪x, y⟫_ℝ ≠ ‖x‖ * ‖y‖ : ⟨ne_of_lt, lt_of_le_of_ne (real_inner_le_norm _ _)⟩
... ↔ ‖y‖ • x ≠ ‖x‖ • y : not_congr inner_eq_norm_mul_iff_real | lemma | inner_lt_norm_mul_iff_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_eq_norm_mul_iff_real",
"real_inner_le_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_lt_one_iff_real_of_norm_one {x y : F} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
⟪x, y⟫_ℝ < 1 ↔ x ≠ y | by { convert inner_lt_norm_mul_iff_real; simp [hx, hy] } | lemma | inner_lt_one_iff_real_of_norm_one | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_lt_norm_mul_iff_real"
] | If the inner product of two unit vectors is strictly less than `1`, then the two vectors are
distinct. One form of the equality case for Cauchy-Schwarz. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_sum_smul_sum_smul_of_sum_eq_zero {ι₁ : Type*} {s₁ : finset ι₁} {w₁ : ι₁ → ℝ}
(v₁ : ι₁ → F) (h₁ : ∑ i in s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : finset ι₂} {w₂ : ι₂ → ℝ}
(v₂ : ι₂ → F) (h₂ : ∑ i in s₂, w₂ i = 0) :
⟪(∑ i₁ in s₁, w₁ i₁ • v₁ i₁), (∑ i₂ in s₂, w₂ i₂ • v₂ i₂)⟫_ℝ =
(-∑ i₁ in s₁, ∑ i₂ in s₂, w₁ i₁ ... | by simp_rw [sum_inner, inner_sum, real_inner_smul_left, real_inner_smul_right,
real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two,
←div_sub_div_same, ←div_add_div_same, mul_sub_left_distrib, left_distrib,
finset.sum_sub_distrib, finset.sum_add_distrib, ←finse... | lemma | inner_sum_smul_sum_smul_of_sum_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"finset",
"finset.mul_sum",
"finset.sum_div",
"inner_sum",
"left_distrib",
"mul_assoc",
"mul_div_assoc",
"mul_sub_left_distrib",
"mul_zero",
"neg_div",
"real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two",
"real_inner_smul_left",
"real_inner_smul_right",
"sum_inner... | The inner product of two weighted sums, where the weights in each
sum add to 0, in terms of the norms of pairwise differences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
innerₛₗ : E →ₗ⋆[𝕜] E →ₗ[𝕜] 𝕜 | linear_map.mk₂'ₛₗ _ _ (λ v w, ⟪v, w⟫) inner_add_left (λ _ _ _, inner_smul_left _ _ _)
inner_add_right (λ _ _ _, inner_smul_right _ _ _) | def | innerₛₗ | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_left",
"inner_add_right",
"inner_smul_left",
"inner_smul_right",
"linear_map.mk₂'ₛₗ"
] | The inner product as a sesquilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
innerₛₗ_apply_coe (v : E) : ⇑(innerₛₗ 𝕜 v) = λ w, ⟪v, w⟫ | rfl | lemma | innerₛₗ_apply_coe | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerₛₗ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
innerₛₗ_apply (v w : E) : innerₛₗ 𝕜 v w = ⟪v, w⟫ | rfl | lemma | innerₛₗ_apply | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerₛₗ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
innerSL : E →L⋆[𝕜] E →L[𝕜] 𝕜 | linear_map.mk_continuous₂ (innerₛₗ 𝕜) 1
(λ x y, by simp only [norm_inner_le_norm, one_mul, innerₛₗ_apply]) | def | innerSL | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerₛₗ",
"innerₛₗ_apply",
"linear_map.mk_continuous₂",
"norm_inner_le_norm",
"one_mul"
] | The inner product as a continuous sesquilinear map. Note that `to_dual_map` (resp. `to_dual`)
in `inner_product_space.dual` is a version of this given as a linear isometry (resp. linear
isometric equivalence). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
innerSL_apply_coe (v : E) : ⇑(innerSL 𝕜 v) = λ w, ⟪v, w⟫ | rfl | lemma | innerSL_apply_coe | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerSL"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
innerSL_apply (v w : E) : innerSL 𝕜 v w = ⟪v, w⟫ | rfl | lemma | innerSL_apply | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerSL"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
innerSL_apply_norm (x : E) : ‖innerSL 𝕜 x‖ = ‖x‖ | begin
refine le_antisymm ((innerSL 𝕜 x).op_norm_le_bound (norm_nonneg _)
(λ y, norm_inner_le_norm _ _)) _,
rcases eq_or_ne x 0 with (rfl | h),
{ simp },
{ refine (mul_le_mul_right (norm_pos_iff.2 h)).mp _,
calc ‖x‖ * ‖x‖ = ‖(⟪x, x⟫ : 𝕜)‖ : by rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow,
norm_... | lemma | innerSL_apply_norm | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"abs_norm",
"eq_or_ne",
"innerSL",
"inner_self_eq_norm_sq_to_K",
"mul_le_mul_right",
"norm_inner_le_norm",
"norm_pow"
] | `innerSL` is an isometry. Note that the associated `linear_isometry` is defined in
`inner_product_space.dual` as `to_dual_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
innerSL_flip : E →L[𝕜] E →L⋆[𝕜] 𝕜 | @continuous_linear_map.flipₗᵢ' 𝕜 𝕜 𝕜 E E 𝕜 _ _ _ _ _ _ _ _ _ (ring_hom.id 𝕜) (star_ring_end 𝕜) _ _
(innerSL 𝕜) | def | innerSL_flip | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous_linear_map.flipₗᵢ'",
"innerSL",
"ring_hom.id",
"star_ring_end"
] | The inner product as a continuous sesquilinear map, with the two arguments flipped. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
innerSL_flip_apply (x y : E) : innerSL_flip 𝕜 x y = ⟪y, x⟫ | rfl | lemma | innerSL_flip_apply | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerSL_flip"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sesq_form : (E →L[𝕜] E') →L[𝕜] E' →L⋆[𝕜] E →L[𝕜] 𝕜 | ↑((continuous_linear_map.flipₗᵢ' E E' 𝕜
(star_ring_end 𝕜) (ring_hom.id 𝕜)).to_continuous_linear_equiv) ∘L
(continuous_linear_map.compSL E E' (E' →L⋆[𝕜] 𝕜) (ring_hom.id 𝕜) (ring_hom.id 𝕜) (innerSL_flip 𝕜)) | def | continuous_linear_map.to_sesq_form | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous_linear_map.compSL",
"continuous_linear_map.flipₗᵢ'",
"innerSL_flip",
"ring_hom.id",
"star_ring_end"
] | Given `f : E →L[𝕜] E'`, construct the continuous sesquilinear form `λ x y, ⟪x, A y⟫`, given
as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_sesq_form_apply_coe (f : E →L[𝕜] E') (x : E') :
to_sesq_form f x = (innerSL 𝕜 x).comp f | rfl | lemma | continuous_linear_map.to_sesq_form_apply_coe | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"innerSL"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sesq_form_apply_norm_le {f : E →L[𝕜] E'} {v : E'} : ‖to_sesq_form f v‖ ≤ ‖f‖ * ‖v‖ | begin
refine op_norm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) _,
intro x,
have h₁ : ‖f x‖ ≤ ‖f‖ * ‖x‖ := le_op_norm _ _,
have h₂ := @norm_inner_le_norm 𝕜 E' _ _ _ v (f x),
calc ‖⟪v, f x⟫‖ ≤ ‖v‖ * ‖f x‖ : h₂
... ≤ ‖v‖ * (‖f‖ * ‖x‖) : mul_le_mul_of_nonneg_left h₁ (norm_nonn... | lemma | continuous_linear_map.to_sesq_form_apply_norm_le | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"mul_le_mul_of_nonneg_left",
"norm_inner_le_norm",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map_inner [normed_space ℝ E] :
is_bounded_bilinear_map ℝ (λ p : E × E, ⟪p.1, p.2⟫) | { add_left := inner_add_left,
smul_left := λ r x y,
by simp only [← algebra_map_smul 𝕜 r x, algebra_map_eq_of_real, inner_smul_real_left],
add_right := inner_add_right,
smul_right := λ r x y,
by simp only [← algebra_map_smul 𝕜 r y, algebra_map_eq_of_real, inner_smul_real_right],
bound := ⟨1, zero_lt_o... | lemma | is_bounded_bilinear_map_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"algebra_map_smul",
"bound",
"inner_add_left",
"inner_add_right",
"inner_smul_real_left",
"inner_smul_real_right",
"is_bounded_bilinear_map",
"norm_inner_le_norm",
"normed_space",
"one_mul",
"zero_lt_one"
] | When an inner product space `E` over `𝕜` is considered as a real normed space, its inner
product satisfies `is_bounded_bilinear_map`.
In order to state these results, we need a `normed_space ℝ E` instance. We will later establish
such an instance by restriction-of-scalars, `inner_product_space.is_R_or_C_to_real 𝕜 E`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.sum_inner_products_le {s : finset ι} (hv : orthonormal 𝕜 v) :
∑ i in s, ‖⟪v i, x⟫‖ ^ 2 ≤ ‖x‖ ^ 2 | begin
have h₂ : ∑ i in s, ∑ j in s, ⟪v i, x⟫ * ⟪x, v j⟫ * ⟪v j, v i⟫
= (∑ k in s, (⟪v k, x⟫ * ⟪x, v k⟫) : 𝕜),
{ exact hv.inner_left_right_finset },
have h₃ : ∀ z : 𝕜, re (z * conj (z)) = ‖z‖ ^ 2,
{ intro z,
simp only [mul_conj, norm_sq_eq_def'],
norm_cast, },
suffices hbf: ‖x - ∑ i in s, ⟪v i, ... | lemma | orthonormal.sum_inner_products_le | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"finset",
"finset.mul_sum",
"inner_conj_symm",
"inner_smul_left",
"inner_smul_right",
"inner_sum",
"norm_sub_sq",
"orthonormal",
"pow_nonneg",
"sum_inner",
"two_mul"
] | Bessel's inequality for finite sums. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.tsum_inner_products_le (hv : orthonormal 𝕜 v) :
∑' i, ‖⟪v i, x⟫‖ ^ 2 ≤ ‖x‖ ^ 2 | begin
refine tsum_le_of_sum_le' _ (λ s, hv.sum_inner_products_le x),
simp only [norm_nonneg, pow_nonneg]
end | lemma | orthonormal.tsum_inner_products_le | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"orthonormal",
"pow_nonneg",
"tsum_le_of_sum_le'"
] | Bessel's inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_products_summable (hv : orthonormal 𝕜 v) : summable (λ i, ‖⟪v i, x⟫‖ ^ 2) | begin
use ⨆ s : finset ι, ∑ i in s, ‖⟪v i, x⟫‖ ^ 2,
apply has_sum_of_is_lub_of_nonneg,
{ intro b,
simp only [norm_nonneg, pow_nonneg], },
{ refine is_lub_csupr _,
use ‖x‖ ^ 2,
rintro y ⟨s, rfl⟩,
exact hv.sum_inner_products_le x }
end | lemma | orthonormal.inner_products_summable | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"finset",
"has_sum_of_is_lub_of_nonneg",
"is_lub_csupr",
"orthonormal",
"pow_nonneg",
"summable"
] | The sum defined in Bessel's inequality is summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_R_or_C.inner_product_space : inner_product_space 𝕜 𝕜 | { inner := λ x y, conj x * y,
norm_sq_eq_inner := λ x,
by { unfold inner, rw [mul_comm, mul_conj, of_real_re, norm_sq_eq_def'] },
conj_symm := λ x y, by simp only [mul_comm, map_mul, star_ring_end_self_apply],
add_left := λ x y z, by simp only [add_mul, map_add],
smul_left := λ x y z, by simp only [mul_asso... | instance | is_R_or_C.inner_product_space | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_product_space",
"map_mul",
"mul_assoc",
"mul_comm",
"smul_eq_mul",
"star_ring_end_self_apply"
] | A field `𝕜` satisfying `is_R_or_C` is itself a `𝕜`-inner product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_R_or_C.inner_apply (x y : 𝕜) : ⟪x, y⟫ = (conj x) * y | rfl | lemma | is_R_or_C.inner_apply | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.inner_product_space (W : submodule 𝕜 E) : inner_product_space 𝕜 W | { inner := λ x y, ⟪(x:E), (y:E)⟫,
conj_symm := λ _ _, inner_conj_symm _ _,
norm_sq_eq_inner := λ x, norm_sq_eq_inner (x : E),
add_left := λ _ _ _, inner_add_left _ _ _,
smul_left := λ _ _ _, inner_smul_left _ _ _,
..submodule.normed_space W } | instance | submodule.inner_product_space | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_add_left",
"inner_conj_symm",
"inner_product_space",
"inner_smul_left",
"submodule",
"submodule.normed_space"
] | Induced inner product on a submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.coe_inner (W : submodule 𝕜 E) (x y : W) : ⟪x, y⟫ = ⟪(x:E), ↑y⟫ | rfl | lemma | submodule.coe_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"submodule"
] | The inner product on submodules is the same as on the ambient space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.cod_restrict {ι : Type*} {v : ι → E} (hv : orthonormal 𝕜 v)
(s : submodule 𝕜 E) (hvs : ∀ i, v i ∈ s) :
@orthonormal 𝕜 s _ _ _ ι (set.cod_restrict v s hvs) | s.subtypeₗᵢ.orthonormal_comp_iff.mp hv | lemma | orthonormal.cod_restrict | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"orthonormal",
"set.cod_restrict",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal_span {ι : Type*} {v : ι → E} (hv : orthonormal 𝕜 v) :
@orthonormal 𝕜 (submodule.span 𝕜 (set.range v)) _ _ _ ι
(λ i : ι, ⟨v i, submodule.subset_span (set.mem_range_self i)⟩) | hv.cod_restrict (submodule.span 𝕜 (set.range v))
(λ i, submodule.subset_span (set.mem_range_self i)) | lemma | orthonormal_span | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"orthonormal",
"set.mem_range_self",
"set.range",
"submodule.span",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family (G : ι → Type*)
[Π i, normed_add_comm_group (G i)] [Π i, inner_product_space 𝕜 (G i)] (V : Π i, G i →ₗᵢ[𝕜] E) :
Prop | ∀ ⦃i j⦄, i ≠ j → ∀ v : G i, ∀ w : G j, ⟪V i v, V j w⟫ = 0 | def | orthogonal_family | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_product_space",
"normed_add_comm_group"
] | An indexed family of mutually-orthogonal subspaces of an inner product space `E`.
The simple way to express this concept would be as a condition on `V : ι → submodule 𝕜 E`. We
We instead implement it as a condition on a family of inner product spaces each equipped with an
isometric embedding into `E`, thus making it... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.orthogonal_family {v : ι → E} (hv : orthonormal 𝕜 v) :
orthogonal_family 𝕜 (λ i : ι, 𝕜)
(λ i, linear_isometry.to_span_singleton 𝕜 E (hv.1 i)) | λ i j hij a b, by simp [inner_smul_left, inner_smul_right, hv.2 hij] | lemma | orthonormal.orthogonal_family | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_smul_left",
"inner_smul_right",
"linear_isometry.to_span_singleton",
"orthogonal_family",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.eq_ite {i j : ι} (v : G i) (w : G j) :
⟪V i v, V j w⟫ = ite (i = j) ⟪V i v, V j w⟫ 0 | begin
split_ifs,
{ refl },
{ exact hV h v w }
end | lemma | orthogonal_family.eq_ite | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.inner_right_dfinsupp (l : ⨁ i, G i) (i : ι) (v : G i) :
⟪V i v, l.sum (λ j, V j)⟫ = ⟪v, l i⟫ | calc ⟪V i v, l.sum (λ j, V j)⟫
= l.sum (λ j, λ w, ⟪V i v, V j w⟫) : dfinsupp.inner_sum (λ j, V j) l (V i v)
... = l.sum (λ j, λ w, ite (i=j) ⟪V i v, V j w⟫ 0) :
congr_arg l.sum $ funext $ λ j, funext $ hV.eq_ite v
... = ⟪v, l i⟫ :
begin
simp only [dfinsupp.sum, submodule.coe_inner, finset.sum_ite_eq, ite_eq_lef... | lemma | orthogonal_family.inner_right_dfinsupp | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"dfinsupp.inner_sum",
"dfinsupp.mem_support_to_fun",
"inner_zero_right",
"ite_eq_left_iff",
"linear_isometry.inner_map_map",
"of_not_not",
"submodule.coe_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.inner_right_fintype [fintype ι] (l : Π i, G i) (i : ι) (v : G i) :
⟪V i v, ∑ j : ι, V j (l j)⟫ = ⟪v, l i⟫ | by classical;
calc ⟪V i v, ∑ j : ι, V j (l j)⟫
= ∑ j : ι, ⟪V i v, V j (l j)⟫: by rw inner_sum
... = ∑ j, ite (i = j) ⟪V i v, V j (l j)⟫ 0 :
congr_arg (finset.sum finset.univ) $ funext $ λ j, (hV.eq_ite v (l j))
... = ⟪v, l i⟫ : by simp only [finset.sum_ite_eq, finset.mem_univ, (V i).inner_map_map, if_true] | lemma | orthogonal_family.inner_right_fintype | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"finset.mem_univ",
"finset.univ",
"fintype",
"inner_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.inner_sum (l₁ l₂ : Π i, G i) (s : finset ι) :
⟪∑ i in s, V i (l₁ i), ∑ j in s, V j (l₂ j)⟫ = ∑ i in s, ⟪l₁ i, l₂ i⟫ | by classical;
calc ⟪∑ i in s, V i (l₁ i), ∑ j in s, V j (l₂ j)⟫
= ∑ j in s, ∑ i in s, ⟪V i (l₁ i), V j (l₂ j)⟫ : by simp only [sum_inner, inner_sum]
... = ∑ j in s, ∑ i in s, ite (i = j) ⟪V i (l₁ i), V j (l₂ j)⟫ 0 :
begin
congr' with i,
congr' with j,
apply hV.eq_ite,
end
... = ∑ i in s, ⟪l₁ i, l₂ i⟫ : by sim... | lemma | orthogonal_family.inner_sum | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"finset",
"imp_self",
"inner_sum",
"linear_isometry.inner_map_map",
"sum_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.norm_sum (l : Π i, G i) (s : finset ι) :
‖∑ i in s, V i (l i)‖ ^ 2 = ∑ i in s, ‖l i‖ ^ 2 | begin
have : (‖∑ i in s, V i (l i)‖ ^ 2 : 𝕜) = ∑ i in s, ‖l i‖ ^ 2,
{ simp only [← inner_self_eq_norm_sq_to_K, hV.inner_sum] },
exact_mod_cast this,
end | lemma | orthogonal_family.norm_sum | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"finset",
"inner_self_eq_norm_sq_to_K"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.comp {γ : Type*} {f : γ → ι} (hf : function.injective f) :
orthogonal_family 𝕜 (λ g, G (f g)) (λ g, V (f g)) | λ i j hij v w, hV (hf.ne hij) v w | lemma | orthogonal_family.comp | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"orthogonal_family"
] | The composition of an orthogonal family of subspaces with an injective function is also an
orthogonal family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family.orthonormal_sigma_orthonormal {α : ι → Type*} {v_family : Π i, (α i) → G i}
(hv_family : ∀ i, orthonormal 𝕜 (v_family i)) :
orthonormal 𝕜 (λ a : Σ i, α i, V a.1 (v_family a.1 a.2)) | begin
split,
{ rintros ⟨i, v⟩,
simpa only [linear_isometry.norm_map] using (hv_family i).left v },
rintros ⟨i, v⟩ ⟨j, w⟩ hvw,
by_cases hij : i = j,
{ subst hij,
have : v ≠ w := λ h, by { subst h, exact hvw rfl },
simpa only [linear_isometry.inner_map_map] using (hv_family i).2 this },
{ exact hV... | lemma | orthogonal_family.orthonormal_sigma_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"linear_isometry.inner_map_map",
"linear_isometry.norm_map",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.norm_sq_diff_sum (f : Π i, G i) (s₁ s₂ : finset ι) :
‖∑ i in s₁, V i (f i) - ∑ i in s₂, V i (f i)‖ ^ 2
= ∑ i in s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2 | begin
rw [← finset.sum_sdiff_sub_sum_sdiff, sub_eq_add_neg, ← finset.sum_neg_distrib],
let F : Π i, G i := λ i, if i ∈ s₁ then f i else - (f i),
have hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i := λ i hi, if_pos (finset.sdiff_subset _ _ hi),
have hF₂ : ∀ i ∈ s₂ \ s₁, F i = - f i := λ i hi, if_neg (finset.mem_sdiff.mp hi).2,... | lemma | orthogonal_family.norm_sq_diff_sum | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"disjoint",
"disjoint_sdiff_sdiff",
"finset",
"finset.sdiff_subset",
"linear_isometry.map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_family.summable_iff_norm_sq_summable [complete_space E] (f : Π i, G i) :
summable (λ i, V i (f i)) ↔ summable (λ i, ‖f i‖ ^ 2) | begin
classical,
simp only [summable_iff_cauchy_seq_finset, normed_add_comm_group.cauchy_seq_iff,
real.norm_eq_abs],
split,
{ intros hf ε hε,
obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε),
use a,
intros s₁ hs₁ s₂ hs₂,
rw ← finset.sum_sdiff_sub_sum_sdiff,
refine (abs_sub _ _).trans_lt _,
ha... | lemma | orthogonal_family.summable_iff_norm_sq_summable | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"abs_lt_of_sq_lt_sq'",
"abs_of_nonneg",
"abs_sub",
"add_tsub_cancel_right",
"complete_space",
"finset.abs_sum_of_nonneg'",
"finset.inter_subset_left",
"finset.inter_subset_right",
"half_pos",
"le_inf",
"real.norm_eq_abs",
"sq_lt_sq",
"sq_nonneg",
"sq_pos_of_pos",
"summable",
"summable_... | A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(λ i, ‖f i‖ ^ 2)` is summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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