statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
inner_eq_zero_symm {x y : E} : ⟪x, y⟫ = 0 ↔ ⟪y, x⟫ = 0 | by { rw [← inner_conj_symm], exact star_eq_zero } | lemma | inner_eq_zero_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"
] | [
"inner_conj_symm",
"star_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_im (x : E) : im ⟪x, x⟫ = 0 | by rw [← @of_real_inj 𝕜, im_eq_conj_sub]; simp | lemma | inner_self_im | 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 | |
inner_add_left (x y z : E) : ⟪x + y, z⟫ = ⟪x, z⟫ + ⟪y, z⟫ | inner_product_space.add_left _ _ _ | lemma | inner_add_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_add_right (x y z : E) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ | by { rw [←inner_conj_symm, inner_add_left, ring_hom.map_add], simp only [inner_conj_symm] } | lemma | inner_add_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_add_left",
"inner_conj_symm",
"ring_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_re_symm (x y : E) : re ⟪x, y⟫ = re ⟪y, x⟫ | by rw [←inner_conj_symm, conj_re] | lemma | inner_re_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_im_symm (x y : E) : im ⟪x, y⟫ = -im ⟪y, x⟫ | by rw [←inner_conj_symm, conj_im] | lemma | inner_im_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_smul_left (x y : E) (r : 𝕜) : ⟪r • x, y⟫ = r† * ⟪x, y⟫ | inner_product_space.smul_left _ _ _ | lemma | inner_smul_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ | inner_smul_left _ _ _ | lemma | real_inner_smul_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"
] | [
"inner_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_smul_real_left (x y : E) (r : ℝ) : ⟪(r : 𝕜) • x, y⟫ = r • ⟪x, y⟫ | by { rw [inner_smul_left, conj_of_real, algebra.smul_def], refl } | lemma | inner_smul_real_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"
] | [
"algebra.smul_def",
"inner_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ | by rw [←inner_conj_symm, inner_smul_left, ring_hom.map_mul, conj_conj, inner_conj_symm] | lemma | inner_smul_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_conj_symm",
"inner_smul_left",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_smul_right (x y : F) (r : ℝ) : ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ | inner_smul_right _ _ _ | lemma | real_inner_smul_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_smul_real_right (x y : E) (r : ℝ) : ⟪x, (r : 𝕜) • y⟫ = r • ⟪x, y⟫ | by { rw [inner_smul_right, algebra.smul_def], refl } | lemma | inner_smul_real_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"
] | [
"algebra.smul_def",
"inner_smul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sesq_form_of_inner : E →ₗ[𝕜] E →ₗ⋆[𝕜] 𝕜 | linear_map.mk₂'ₛₗ (ring_hom.id 𝕜) (star_ring_end _)
(λ x y, ⟪y, x⟫)
(λ x y z, inner_add_right _ _ _)
(λ r x y, inner_smul_right _ _ _)
(λ x y z, inner_add_left _ _ _)
(λ r x y, inner_smul_left _ _ _) | def | sesq_form_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"
] | [
"inner_add_left",
"inner_add_right",
"inner_smul_left",
"inner_smul_right",
"linear_map.mk₂'ₛₗ",
"ring_hom.id",
"star_ring_end"
] | The inner product as a sesquilinear form.
Note that in the case `𝕜 = ℝ` this is a bilinear form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bilin_form_of_real_inner : bilin_form ℝ F | { bilin := inner,
bilin_add_left := inner_add_left,
bilin_smul_left := λ a x y, inner_smul_left _ _ _,
bilin_add_right := inner_add_right,
bilin_smul_right := λ a x y, inner_smul_right _ _ _ } | def | bilin_form_of_real_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"
] | [
"bilin_form",
"inner_add_left",
"inner_add_right",
"inner_smul_left",
"inner_smul_right"
] | The real inner product as a bilinear form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_inner {ι : Type*} (s : finset ι) (f : ι → E) (x : E) :
⟪∑ i in s, f i, x⟫ = ∑ i in s, ⟪f i, x⟫ | (sesq_form_of_inner x).map_sum | lemma | sum_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"
] | [
"finset",
"sesq_form_of_inner"
] | An inner product with a sum on the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_sum {ι : Type*} (s : finset ι) (f : ι → E) (x : E) :
⟪x, ∑ i in s, f i⟫ = ∑ i in s, ⟪x, f i⟫ | (linear_map.flip sesq_form_of_inner x).map_sum | lemma | 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",
"linear_map.flip",
"sesq_form_of_inner"
] | An inner product with a sum on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finsupp.sum_inner {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
⟪l.sum (λ (i : ι) (a : 𝕜), a • v i), x⟫
= l.sum (λ (i : ι) (a : 𝕜), (conj a) • ⟪v i, x⟫) | by { convert sum_inner l.support (λ a, l a • v a) x,
simp only [inner_smul_left, finsupp.sum, smul_eq_mul] } | lemma | finsupp.sum_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"
] | [
"inner_smul_left",
"smul_eq_mul",
"sum_inner"
] | An inner product with a sum on the left, `finsupp` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finsupp.inner_sum {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
⟪x, l.sum (λ (i : ι) (a : 𝕜), a • v i)⟫ = l.sum (λ (i : ι) (a : 𝕜), a • ⟪x, v i⟫) | by { convert inner_sum l.support (λ a, l a • v a) x,
simp only [inner_smul_right, finsupp.sum, smul_eq_mul] } | lemma | finsupp.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"
] | [
"inner_smul_right",
"inner_sum",
"smul_eq_mul"
] | An inner product with a sum on the right, `finsupp` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dfinsupp.sum_inner {ι : Type*} [dec : decidable_eq ι] {α : ι → Type*}
[Π i, add_zero_class (α i)] [Π i (x : α i), decidable (x ≠ 0)]
(f : Π i, α i → E) (l : Π₀ i, α i) (x : E) :
⟪l.sum f, x⟫ = l.sum (λ i a, ⟪f i a, x⟫) | by simp only [dfinsupp.sum, sum_inner, smul_eq_mul] {contextual := tt} | lemma | dfinsupp.sum_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"
] | [
"add_zero_class",
"smul_eq_mul",
"sum_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dfinsupp.inner_sum {ι : Type*} [dec : decidable_eq ι] {α : ι → Type*}
[Π i, add_zero_class (α i)] [Π i (x : α i), decidable (x ≠ 0)]
(f : Π i, α i → E) (l : Π₀ i, α i) (x : E) :
⟪x, l.sum f⟫ = l.sum (λ i a, ⟪x, f i a⟫) | by simp only [dfinsupp.sum, inner_sum, smul_eq_mul] {contextual := tt} | lemma | dfinsupp.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"
] | [
"add_zero_class",
"inner_sum",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_zero_left (x : E) : ⟪0, x⟫ = 0 | by rw [← zero_smul 𝕜 (0:E), inner_smul_left, ring_hom.map_zero, zero_mul] | lemma | inner_zero_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"
] | [
"inner_smul_left",
"ring_hom.map_zero",
"zero_mul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_re_zero_left (x : E) : re ⟪0, x⟫ = 0 | by simp only [inner_zero_left, add_monoid_hom.map_zero] | lemma | inner_re_zero_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"
] | [
"inner_zero_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_zero_right (x : E) : ⟪x, 0⟫ = 0 | by rw [←inner_conj_symm, inner_zero_left, ring_hom.map_zero] | lemma | inner_zero_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_zero_left",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_re_zero_right (x : E) : re ⟪x, 0⟫ = 0 | by simp only [inner_zero_right, add_monoid_hom.map_zero] | lemma | inner_re_zero_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_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_nonneg {x : E} : 0 ≤ re ⟪x, x⟫ | inner_product_space.to_core.nonneg_re x | lemma | inner_self_nonneg | 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 | |
real_inner_self_nonneg {x : F} : 0 ≤ ⟪x, x⟫_ℝ | @inner_self_nonneg ℝ F _ _ _ x | lemma | real_inner_self_nonneg | 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_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_re_to_K (x : E) : (re ⟪x, x⟫ : 𝕜) = ⟪x, x⟫ | ((is_R_or_C.is_real_tfae (⟪x, x⟫ : 𝕜)).out 2 3).2 (inner_self_im _) | lemma | inner_self_re_to_K | 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_im",
"is_R_or_C.is_real_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_eq_norm_sq_to_K (x : E) : ⟪x, x⟫ = (‖x‖ ^ 2 : 𝕜) | by rw [← inner_self_re_to_K, ← norm_sq_eq_inner, of_real_pow] | lemma | inner_self_eq_norm_sq_to_K | 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_re_to_K"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_re_eq_norm (x : E) : re ⟪x, x⟫ = ‖⟪x, x⟫‖ | begin
conv_rhs { rw [←inner_self_re_to_K] },
symmetry,
exact norm_of_nonneg inner_self_nonneg,
end | lemma | inner_self_re_eq_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"
] | [
"inner_self_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_norm_to_K (x : E) : (‖⟪x, x⟫‖ : 𝕜) = ⟪x, x⟫ | by { rw [←inner_self_re_eq_norm], exact inner_self_re_to_K _ } | lemma | inner_self_norm_to_K | 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_re_to_K"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_self_abs (x : F) : |⟪x, x⟫_ℝ| = ⟪x, x⟫_ℝ | @inner_self_norm_to_K ℝ F _ _ _ x | lemma | real_inner_self_abs | 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_norm_to_K"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_eq_zero {x : E} : ⟪x, x⟫ = 0 ↔ x = 0 | by rw [inner_self_eq_norm_sq_to_K, sq_eq_zero_iff, of_real_eq_zero, norm_eq_zero] | lemma | inner_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_self_eq_norm_sq_to_K",
"norm_eq_zero",
"sq_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_ne_zero {x : E} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 | inner_self_eq_zero.not | lemma | inner_self_ne_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_nonpos {x : E} : re ⟪x, x⟫ ≤ 0 ↔ x = 0 | by rw [← norm_sq_eq_inner, (sq_nonneg _).le_iff_eq, sq_eq_zero_iff, norm_eq_zero] | lemma | inner_self_nonpos | 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_zero",
"sq_eq_zero_iff",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_self_nonpos {x : F} : ⟪x, x⟫_ℝ ≤ 0 ↔ x = 0 | @inner_self_nonpos ℝ F _ _ _ x | lemma | real_inner_self_nonpos | 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_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inner_symm (x y : E) : ‖⟪x, y⟫‖ = ‖⟪y, x⟫‖ | by rw [←inner_conj_symm, norm_conj] | lemma | norm_inner_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_neg_left (x y : E) : ⟪-x, y⟫ = -⟪x, y⟫ | by { rw [← neg_one_smul 𝕜 x, inner_smul_left], simp } | lemma | inner_neg_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"
] | [
"inner_smul_left",
"neg_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_neg_right (x y : E) : ⟪x, -y⟫ = -⟪x, y⟫ | by rw [←inner_conj_symm, inner_neg_left]; simp only [ring_hom.map_neg, inner_conj_symm] | lemma | inner_neg_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_conj_symm",
"inner_neg_left",
"ring_hom.map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_neg_neg (x y : E) : ⟪-x, -y⟫ = ⟪x, y⟫ | by simp | lemma | inner_neg_neg | 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 | |
inner_self_conj (x : E) : ⟪x, x⟫† = ⟪x, x⟫ | by rw [is_R_or_C.ext_iff]; exact ⟨by rw [conj_re], by rw [conj_im, inner_self_im, neg_zero]⟩ | lemma | inner_self_conj | 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_im",
"is_R_or_C.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_sub_left (x y z : E) : ⟪x - y, z⟫ = ⟪x, z⟫ - ⟪y, z⟫ | by { simp [sub_eq_add_neg, inner_add_left] } | lemma | inner_sub_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"
] | [
"inner_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_sub_right (x y z : E) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ | by { simp [sub_eq_add_neg, inner_add_right] } | lemma | inner_sub_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_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_mul_symm_re_eq_norm (x y : E) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ | by { rw [←inner_conj_symm, mul_comm], exact re_eq_norm_of_mul_conj (inner y x), } | lemma | inner_mul_symm_re_eq_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"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_add_add_self (x y : E) : ⟪x + y, x + y⟫ = ⟪x, x⟫ + ⟪x, y⟫ + ⟪y, x⟫ + ⟪y, y⟫ | by simp only [inner_add_left, inner_add_right]; ring | lemma | inner_add_add_self | 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",
"ring"
] | Expand `⟪x + y, x + y⟫` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_add_add_self (x y : F) : ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ | begin
have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [←inner_conj_symm]; refl,
simp only [inner_add_add_self, this, add_left_inj],
ring,
end | lemma | real_inner_add_add_self | 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_add_self",
"ring"
] | Expand `⟪x + y, x + y⟫_ℝ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_sub_sub_self (x y : E) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ | by simp only [inner_sub_left, inner_sub_right]; ring | lemma | inner_sub_sub_self | 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",
"inner_sub_right",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫_ℝ = ⟪x, x⟫_ℝ - 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ | begin
have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [←inner_conj_symm]; refl,
simp only [inner_sub_sub_self, this, add_left_inj],
ring,
end | lemma | real_inner_sub_sub_self | 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_sub_self",
"ring"
] | Expand `⟪x - y, x - y⟫_ℝ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_inner_left {x y : E} (h : ∀ v, ⟪v, x⟫ = ⟪v, y⟫) : x = y | by rw [←sub_eq_zero, ←@inner_self_eq_zero 𝕜, inner_sub_right, sub_eq_zero, h (x - y)] | lemma | ext_inner_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"
] | [
"inner_self_eq_zero",
"inner_sub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_inner_right {x y : E} (h : ∀ v, ⟪x, v⟫ = ⟪y, v⟫) : x = y | by rw [←sub_eq_zero, ←@inner_self_eq_zero 𝕜, inner_sub_left, sub_eq_zero, h (x - y)] | lemma | ext_inner_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_self_eq_zero",
"inner_sub_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallelogram_law {x y : E} :
⟪x + y, x + y⟫ + ⟪x - y, x - y⟫ = 2 * (⟪x, x⟫ + ⟪y, y⟫) | by simp [inner_add_add_self, inner_sub_sub_self, two_mul, sub_eq_add_neg, add_comm, add_left_comm] | lemma | parallelogram_law | 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_add_self",
"inner_sub_sub_self",
"two_mul"
] | Parallelogram law | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_mul_inner_self_le (x y : E) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ | begin
letI c : inner_product_space.core 𝕜 E := inner_product_space.to_core,
exact inner_product_space.core.inner_mul_inner_self_le x y
end | lemma | inner_mul_inner_self_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"
] | [
"inner_product_space.core",
"inner_product_space.core.inner_mul_inner_self_le",
"inner_product_space.to_core"
] | **Cauchy–Schwarz inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_mul_inner_self_le (x y : F) : ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ | calc ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ‖⟪x, y⟫_ℝ‖ * ‖⟪y, x⟫_ℝ‖ :
by { rw [real_inner_comm y, ← norm_mul], exact le_abs_self _ }
... ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ : @inner_mul_inner_self_le ℝ _ _ _ _ x y | lemma | real_inner_mul_inner_self_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"
] | [
"inner_mul_inner_self_le",
"le_abs_self",
"norm_mul",
"real_inner_comm"
] | Cauchy–Schwarz inequality for real inner products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_independent_of_ne_zero_of_inner_eq_zero {ι : Type*} {v : ι → E}
(hz : ∀ i, v i ≠ 0) (ho : ∀ i j, i ≠ j → ⟪v i, v j⟫ = 0) : linear_independent 𝕜 v | begin
rw linear_independent_iff',
intros s g hg i hi,
have h' : g i * inner (v i) (v i) = inner (v i) (∑ j in s, g j • v j),
{ rw inner_sum,
symmetry,
convert finset.sum_eq_single i _ _,
{ rw inner_smul_right },
{ intros j hj hji,
rw [inner_smul_right, ho i j hji.symm, mul_zero] },
{ e... | lemma | linear_independent_of_ne_zero_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"
] | [
"inner_smul_right",
"inner_sum",
"linear_independent",
"linear_independent_iff'",
"mul_zero"
] | A family of vectors is linearly independent if they are nonzero
and orthogonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal (v : ι → E) : Prop | (∀ i, ‖v i‖ = 1) ∧ (∀ {i j}, i ≠ j → ⟪v i, v j⟫ = 0) | def | 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"
] | [] | An orthonormal set of vectors in an `inner_product_space` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_iff_ite {v : ι → E} :
orthonormal 𝕜 v ↔ ∀ i j, ⟪v i, v j⟫ = if i = j then (1:𝕜) else (0:𝕜) | begin
split,
{ intros hv i j,
split_ifs,
{ simp [h, inner_self_eq_norm_sq_to_K, hv.1] },
{ exact hv.2 h } },
{ intros h,
split,
{ intros i,
have h' : ‖v i‖ ^ 2 = 1 ^ 2 := by simp [@norm_sq_eq_inner 𝕜, h i i],
have h₁ : 0 ≤ ‖v i‖ := norm_nonneg _,
have h₂ : (0:ℝ) ≤ 1 := zero_... | lemma | orthonormal_iff_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"
] | [
"inner_self_eq_norm_sq_to_K",
"orthonormal",
"sq_eq_sq",
"zero_le_one"
] | `if ... then ... else` characterization of an indexed set of vectors being orthonormal. (Inner
product equals Kronecker delta.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_subtype_iff_ite {s : set E} :
orthonormal 𝕜 (coe : s → E) ↔
(∀ v ∈ s, ∀ w ∈ s, ⟪v, w⟫ = if v = w then 1 else 0) | begin
rw orthonormal_iff_ite,
split,
{ intros h v hv w hw,
convert h ⟨v, hv⟩ ⟨w, hw⟩ using 1,
simp },
{ rintros h ⟨v, hv⟩ ⟨w, hw⟩,
convert h v hv w hw using 1,
simp }
end | theorem | orthonormal_subtype_iff_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"
] | [
"orthonormal",
"orthonormal_iff_ite"
] | `if ... then ... else` characterization of a set of vectors being orthonormal. (Inner product
equals Kronecker delta.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_right_finsupp {v : ι → E} (hv : orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
⟪v i, finsupp.total ι E 𝕜 v l⟫ = l i | by classical; simp [finsupp.total_apply, finsupp.inner_sum, orthonormal_iff_ite.mp hv] | lemma | orthonormal.inner_right_finsupp | 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"
] | [
"finsupp.inner_sum",
"finsupp.total",
"finsupp.total_apply",
"orthonormal"
] | The inner product of a linear combination of a set of orthonormal vectors with one of those
vectors picks out the coefficient of that vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_right_sum
{v : ι → E} (hv : orthonormal 𝕜 v) (l : ι → 𝕜) {s : finset ι} {i : ι} (hi : i ∈ s) :
⟪v i, ∑ i in s, (l i) • (v i)⟫ = l i | by classical; simp [inner_sum, inner_smul_right, orthonormal_iff_ite.mp hv, hi] | lemma | orthonormal.inner_right_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_smul_right",
"inner_sum",
"orthonormal"
] | The inner product of a linear combination of a set of orthonormal vectors with one of those
vectors picks out the coefficient of that vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_right_fintype [fintype ι]
{v : ι → E} (hv : orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι) :
⟪v i, ∑ i : ι, (l i) • (v i)⟫ = l i | hv.inner_right_sum l (finset.mem_univ _) | lemma | orthonormal.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",
"fintype",
"orthonormal"
] | The inner product of a linear combination of a set of orthonormal vectors with one of those
vectors picks out the coefficient of that vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_left_finsupp {v : ι → E} (hv : orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
⟪finsupp.total ι E 𝕜 v l, v i⟫ = conj (l i) | by rw [← inner_conj_symm, hv.inner_right_finsupp] | lemma | orthonormal.inner_left_finsupp | 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_conj_symm",
"orthonormal"
] | The inner product of a linear combination of a set of orthonormal vectors with one of those
vectors picks out the coefficient of that vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_left_sum
{v : ι → E} (hv : orthonormal 𝕜 v) (l : ι → 𝕜) {s : finset ι} {i : ι} (hi : i ∈ s) :
⟪∑ i in s, (l i) • (v i), v i⟫ = conj (l i) | by classical; simp only
[sum_inner, inner_smul_left, orthonormal_iff_ite.mp hv, hi, mul_boole, finset.sum_ite_eq', if_true] | lemma | orthonormal.inner_left_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_smul_left",
"mul_boole",
"orthonormal",
"sum_inner"
] | The inner product of a linear combination of a set of orthonormal vectors with one of those
vectors picks out the coefficient of that vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_left_fintype [fintype ι]
{v : ι → E} (hv : orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι) :
⟪∑ i : ι, (l i) • (v i), v i⟫ = conj (l i) | hv.inner_left_sum l (finset.mem_univ _) | lemma | orthonormal.inner_left_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",
"fintype",
"orthonormal"
] | The inner product of a linear combination of a set of orthonormal vectors with one of those
vectors picks out the coefficient of that vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_finsupp_eq_sum_left
{v : ι → E} (hv : orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
⟪finsupp.total ι E 𝕜 v l₁, finsupp.total ι E 𝕜 v l₂⟫ = l₁.sum (λ i y, conj y * l₂ i) | by simp only [l₁.total_apply _, finsupp.sum_inner, hv.inner_right_finsupp, smul_eq_mul] | lemma | orthonormal.inner_finsupp_eq_sum_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"
] | [
"finsupp.sum_inner",
"finsupp.total",
"orthonormal",
"smul_eq_mul"
] | The inner product of two linear combinations of a set of orthonormal vectors, expressed as
a sum over the first `finsupp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_finsupp_eq_sum_right
{v : ι → E} (hv : orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
⟪finsupp.total ι E 𝕜 v l₁, finsupp.total ι E 𝕜 v l₂⟫ = l₂.sum (λ i y, conj (l₁ i) * y) | by simp only [l₂.total_apply _, finsupp.inner_sum, hv.inner_left_finsupp, mul_comm, smul_eq_mul] | lemma | orthonormal.inner_finsupp_eq_sum_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"
] | [
"finsupp.inner_sum",
"finsupp.total",
"mul_comm",
"orthonormal",
"smul_eq_mul"
] | The inner product of two linear combinations of a set of orthonormal vectors, expressed as
a sum over the second `finsupp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_sum
{v : ι → E} (hv : orthonormal 𝕜 v) (l₁ l₂ : ι → 𝕜) (s : finset ι) :
⟪∑ i in s, l₁ i • v i, ∑ i in s, l₂ i • v i⟫ = ∑ i in s, conj (l₁ i) * l₂ i | begin
simp_rw [sum_inner, inner_smul_left],
refine finset.sum_congr rfl (λ i hi, _),
rw hv.inner_right_sum l₂ hi
end | lemma | orthonormal.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",
"inner_smul_left",
"orthonormal",
"sum_inner"
] | The inner product of two linear combinations of a set of orthonormal vectors, expressed as
a sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_left_right_finset {s : finset ι} {v : ι → E} (hv : orthonormal 𝕜 v)
{a : ι → ι → 𝕜} : ∑ i in s, ∑ j in s, (a i j) • ⟪v j, v i⟫ = ∑ k in s, a k k | by classical; simp [orthonormal_iff_ite.mp hv, finset.sum_ite_of_true] | lemma | orthonormal.inner_left_right_finset | 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",
"orthonormal"
] | The double sum of weighted inner products of pairs of vectors from an orthonormal sequence is the
sum of the weights. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.linear_independent {v : ι → E} (hv : orthonormal 𝕜 v) :
linear_independent 𝕜 v | begin
rw linear_independent_iff,
intros l hl,
ext i,
have key : ⟪v i, finsupp.total ι E 𝕜 v l⟫ = ⟪v i, 0⟫ := by rw hl,
simpa only [hv.inner_right_finsupp, inner_zero_right] using key
end | lemma | orthonormal.linear_independent | 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"
] | [
"finsupp.total",
"inner_zero_right",
"linear_independent",
"linear_independent_iff",
"orthonormal"
] | An orthonormal set is linearly independent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.comp
{ι' : Type*} {v : ι → E} (hv : orthonormal 𝕜 v) (f : ι' → ι) (hf : function.injective f) :
orthonormal 𝕜 (v ∘ f) | begin
classical,
rw orthonormal_iff_ite at ⊢ hv,
intros i j,
convert hv (f i) (f j) using 1,
simp [hf.eq_iff]
end | lemma | orthonormal.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"
] | [
"orthonormal",
"orthonormal_iff_ite"
] | A subfamily of an orthonormal family (i.e., a composition with an injective map) is an
orthonormal family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_subtype_range {v : ι → E} (hv : function.injective v) :
orthonormal 𝕜 (coe : set.range v → E) ↔ orthonormal 𝕜 v | begin
let f : ι ≃ set.range v := equiv.of_injective v hv,
refine ⟨λ h, h.comp f f.injective, λ h, _⟩,
rw ← equiv.self_comp_of_injective_symm hv,
exact h.comp f.symm f.symm.injective,
end | lemma | orthonormal_subtype_range | 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"
] | [
"equiv.of_injective",
"equiv.self_comp_of_injective_symm",
"orthonormal",
"set.range"
] | An injective family `v : ι → E` is orthonormal if and only if `coe : (range v) → E` is
orthonormal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.to_subtype_range {v : ι → E} (hv : orthonormal 𝕜 v) :
orthonormal 𝕜 (coe : set.range v → E) | (orthonormal_subtype_range hv.linear_independent.injective).2 hv | lemma | orthonormal.to_subtype_range | 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",
"orthonormal_subtype_range",
"set.range"
] | If `v : ι → E` is an orthonormal family, then `coe : (range v) → E` is an orthonormal
family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.inner_finsupp_eq_zero
{v : ι → E} (hv : orthonormal 𝕜 v) {s : set ι} {i : ι} (hi : i ∉ s) {l : ι →₀ 𝕜}
(hl : l ∈ finsupp.supported 𝕜 𝕜 s) :
⟪finsupp.total ι E 𝕜 v l, v i⟫ = 0 | begin
rw finsupp.mem_supported' at hl,
simp only [hv.inner_left_finsupp, hl i hi, map_zero],
end | lemma | orthonormal.inner_finsupp_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"
] | [
"finsupp.mem_supported'",
"finsupp.supported",
"orthonormal"
] | A linear combination of some subset of an orthonormal set is orthogonal to other members of the
set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.orthonormal_of_forall_eq_or_eq_neg {v w : ι → E} (hv : orthonormal 𝕜 v)
(hw : ∀ i, w i = v i ∨ w i = -(v i)) : orthonormal 𝕜 w | begin
classical,
rw orthonormal_iff_ite at *,
intros i j,
cases hw i with hi hi; cases hw j with hj hj; split_ifs with h;
simpa only [hi, hj, h, inner_neg_right, inner_neg_left,
neg_neg, eq_self_iff_true, neg_eq_zero] using hv i j
end | lemma | orthonormal.orthonormal_of_forall_eq_or_eq_neg | 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_left",
"inner_neg_right",
"orthonormal",
"orthonormal_iff_ite"
] | Given an orthonormal family, a second family of vectors is orthonormal if every vector equals
the corresponding vector in the original family or its negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_empty : orthonormal 𝕜 (λ x, x : (∅ : set E) → E) | by classical; simp [orthonormal_subtype_iff_ite] | lemma | orthonormal_empty | 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",
"orthonormal_subtype_iff_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal_Union_of_directed
{η : Type*} {s : η → set E} (hs : directed (⊆) s) (h : ∀ i, orthonormal 𝕜 (λ x, x : s i → E)) :
orthonormal 𝕜 (λ x, x : (⋃ i, s i) → E) | begin
classical,
rw orthonormal_subtype_iff_ite,
rintros x ⟨_, ⟨i, rfl⟩, hxi⟩ y ⟨_, ⟨j, rfl⟩, hyj⟩,
obtain ⟨k, hik, hjk⟩ := hs i j,
have h_orth : orthonormal 𝕜 (λ x, x : (s k) → E) := h k,
rw orthonormal_subtype_iff_ite at h_orth,
exact h_orth x (hik hxi) y (hjk hyj)
end | lemma | orthonormal_Union_of_directed | 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"
] | [
"directed",
"orthonormal",
"orthonormal_subtype_iff_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal_sUnion_of_directed
{s : set (set E)} (hs : directed_on (⊆) s)
(h : ∀ a ∈ s, orthonormal 𝕜 (λ x, x : (a : set E) → E)) :
orthonormal 𝕜 (λ x, x : (⋃₀ s) → E) | by rw set.sUnion_eq_Union; exact orthonormal_Union_of_directed hs.directed_coe (by simpa using h) | lemma | orthonormal_sUnion_of_directed | 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"
] | [
"directed_on",
"orthonormal",
"orthonormal_Union_of_directed",
"set.sUnion_eq_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_maximal_orthonormal {s : set E} (hs : orthonormal 𝕜 (coe : s → E)) :
∃ w ⊇ s, orthonormal 𝕜 (coe : w → E) ∧ ∀ u ⊇ w, orthonormal 𝕜 (coe : u → E) → u = w | begin
obtain ⟨b, bi, sb, h⟩ := zorn_subset_nonempty {b | orthonormal 𝕜 (coe : b → E)} _ _ hs,
{ refine ⟨b, sb, bi, _⟩,
exact λ u hus hu, h u hu hus },
{ refine λ c hc cc c0, ⟨⋃₀ c, _, _⟩,
{ exact orthonormal_sUnion_of_directed cc.directed_on (λ x xc, hc xc) },
{ exact λ _, set.subset_sUnion_of_mem } ... | lemma | exists_maximal_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"
] | [
"orthonormal",
"orthonormal_sUnion_of_directed",
"set.subset_sUnion_of_mem",
"zorn_subset_nonempty"
] | Given an orthonormal set `v` of vectors in `E`, there exists a maximal orthonormal set
containing it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.ne_zero {v : ι → E} (hv : orthonormal 𝕜 v) (i : ι) : v i ≠ 0 | begin
have : ‖v i‖ ≠ 0,
{ rw hv.1 i,
norm_num },
simpa using this
end | lemma | orthonormal.ne_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"
] | [
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_of_orthonormal_of_card_eq_finrank [fintype ι] [nonempty ι] {v : ι → E}
(hv : orthonormal 𝕜 v) (card_eq : fintype.card ι = finrank 𝕜 E) :
basis ι 𝕜 E | basis_of_linear_independent_of_card_eq_finrank hv.linear_independent card_eq | def | basis_of_orthonormal_of_card_eq_finrank | 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_of_linear_independent_of_card_eq_finrank",
"fintype",
"fintype.card",
"orthonormal"
] | A family of orthonormal vectors with the correct cardinality forms a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_basis_of_orthonormal_of_card_eq_finrank [fintype ι] [nonempty ι] {v : ι → E}
(hv : orthonormal 𝕜 v) (card_eq : fintype.card ι = finrank 𝕜 E) :
(basis_of_orthonormal_of_card_eq_finrank hv card_eq : ι → E) = v | coe_basis_of_linear_independent_of_card_eq_finrank _ _ | lemma | coe_basis_of_orthonormal_of_card_eq_finrank | 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_of_orthonormal_of_card_eq_finrank",
"coe_basis_of_linear_independent_of_card_eq_finrank",
"fintype",
"fintype.card",
"orthonormal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_sqrt_inner (x : E) : ‖x‖ = sqrt (re ⟪x, x⟫) | calc ‖x‖ = sqrt (‖x‖ ^ 2) : (sqrt_sq (norm_nonneg _)).symm
... = sqrt (re ⟪x, x⟫) : congr_arg _ (norm_sq_eq_inner _) | lemma | norm_eq_sqrt_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 | |
norm_eq_sqrt_real_inner (x : F) : ‖x‖ = sqrt ⟪x, x⟫_ℝ | @norm_eq_sqrt_inner ℝ _ _ _ _ x | lemma | norm_eq_sqrt_real_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_eq_norm_mul_norm (x : E) : re ⟪x, x⟫ = ‖x‖ * ‖x‖ | by rw [@norm_eq_sqrt_inner 𝕜, ←sqrt_mul inner_self_nonneg (re ⟪x, x⟫),
sqrt_mul_self inner_self_nonneg] | lemma | inner_self_eq_norm_mul_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"
] | [
"inner_self_nonneg",
"norm_eq_sqrt_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self_eq_norm_sq (x : E) : re ⟪x, x⟫ = ‖x‖^2 | by rw [pow_two, inner_self_eq_norm_mul_norm] | lemma | inner_self_eq_norm_sq | 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_mul_norm",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_self_eq_norm_mul_norm (x : F) : ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖ | by { have h := @inner_self_eq_norm_mul_norm ℝ F _ _ _ x, simpa using h } | lemma | real_inner_self_eq_norm_mul_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"
] | [
"inner_self_eq_norm_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_self_eq_norm_sq (x : F) : ⟪x, x⟫_ℝ = ‖x‖^2 | by rw [pow_two, real_inner_self_eq_norm_mul_norm] | lemma | real_inner_self_eq_norm_sq | 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"
] | [
"pow_two",
"real_inner_self_eq_norm_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_add_sq (x y : E) : ‖x + y‖^2 = ‖x‖^2 + 2 * (re ⟪x, y⟫) + ‖y‖^2 | begin
repeat {rw [sq, ←@inner_self_eq_norm_mul_norm 𝕜]},
rw [inner_add_add_self, two_mul],
simp only [add_assoc, add_left_inj, add_right_inj, add_monoid_hom.map_add],
rw [←inner_conj_symm, conj_re],
end | lemma | norm_add_sq | 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_add_self",
"inner_self_eq_norm_mul_norm",
"two_mul"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_sq_real (x y : F) : ‖x + y‖^2 = ‖x‖^2 + 2 * ⟪x, y⟫_ℝ + ‖y‖^2 | by { have h := @norm_add_sq ℝ _ _ _ _ x y, simpa using h } | lemma | norm_add_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"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_mul_self (x y : E) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * (re ⟪x, y⟫) + ‖y‖ * ‖y‖ | by { repeat {rw [← sq]}, exact norm_add_sq _ _ } | lemma | norm_add_mul_self | 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"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add_mul_self_real (x y : F) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ | by { have h := @norm_add_mul_self ℝ _ _ _ _ x y, simpa using h } | lemma | norm_add_mul_self_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_mul_self"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_sq (x y : E) : ‖x - y‖^2 = ‖x‖^2 - 2 * (re ⟪x, y⟫) + ‖y‖^2 | by rw [sub_eq_add_neg, @norm_add_sq 𝕜 _ _ _ _ x (-y), norm_neg, inner_neg_right, map_neg, mul_neg,
sub_eq_add_neg] | lemma | norm_sub_sq | 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",
"mul_neg",
"norm_add_sq"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_sq_real (x y : F) : ‖x - y‖^2 = ‖x‖^2 - 2 * ⟪x, y⟫_ℝ + ‖y‖^2 | @norm_sub_sq ℝ _ _ _ _ _ _ | lemma | norm_sub_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"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_mul_self (x y : E) : ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ | by { repeat {rw [← sq]}, exact norm_sub_sq _ _ } | lemma | norm_sub_mul_self | 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"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_mul_self_real (x y : F) : ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ | by { have h := @norm_sub_mul_self ℝ _ _ _ _ x y, simpa using h } | lemma | norm_sub_mul_self_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_mul_self"
] | Expand the square | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_inner_le_norm (x y : E) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ | begin
rw [norm_eq_sqrt_inner x, norm_eq_sqrt_inner y],
letI : inner_product_space.core 𝕜 E := inner_product_space.to_core,
exact inner_product_space.core.norm_inner_le_norm x y
end | lemma | norm_inner_le_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"
] | [
"inner_product_space.core",
"inner_product_space.core.norm_inner_le_norm",
"inner_product_space.to_core",
"norm_eq_sqrt_inner"
] | Cauchy–Schwarz inequality with norm | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_inner_le_nnnorm (x y : E) : ‖⟪x, y⟫‖₊ ≤ ‖x‖₊ * ‖y‖₊ | norm_inner_le_norm x y | lemma | nnnorm_inner_le_nnnorm | 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_le_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_inner_le_norm (x y : E) : re ⟪x, y⟫ ≤ ‖x‖ * ‖y‖ | le_trans (re_le_norm (inner x y)) (norm_inner_le_norm x y) | lemma | re_inner_le_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"
] | [
"norm_inner_le_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_real_inner_le_norm (x y : F) : |⟪x, y⟫_ℝ| ≤ ‖x‖ * ‖y‖ | (real.norm_eq_abs _).ge.trans (norm_inner_le_norm x y) | lemma | abs_real_inner_le_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"
] | [
"norm_inner_le_norm",
"real.norm_eq_abs"
] | Cauchy–Schwarz inequality with norm | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_le_norm (x y : F) : ⟪x, y⟫_ℝ ≤ ‖x‖ * ‖y‖ | le_trans (le_abs_self _) (abs_real_inner_le_norm _ _) | lemma | real_inner_le_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_real_inner_le_norm",
"le_abs_self"
] | Cauchy–Schwarz inequality with norm | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parallelogram_law_with_norm (x y : E) :
‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) | begin
simp only [← @inner_self_eq_norm_mul_norm 𝕜],
rw [← re.map_add, parallelogram_law, two_mul, two_mul],
simp only [re.map_add],
end | lemma | parallelogram_law_with_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"
] | [
"inner_self_eq_norm_mul_norm",
"parallelogram_law",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallelogram_law_with_nnnorm (x y : E) :
‖x + y‖₊ * ‖x + y‖₊ + ‖x - y‖₊ * ‖x - y‖₊ = 2 * (‖x‖₊ * ‖x‖₊ + ‖y‖₊ * ‖y‖₊) | subtype.ext $ parallelogram_law_with_norm 𝕜 x y | lemma | parallelogram_law_with_nnnorm | 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",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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