statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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complex.orthonormal_basis_one_I_repr_symm_apply (x : euclidean_space ℝ (fin 2)) :
complex.orthonormal_basis_one_I.repr.symm x = (x 0) + (x 1) * I | rfl | lemma | complex.orthonormal_basis_one_I_repr_symm_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.to_basis_orthonormal_basis_one_I :
complex.orthonormal_basis_one_I.to_basis = complex.basis_one_I | basis.to_basis_to_orthonormal_basis _ _ | lemma | complex.to_basis_orthonormal_basis_one_I | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"basis.to_basis_to_orthonormal_basis",
"complex.basis_one_I"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.coe_orthonormal_basis_one_I :
(complex.orthonormal_basis_one_I : (fin 2) → ℂ) = ![1, I] | by simp [complex.orthonormal_basis_one_I] | lemma | complex.coe_orthonormal_basis_one_I | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complex.orthonormal_basis_one_I"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.isometry_of_orthonormal (v : orthonormal_basis (fin 2) ℝ F) : ℂ ≃ₗᵢ[ℝ] F | complex.orthonormal_basis_one_I.repr.trans v.repr.symm | def | complex.isometry_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthonormal_basis"
] | The isometry between `ℂ` and a two-dimensional real inner product space given by a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.map_isometry_of_orthonormal (v : orthonormal_basis (fin 2) ℝ F)
(f : F ≃ₗᵢ[ℝ] F') :
complex.isometry_of_orthonormal (v.map f) =
(complex.isometry_of_orthonormal v).trans f | by simp [complex.isometry_of_orthonormal, linear_isometry_equiv.trans_assoc, orthonormal_basis.map] | lemma | complex.map_isometry_of_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complex.isometry_of_orthonormal",
"linear_isometry_equiv.trans_assoc",
"orthonormal_basis",
"orthonormal_basis.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.isometry_of_orthonormal_symm_apply
(v : orthonormal_basis (fin 2) ℝ F) (f : F) :
(complex.isometry_of_orthonormal v).symm f
= (v.to_basis.coord 0 f : ℂ) + (v.to_basis.coord 1 f : ℂ) * I | by simp [complex.isometry_of_orthonormal] | lemma | complex.isometry_of_orthonormal_symm_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complex.isometry_of_orthonormal",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.isometry_of_orthonormal_apply
(v : orthonormal_basis (fin 2) ℝ F) (z : ℂ) :
complex.isometry_of_orthonormal v z = z.re • v 0 + z.im • v 1 | by simp [complex.isometry_of_orthonormal, ← v.sum_repr_symm] | lemma | complex.isometry_of_orthonormal_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complex.isometry_of_orthonormal",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal_basis.to_matrix_orthonormal_basis_mem_unitary :
a.to_basis.to_matrix b ∈ matrix.unitary_group ι 𝕜 | begin
rw matrix.mem_unitary_group_iff',
ext i j,
convert a.repr.inner_map_map (b i) (b j),
rw orthonormal_iff_ite.mp b.orthonormal i j,
refl,
end | lemma | orthonormal_basis.to_matrix_orthonormal_basis_mem_unitary | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"matrix.mem_unitary_group_iff'",
"matrix.unitary_group"
] | The change-of-basis matrix between two orthonormal bases `a`, `b` is a unitary matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_basis.det_to_matrix_orthonormal_basis :
‖a.to_basis.det b‖ = 1 | begin
have : (norm_sq (a.to_basis.det b) : 𝕜) = 1,
{ simpa [is_R_or_C.mul_conj]
using (matrix.det_of_mem_unitary (a.to_matrix_orthonormal_basis_mem_unitary b)).2 },
norm_cast at this,
rwa [← sqrt_norm_sq_eq_norm, sqrt_eq_one],
end | lemma | orthonormal_basis.det_to_matrix_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"is_R_or_C.mul_conj",
"matrix.det_of_mem_unitary"
] | The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` has
unit length. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_basis.to_matrix_orthonormal_basis_mem_orthogonal :
a.to_basis.to_matrix b ∈ matrix.orthogonal_group ι ℝ | a.to_matrix_orthonormal_basis_mem_unitary b | lemma | orthonormal_basis.to_matrix_orthonormal_basis_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"matrix.orthogonal_group"
] | The change-of-basis matrix between two orthonormal bases `a`, `b` is an orthogonal matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_basis.det_to_matrix_orthonormal_basis_real :
a.to_basis.det b = 1 ∨ a.to_basis.det b = -1 | begin
rw ← sq_eq_one_iff,
simpa [unitary, sq] using matrix.det_of_mem_unitary (a.to_matrix_orthonormal_basis_mem_unitary b)
end | lemma | orthonormal_basis.det_to_matrix_orthonormal_basis_real | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"matrix.det_of_mem_unitary",
"sq_eq_one_iff",
"unitary"
] | The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` is ±1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.collected_orthonormal_basis
(hV : orthogonal_family 𝕜 (λ i, A i) (λ i, (A i).subtypeₗᵢ))
[decidable_eq ι] (hV_sum : direct_sum.is_internal (λ i, A i)) {α : ι → Type*}
[Π i, fintype (α i)] (v_family : Π i, orthonormal_basis (α i) 𝕜 (A i)) :
orthonormal_basis (Σ i, α i) 𝕜 E | (hV_sum.collected_basis (λ i, (v_family i).to_basis)).to_orthonormal_basis $
by simpa using hV.orthonormal_sigma_orthonormal
(show (∀ i, orthonormal 𝕜 (v_family i).to_basis), by simp) | def | direct_sum.is_internal.collected_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"direct_sum.is_internal",
"fintype",
"orthogonal_family",
"orthonormal",
"orthonormal_basis"
] | Given an internal direct sum decomposition of a module `M`, and an orthonormal basis for each
of the components of the direct sum, the disjoint union of these orthonormal bases is an
orthonormal basis for `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.collected_orthonormal_basis_mem [decidable_eq ι]
(h : direct_sum.is_internal A) {α : ι → Type*}
[Π i, fintype (α i)] (hV : orthogonal_family 𝕜 (λ i, A i) (λ i, (A i).subtypeₗᵢ))
(v : Π i, orthonormal_basis (α i) 𝕜 (A i)) (a : Σ i, α i) :
h.collected_orthonormal_basis hV v a ∈ A a.1 | by simp [direct_sum.is_internal.collected_orthonormal_basis] | lemma | direct_sum.is_internal.collected_orthonormal_basis_mem | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"direct_sum.is_internal",
"direct_sum.is_internal.collected_orthonormal_basis",
"fintype",
"orthogonal_family",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.orthonormal.exists_orthonormal_basis_extension (hv : orthonormal 𝕜 (coe : v → E)) :
∃ {u : finset E} (b : orthonormal_basis u 𝕜 E), v ⊆ u ∧ ⇑b = coe | begin
obtain ⟨u₀, hu₀s, hu₀, hu₀_max⟩ := exists_maximal_orthonormal hv,
rw maximal_orthonormal_iff_orthogonal_complement_eq_bot hu₀ at hu₀_max,
have hu₀_finite : u₀.finite := hu₀.linear_independent.finite,
let u : finset E := hu₀_finite.to_finset,
let fu : ↥u ≃ ↥u₀ := equiv.cast (congr_arg coe_sort hu₀_finite... | lemma | orthonormal.exists_orthonormal_basis_extension | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"equiv.cast",
"exists_maximal_orthonormal",
"finset",
"maximal_orthonormal_iff_orthogonal_complement_eq_bot",
"orthonormal",
"orthonormal_basis"
] | In a finite-dimensional `inner_product_space`, any orthonormal subset can be extended to an
orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.orthonormal.exists_orthonormal_basis_extension_of_card_eq
{ι : Type*} [fintype ι] (card_ι : finrank 𝕜 E = fintype.card ι) {v : ι → E} {s : set ι}
(hv : orthonormal 𝕜 (s.restrict v)) :
∃ b : orthonormal_basis ι 𝕜 E, ∀ i ∈ s, b i = v i | begin
have hsv : injective (s.restrict v) := hv.linear_independent.injective,
have hX : orthonormal 𝕜 (coe : set.range (s.restrict v) → E),
{ rwa orthonormal_subtype_range hsv },
obtain ⟨Y, b₀, hX, hb₀⟩ := hX.exists_orthonormal_basis_extension,
have hιY : fintype.card ι = Y.card,
{ refine (card_ι.symm.tran... | lemma | orthonormal.exists_orthonormal_basis_extension_of_card_eq | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"finite_dimensional.finrank_eq_card_finset_basis",
"fintype",
"fintype.card",
"orthonormal",
"orthonormal_basis",
"orthonormal_subtype_range",
"set.inj_on",
"set.inj_on_iff_injective",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.exists_orthonormal_basis :
∃ (w : finset E) (b : orthonormal_basis w 𝕜 E), ⇑b = (coe : w → E) | let ⟨w, hw, hw', hw''⟩ := (orthonormal_empty 𝕜 E).exists_orthonormal_basis_extension in
⟨w, hw, hw''⟩ | lemma | exists_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"finset",
"orthonormal_basis",
"orthonormal_empty"
] | A finite-dimensional inner product space admits an orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
std_orthonormal_basis : orthonormal_basis (fin (finrank 𝕜 E)) 𝕜 E | begin
let b := classical.some (classical.some_spec $ exists_orthonormal_basis 𝕜 E),
rw [finrank_eq_card_basis b.to_basis],
exact b.reindex (fintype.equiv_fin_of_card_eq rfl),
end | def | std_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"exists_orthonormal_basis",
"fintype.equiv_fin_of_card_eq",
"orthonormal_basis"
] | A finite-dimensional `inner_product_space` has an orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_basis_one_dim (b : orthonormal_basis ι ℝ ℝ) :
⇑b = (λ _, (1 : ℝ)) ∨ ⇑b = (λ _, (-1 : ℝ)) | begin
haveI : unique ι, from b.to_basis.unique,
have : b default = 1 ∨ b default = - 1,
{ have : ‖b default‖ = 1, from b.orthonormal.1 _,
rwa [real.norm_eq_abs, abs_eq (zero_le_one : (0 : ℝ) ≤ 1)] at this },
rw eq_const_of_unique b,
refine this.imp _ _; simp,
end | lemma | orthonormal_basis_one_dim | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"abs_eq",
"eq_const_of_unique",
"orthonormal_basis",
"real.norm_eq_abs",
"unique",
"zero_le_one"
] | An orthonormal basis of `ℝ` is made either of the vector `1`, or of the vector `-1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.sigma_orthonormal_basis_index_equiv
(hV' : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) :
(Σ i, fin (finrank 𝕜 (V i))) ≃ fin n | let b := hV.collected_orthonormal_basis hV' (λ i, (std_orthonormal_basis 𝕜 (V i))) in
fintype.equiv_fin_of_card_eq $ (finite_dimensional.finrank_eq_card_basis b.to_basis).symm.trans hn | def | direct_sum.is_internal.sigma_orthonormal_basis_index_equiv | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"finite_dimensional.finrank_eq_card_basis",
"fintype.equiv_fin_of_card_eq",
"orthogonal_family",
"std_orthonormal_basis"
] | Exhibit a bijection between `fin n` and the index set of a certain basis of an `n`-dimensional
inner product space `E`. This should not be accessed directly, but only via the subsequent API. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.subordinate_orthonormal_basis
(hV' : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) :
orthonormal_basis (fin n) 𝕜 E | ((hV.collected_orthonormal_basis hV' (λ i, (std_orthonormal_basis 𝕜 (V i)))).reindex
(hV.sigma_orthonormal_basis_index_equiv hn hV')) | def | direct_sum.is_internal.subordinate_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthogonal_family",
"orthonormal_basis",
"reindex",
"std_orthonormal_basis"
] | An `n`-dimensional `inner_product_space` equipped with a decomposition as an internal direct
sum has an orthonormal basis indexed by `fin n` and subordinate to that direct sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.subordinate_orthonormal_basis_index
(a : fin n) (hV' : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) : ι | ((hV.sigma_orthonormal_basis_index_equiv hn hV').symm a).1 | def | direct_sum.is_internal.subordinate_orthonormal_basis_index | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"orthogonal_family"
] | An `n`-dimensional `inner_product_space` equipped with a decomposition as an internal direct
sum has an orthonormal basis indexed by `fin n` and subordinate to that direct sum. This function
provides the mapping by which it is subordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.subordinate_orthonormal_basis_subordinate
(a : fin n) (hV' : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) :
(hV.subordinate_orthonormal_basis hn hV' a) ∈
V (hV.subordinate_orthonormal_basis_index hn a hV') | by simpa only [direct_sum.is_internal.subordinate_orthonormal_basis,
orthonormal_basis.coe_reindex, direct_sum.is_internal.subordinate_orthonormal_basis_index]
using hV.collected_orthonormal_basis_mem hV' (λ i, (std_orthonormal_basis 𝕜 (V i)))
((hV.sigma_orthonormal_basis_index_equiv hn hV').symm a) | lemma | direct_sum.is_internal.subordinate_orthonormal_basis_subordinate | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"direct_sum.is_internal.subordinate_orthonormal_basis",
"direct_sum.is_internal.subordinate_orthonormal_basis_index",
"orthogonal_family",
"std_orthonormal_basis"
] | The basis constructed in `orthogonal_family.subordinate_orthonormal_basis` is subordinate to
the `orthogonal_family` in question. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_basis.from_orthogonal_span_singleton
(n : ℕ) [fact (finrank 𝕜 E = n + 1)] {v : E} (hv : v ≠ 0) :
orthonormal_basis (fin n) 𝕜 (𝕜 ∙ v)ᗮ | (std_orthonormal_basis _ _).reindex $ fin_congr $ finrank_orthogonal_span_singleton hv | def | orthonormal_basis.from_orthogonal_span_singleton | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"fact",
"fin_congr",
"finrank_orthogonal_span_singleton",
"orthonormal_basis",
"reindex",
"std_orthonormal_basis"
] | Given a natural number `n` one less than the `finrank` of a finite-dimensional inner product
space, there exists an isometry from the orthogonal complement of a nonzero singleton to
`euclidean_space 𝕜 (fin n)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.extend (L : S →ₗᵢ[𝕜] V): V →ₗᵢ[𝕜] V | begin
-- Build an isometry from Sᗮ to L(S)ᗮ through euclidean_space
let d := finrank 𝕜 Sᗮ,
have dim_S_perp : finrank 𝕜 Sᗮ = d := rfl,
let LS := L.to_linear_map.range,
have E : Sᗮ ≃ₗᵢ[𝕜] LSᗮ,
{ have dim_LS_perp : finrank 𝕜 LSᗮ = d,
calc finrank 𝕜 LSᗮ = finrank 𝕜 V - finrank 𝕜 LS : by simp only
... | def | linear_isometry.extend | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"add_tsub_cancel_left",
"complete_space",
"continuous_linear_map.coe_coe",
"continuous_linear_map.to_linear_map_eq_coe",
"fin_congr",
"finite_dimensional.complete",
"linear_isometry.coe_comp",
"linear_isometry.coe_to_linear_map",
"linear_isometry.norm_map",
"linear_map.add_apply",
"linear_map.co... | Let `S` be a subspace of a finite-dimensional complex inner product space `V`. A linear
isometry mapping `S` into `V` can be extended to a full isometry of `V`.
TODO: The case when `S` is a finite-dimensional subspace of an infinite-dimensional `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.extend_apply (L : S →ₗᵢ[𝕜] V) (s : S):
L.extend s = L s | begin
haveI : complete_space S := finite_dimensional.complete 𝕜 S,
simp only [linear_isometry.extend, continuous_linear_map.to_linear_map_eq_coe,
←linear_isometry.coe_to_linear_map],
simp only [add_right_eq_self, linear_isometry.coe_to_linear_map,
linear_isometry_equiv.coe_to_linear_isometry, linear_isom... | lemma | linear_isometry.extend_apply | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"complete_space",
"continuous_linear_map.coe_coe",
"continuous_linear_map.to_linear_map_eq_coe",
"finite_dimensional.complete",
"linear_isometry.coe_comp",
"linear_isometry.coe_to_linear_map",
"linear_isometry.extend",
"linear_isometry_equiv.coe_to_linear_isometry",
"linear_isometry_equiv.map_eq_zer... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_euclidean_lin : matrix m n 𝕜 ≃ₗ[𝕜] (euclidean_space 𝕜 n →ₗ[𝕜] euclidean_space 𝕜 m) | matrix.to_lin' ≪≫ₗ linear_equiv.arrow_congr
(pi_Lp.linear_equiv _ 𝕜 (λ _ : n, 𝕜)).symm (pi_Lp.linear_equiv _ 𝕜 (λ _ : m, 𝕜)).symm | def | matrix.to_euclidean_lin | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"linear_equiv.arrow_congr",
"matrix",
"matrix.to_lin'",
"pi_Lp.linear_equiv"
] | `matrix.to_lin'` adapted for `euclidean_space 𝕜 _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_euclidean_lin_pi_Lp_equiv_symm (A : matrix m n 𝕜) (x : n → 𝕜) :
A.to_euclidean_lin ((pi_Lp.equiv _ _).symm x) = (pi_Lp.equiv _ _).symm (A.to_lin' x) | rfl | lemma | matrix.to_euclidean_lin_pi_Lp_equiv_symm | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"matrix",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_Lp_equiv_to_euclidean_lin (A : matrix m n 𝕜) (x : euclidean_space 𝕜 n) :
pi_Lp.equiv _ _ (A.to_euclidean_lin x) = A.to_lin' (pi_Lp.equiv _ _ x) | rfl | lemma | matrix.pi_Lp_equiv_to_euclidean_lin | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space",
"matrix",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_euclidean_lin_eq_to_lin :
(to_euclidean_lin : matrix m n 𝕜 ≃ₗ[𝕜] _) =
matrix.to_lin (pi_Lp.basis_fun _ _ _) (pi_Lp.basis_fun _ _ _) | rfl | lemma | matrix.to_euclidean_lin_eq_to_lin | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"matrix",
"matrix.to_lin",
"pi_Lp.basis_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_matrix_row_row [fintype n] (A B : matrix m n 𝕜) (i j : m) :
⟪A i, B j⟫ₑ = (B ⬝ Aᴴ) j i | by simp_rw [euclidean_space.inner_pi_Lp_equiv_symm, matrix.mul_apply', matrix.dot_product_comm,
matrix.conj_transpose_apply, pi.star_def] | lemma | inner_matrix_row_row | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"euclidean_space.inner_pi_Lp_equiv_symm",
"fintype",
"matrix",
"matrix.conj_transpose_apply",
"matrix.dot_product_comm",
"matrix.mul_apply'",
"pi.star_def"
] | The inner product of a row of `A` and a row of `B` is an entry of `B ⬝ Aᴴ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_matrix_col_col [fintype m] (A B : matrix m n 𝕜) (i j : n) :
⟪Aᵀ i, Bᵀ j⟫ₑ = (Aᴴ ⬝ B) i j | rfl | lemma | inner_matrix_col_col | analysis.inner_product_space | src/analysis/inner_product_space/pi_L2.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.pi_Lp",
"linear_algebra.finite_dimensional",
"linear_algebra.unitary_group"
] | [
"fintype",
"matrix"
] | The inner product of a column of `A` and a column of `B` is an entry of `Aᴴ ⬝ B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_positive (T : E →L[𝕜] E) : Prop | is_self_adjoint T ∧ ∀ x, 0 ≤ T.re_apply_inner_self x | def | continuous_linear_map.is_positive | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"is_self_adjoint"
] | A continuous linear endomorphism `T` of a Hilbert space is **positive** if it is self adjoint
and `∀ x, 0 ≤ re ⟪T x, x⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_positive.is_self_adjoint {T : E →L[𝕜] E} (hT : is_positive T) :
is_self_adjoint T | hT.1 | lemma | continuous_linear_map.is_positive.is_self_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.inner_nonneg_left {T : E →L[𝕜] E} (hT : is_positive T) (x : E) :
0 ≤ re ⟪T x, x⟫ | hT.2 x | lemma | continuous_linear_map.is_positive.inner_nonneg_left | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.inner_nonneg_right {T : E →L[𝕜] E} (hT : is_positive T) (x : E) :
0 ≤ re ⟪x, T x⟫ | by rw inner_re_symm; exact hT.inner_nonneg_left x | lemma | continuous_linear_map.is_positive.inner_nonneg_right | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"inner_re_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive_zero : is_positive (0 : E →L[𝕜] E) | begin
refine ⟨is_self_adjoint_zero _, λ x, _⟩,
change 0 ≤ re ⟪_, _⟫,
rw [zero_apply, inner_zero_left, zero_hom_class.map_zero]
end | lemma | continuous_linear_map.is_positive_zero | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"inner_zero_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive_one : is_positive (1 : E →L[𝕜] E) | ⟨is_self_adjoint_one _, λ x, inner_self_nonneg⟩ | lemma | continuous_linear_map.is_positive_one | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.add {T S : E →L[𝕜] E} (hT : T.is_positive)
(hS : S.is_positive) : (T + S).is_positive | begin
refine ⟨hT.is_self_adjoint.add hS.is_self_adjoint, λ x, _⟩,
rw [re_apply_inner_self, add_apply, inner_add_left, map_add],
exact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)
end | lemma | continuous_linear_map.is_positive.add | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"inner_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.conj_adjoint {T : E →L[𝕜] E}
(hT : T.is_positive) (S : E →L[𝕜] F) : (S ∘L T ∘L S†).is_positive | begin
refine ⟨hT.is_self_adjoint.conj_adjoint S, λ x, _⟩,
rw [re_apply_inner_self, comp_apply, ← adjoint_inner_right],
exact hT.inner_nonneg_left _
end | lemma | continuous_linear_map.is_positive.conj_adjoint | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.adjoint_conj {T : E →L[𝕜] E}
(hT : T.is_positive) (S : F →L[𝕜] E) : (S† ∘L T ∘L S).is_positive | begin
convert hT.conj_adjoint (S†),
rw adjoint_adjoint
end | lemma | continuous_linear_map.is_positive.adjoint_conj | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.conj_orthogonal_projection (U : submodule 𝕜 E) {T : E →L[𝕜] E}
(hT : T.is_positive) [complete_space U] :
(U.subtypeL ∘L orthogonal_projection U ∘L T ∘L U.subtypeL ∘L
orthogonal_projection U).is_positive | begin
have := hT.conj_adjoint (U.subtypeL ∘L orthogonal_projection U),
rwa (orthogonal_projection_is_self_adjoint U).adjoint_eq at this
end | lemma | continuous_linear_map.is_positive.conj_orthogonal_projection | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"complete_space",
"orthogonal_projection",
"orthogonal_projection_is_self_adjoint",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive.orthogonal_projection_comp {T : E →L[𝕜] E}
(hT : T.is_positive) (U : submodule 𝕜 E) [complete_space U] :
(orthogonal_projection U ∘L T ∘L U.subtypeL).is_positive | begin
have := hT.conj_adjoint (orthogonal_projection U : E →L[𝕜] U),
rwa [U.adjoint_orthogonal_projection] at this,
end | lemma | continuous_linear_map.is_positive.orthogonal_projection_comp | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"complete_space",
"orthogonal_projection",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_positive_iff_complex (T : E' →L[ℂ] E') :
is_positive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ | begin
simp_rw [is_positive, forall_and_distrib, is_self_adjoint_iff_is_symmetric,
linear_map.is_symmetric_iff_inner_map_self_real, conj_eq_iff_re],
refl
end | lemma | continuous_linear_map.is_positive_iff_complex | analysis.inner_product_space | src/analysis/inner_product_space/positive.lean | [
"analysis.inner_product_space.adjoint"
] | [
"forall_and_distrib",
"linear_map.is_symmetric_iff_inner_map_self_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_norm_eq_infi_of_complete_convex {K : set F} (ne : K.nonempty) (h₁ : is_complete K)
(h₂ : convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ | assume u,
begin
let δ := ⨅ w : K, ‖u - w‖,
letI : nonempty K := ne.to_subtype,
have zero_le_δ : 0 ≤ δ := le_cinfi (λ _, norm_nonneg _),
have δ_le : ∀ w : K, δ ≤ ‖u - w‖,
from cinfi_le ⟨0, set.forall_range_iff.2 $ λ _, norm_nonneg _⟩,
have δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖ := assume w hw, δ_le ⟨w, hw⟩,
-- Ste... | theorem | exists_norm_eq_infi_of_complete_convex | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"add_halves",
"add_smul",
"cauchy_seq",
"cauchy_seq_iff_le_tendsto_0",
"cauchy_seq_tendsto_of_is_complete",
"cinfi_le",
"continuous",
"continuous.comp",
"continuous_const",
"continuous_id",
"convex",
"exists_lt_of_cinfi_lt",
"is_complete",
"le_cinfi",
"le_rfl",
"mul_assoc",
"mul_le_m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_infi_iff_real_inner_le_zero {K : set F} (h : convex ℝ K) {u : F} {v : F}
(hv : v ∈ K) : ‖u - v‖ = (⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 | iff.intro
begin
assume eq w hw,
let δ := ⨅ w : K, ‖u - w‖, let p := ⟪u - v, w - v⟫_ℝ, let q := ‖w - v‖^2,
letI : nonempty K := ⟨⟨v, hv⟩⟩,
have zero_le_δ : 0 ≤ δ,
apply le_cinfi, intro, exact norm_nonneg _,
have δ_le : ∀ w : K, δ ≤ ‖u - w‖,
assume w, apply cinfi_le, use (0:ℝ), rintros _ ⟨_, rfl⟩, exact... | theorem | norm_eq_infi_iff_real_inner_le_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"abs_of_pos",
"by_contradiction",
"cinfi_le",
"convex",
"div_mul_cancel",
"div_pos",
"inner_smul_right",
"le_cinfi",
"mul_le_mul_of_nonneg_right",
"mul_self_le_mul_self",
"nonneg_le_nonneg_of_sq_le_sq",
"nonneg_of_mul_nonneg_right",
"norm_smul",
"norm_sub_sq",
"one_smul",
"ring",
"sm... | Characterization of minimizers for the projection on a convex set in a real inner product
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_norm_eq_infi_of_complete_subspace
(h : is_complete (↑K : set E)) : ∀ u : E, ∃ v ∈ K, ‖u - v‖ = ⨅ w : (K : set E), ‖u - w‖ | begin
letI : inner_product_space ℝ E := inner_product_space.is_R_or_C_to_real 𝕜 E,
letI : module ℝ E := restrict_scalars.module ℝ 𝕜 E,
let K' : submodule ℝ E := submodule.restrict_scalars ℝ K,
exact exists_norm_eq_infi_of_complete_convex ⟨0, K'.zero_mem⟩ h K'.convex
end | theorem | exists_norm_eq_infi_of_complete_subspace | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"exists_norm_eq_infi_of_complete_convex",
"inner_product_space",
"inner_product_space.is_R_or_C_to_real",
"is_complete",
"module",
"submodule",
"submodule.restrict_scalars"
] | Existence of projections on complete subspaces.
Let `u` be a point in an inner product space, and let `K` be a nonempty complete subspace.
Then there exists a (unique) `v` in `K` that minimizes the distance `‖u - v‖` to `u`.
This point `v` is usually called the orthogonal projection of `u` onto `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_infi_iff_real_inner_eq_zero (K : submodule ℝ F) {u : F} {v : F}
(hv : v ∈ K) : ‖u - v‖ = (⨅ w : (↑K : set F), ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0 | iff.intro
begin
assume h,
have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0,
{ rwa [norm_eq_infi_iff_real_inner_le_zero] at h, exacts [K.convex, hv] },
assume w hw,
have le : ⟪u - v, w⟫_ℝ ≤ 0,
let w' := w + v,
have : w' ∈ K := submodule.add_mem _ hw hv,
have h₁ := h w' this,
have h₂ : w' - v = w, simp on... | theorem | norm_eq_infi_iff_real_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"inner_neg_right",
"norm_eq_infi_iff_real_inner_le_zero",
"submodule",
"submodule.add_mem",
"submodule.convex",
"submodule.neg_mem",
"submodule.sub_mem"
] | Characterization of minimizers in the projection on a subspace, in the real case.
Let `u` be a point in a real inner product space, and let `K` be a nonempty subspace.
Then point `v` minimizes the distance `‖u - v‖` over points in `K` if and only if
for all `w ∈ K`, `⟪u - v, w⟫ = 0` (i.e., `u - v` is orthogonal to the ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_infi_iff_inner_eq_zero {u : E} {v : E}
(hv : v ∈ K) : ‖u - v‖ = (⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫ = 0 | begin
letI : inner_product_space ℝ E := inner_product_space.is_R_or_C_to_real 𝕜 E,
letI : module ℝ E := restrict_scalars.module ℝ 𝕜 E,
let K' : submodule ℝ E := K.restrict_scalars ℝ,
split,
{ assume H,
have A : ∀ w ∈ K, re ⟪u - v, w⟫ = 0 := (norm_eq_infi_iff_real_inner_eq_zero K' hv).1 H,
assume w h... | theorem | norm_eq_infi_iff_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"inner_product_space",
"inner_product_space.is_R_or_C_to_real",
"inner_smul_right",
"module",
"norm_eq_infi_iff_real_inner_eq_zero",
"real_inner_eq_re_inner",
"submodule"
] | Characterization of minimizers in the projection on a subspace.
Let `u` be a point in an inner product space, and let `K` be a nonempty subspace.
Then point `v` minimizes the distance `‖u - v‖` over points in `K` if and only if
for all `w ∈ K`, `⟪u - v, w⟫ = 0` (i.e., `u - v` is orthogonal to the subspace `K`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_fn (v : E) | (exists_norm_eq_infi_of_complete_subspace K (complete_space_coe_iff_is_complete.mp ‹_›) v).some | def | orthogonal_projection_fn | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"exists_norm_eq_infi_of_complete_subspace"
] | The orthogonal projection onto a complete subspace, as an
unbundled function. This definition is only intended for use in
setting up the bundled version `orthogonal_projection` and should not
be used once that is defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_fn_mem (v : E) : orthogonal_projection_fn K v ∈ K | (exists_norm_eq_infi_of_complete_subspace K
(complete_space_coe_iff_is_complete.mp ‹_›) v).some_spec.some | lemma | orthogonal_projection_fn_mem | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"exists_norm_eq_infi_of_complete_subspace",
"orthogonal_projection_fn"
] | The unbundled orthogonal projection is in the given subspace.
This lemma is only intended for use in setting up the bundled version
and should not be used once that is defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_fn_inner_eq_zero (v : E) :
∀ w ∈ K, ⟪v - orthogonal_projection_fn K v, w⟫ = 0 | begin
rw ←norm_eq_infi_iff_inner_eq_zero K (orthogonal_projection_fn_mem v),
exact (exists_norm_eq_infi_of_complete_subspace K
(complete_space_coe_iff_is_complete.mp ‹_›) v).some_spec.some_spec
end | lemma | orthogonal_projection_fn_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"exists_norm_eq_infi_of_complete_subspace",
"orthogonal_projection_fn",
"orthogonal_projection_fn_mem"
] | The characterization of the unbundled orthogonal projection. This
lemma is only intended for use in setting up the bundled version
and should not be used once that is defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero
{u v : E} (hvm : v ∈ K) (hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0) :
orthogonal_projection_fn K u = v | begin
rw [←sub_eq_zero, ←@inner_self_eq_zero 𝕜],
have hvs : orthogonal_projection_fn K u - v ∈ K :=
submodule.sub_mem K (orthogonal_projection_fn_mem u) hvm,
have huo : ⟪u - orthogonal_projection_fn K u, orthogonal_projection_fn K u - v⟫ = 0 :=
orthogonal_projection_fn_inner_eq_zero u _ hvs,
have huv :... | lemma | eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"inner_self_eq_zero",
"inner_sub_left",
"orthogonal_projection_fn",
"orthogonal_projection_fn_inner_eq_zero",
"orthogonal_projection_fn_mem",
"submodule.sub_mem"
] | The unbundled orthogonal projection is the unique point in `K`
with the orthogonality property. This lemma is only intended for use
in setting up the bundled version and should not be used once that is
defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_fn_norm_sq (v : E) :
‖v‖ * ‖v‖ = ‖v - (orthogonal_projection_fn K v)‖ * ‖v - (orthogonal_projection_fn K v)‖
+ ‖orthogonal_projection_fn K v‖ * ‖orthogonal_projection_fn K v‖ | begin
set p := orthogonal_projection_fn K v,
have h' : ⟪v - p, p⟫ = 0,
{ exact orthogonal_projection_fn_inner_eq_zero _ _ (orthogonal_projection_fn_mem v) },
convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2;
simp,
end | lemma | orthogonal_projection_fn_norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero",
"orthogonal_projection_fn",
"orthogonal_projection_fn_inner_eq_zero",
"orthogonal_projection_fn_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_projection : E →L[𝕜] K | linear_map.mk_continuous
{ to_fun := λ v, ⟨orthogonal_projection_fn K v, orthogonal_projection_fn_mem v⟩,
map_add' := λ x y, begin
have hm : orthogonal_projection_fn K x + orthogonal_projection_fn K y ∈ K :=
submodule.add_mem K (orthogonal_projection_fn_mem x) (orthogonal_projection_fn_mem y),
... | def | orthogonal_projection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero",
"inner_add_left",
"inner_smul_left",
"le_of_pow_le_pow",
"linear_map.coe_mk",
"linear_map.mk_continuous",
"mul_zero",
"one_mul",
"orthogonal_projection_fn",
"orthogonal_projection_fn_inner_eq_zero",
"orthogonal_projection_fn_mem",
"orthogo... | The orthogonal projection onto a complete subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_fn_eq (v : E) :
orthogonal_projection_fn K v = (orthogonal_projection K v : E) | rfl | lemma | orthogonal_projection_fn_eq | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"orthogonal_projection",
"orthogonal_projection_fn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_projection_inner_eq_zero (v : E) :
∀ w ∈ K, ⟪v - orthogonal_projection K v, w⟫ = 0 | orthogonal_projection_fn_inner_eq_zero v | lemma | orthogonal_projection_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"orthogonal_projection",
"orthogonal_projection_fn_inner_eq_zero"
] | The characterization of the orthogonal projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_orthogonal_projection_mem_orthogonal (v : E) :
v - orthogonal_projection K v ∈ Kᗮ | begin
intros w hw,
rw inner_eq_zero_symm,
exact orthogonal_projection_inner_eq_zero _ _ hw
end | lemma | sub_orthogonal_projection_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"inner_eq_zero_symm",
"orthogonal_projection",
"orthogonal_projection_inner_eq_zero"
] | The difference of `v` from its orthogonal projection onto `K` is in `Kᗮ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_orthogonal_projection_of_mem_of_inner_eq_zero
{u v : E} (hvm : v ∈ K) (hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0) :
(orthogonal_projection K u : E) = v | eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hvm hvo | lemma | eq_orthogonal_projection_of_mem_of_inner_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero",
"orthogonal_projection"
] | The orthogonal projection is the unique point in `K` with the
orthogonality property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_minimal {U : submodule 𝕜 E} [complete_space U] (y : E) :
‖y - orthogonal_projection U y‖ = ⨅ x : U, ‖y - x‖ | begin
rw norm_eq_infi_iff_inner_eq_zero _ (submodule.coe_mem _),
exact orthogonal_projection_inner_eq_zero _
end | lemma | orthogonal_projection_minimal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"norm_eq_infi_iff_inner_eq_zero",
"orthogonal_projection",
"orthogonal_projection_inner_eq_zero",
"submodule",
"submodule.coe_mem"
] | The orthogonal projection of `y` on `U` minimizes the distance `‖y - x‖` for `x ∈ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_orthogonal_projection_of_eq_submodule
{K' : submodule 𝕜 E} [complete_space K'] (h : K = K') (u : E) :
(orthogonal_projection K u : E) = (orthogonal_projection K' u : E) | begin
change orthogonal_projection_fn K u = orthogonal_projection_fn K' u,
congr,
exact h
end | lemma | eq_orthogonal_projection_of_eq_submodule | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"orthogonal_projection",
"orthogonal_projection_fn",
"submodule"
] | The orthogonal projections onto equal subspaces are coerced back to the same point in `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_mem_subspace_eq_self (v : K) : orthogonal_projection K v = v | by { ext, apply eq_orthogonal_projection_of_mem_of_inner_eq_zero; simp } | lemma | orthogonal_projection_mem_subspace_eq_self | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"eq_orthogonal_projection_of_mem_of_inner_eq_zero",
"orthogonal_projection"
] | The orthogonal projection sends elements of `K` to themselves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_eq_self_iff {v : E} :
(orthogonal_projection K v : E) = v ↔ v ∈ K | begin
refine ⟨λ h, _, λ h, eq_orthogonal_projection_of_mem_of_inner_eq_zero h _⟩,
{ rw ← h,
simp },
{ simp }
end | lemma | orthogonal_projection_eq_self_iff | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"eq_orthogonal_projection_of_mem_of_inner_eq_zero",
"orthogonal_projection"
] | A point equals its orthogonal projection if and only if it lies in the subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.map_orthogonal_projection {E E' : Type*}
[normed_add_comm_group E] [normed_add_comm_group E']
[inner_product_space 𝕜 E] [inner_product_space 𝕜 E']
(f : E →ₗᵢ[𝕜] E') (p : submodule 𝕜 E) [complete_space p]
(x : E) :
f (orthogonal_projection p x) = orthogonal_projection (p.map f.to_linear_map... | begin
refine (eq_orthogonal_projection_of_mem_of_inner_eq_zero _ $
λ y hy, _).symm,
refine submodule.apply_coe_mem_map _ _,
rcases hy with ⟨x', hx', rfl : f x' = y⟩,
rw [← f.map_sub, f.inner_map_map, orthogonal_projection_inner_eq_zero x x' hx']
end | lemma | linear_isometry.map_orthogonal_projection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_orthogonal_projection_of_mem_of_inner_eq_zero",
"inner_product_space",
"normed_add_comm_group",
"orthogonal_projection",
"orthogonal_projection_inner_eq_zero",
"submodule",
"submodule.apply_coe_mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.map_orthogonal_projection' {E E' : Type*}
[normed_add_comm_group E] [normed_add_comm_group E']
[inner_product_space 𝕜 E] [inner_product_space 𝕜 E']
(f : E →ₗᵢ[𝕜] E') (p : submodule 𝕜 E) [complete_space p]
(x : E) :
f (orthogonal_projection p x) = orthogonal_projection (p.map f) (f x) | begin
refine (eq_orthogonal_projection_of_mem_of_inner_eq_zero _ $
λ y hy, _).symm,
refine submodule.apply_coe_mem_map _ _,
rcases hy with ⟨x', hx', rfl : f x' = y⟩,
rw [← f.map_sub, f.inner_map_map, orthogonal_projection_inner_eq_zero x x' hx']
end | lemma | linear_isometry.map_orthogonal_projection' | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_orthogonal_projection_of_mem_of_inner_eq_zero",
"inner_product_space",
"normed_add_comm_group",
"orthogonal_projection",
"orthogonal_projection_inner_eq_zero",
"submodule",
"submodule.apply_coe_mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_projection_map_apply {E E' : Type*}
[normed_add_comm_group E] [normed_add_comm_group E']
[inner_product_space 𝕜 E] [inner_product_space 𝕜 E']
(f : E ≃ₗᵢ[𝕜] E') (p : submodule 𝕜 E) [complete_space p]
(x : E') :
(orthogonal_projection (p.map (f.to_linear_equiv : E →ₗ[𝕜] E')) x : E')
= f (ortho... | by simpa only [f.coe_to_linear_isometry, f.apply_symm_apply]
using (f.to_linear_isometry.map_orthogonal_projection p (f.symm x)).symm | lemma | orthogonal_projection_map_apply | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_product_space",
"normed_add_comm_group",
"orthogonal_projection",
"submodule"
] | Orthogonal projection onto the `submodule.map` of a subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_bot : orthogonal_projection (⊥ : submodule 𝕜 E) = 0 | by ext | lemma | orthogonal_projection_bot | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"orthogonal_projection",
"submodule"
] | The orthogonal projection onto the trivial submodule is the zero map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_norm_le : ‖orthogonal_projection K‖ ≤ 1 | linear_map.mk_continuous_norm_le _ (by norm_num) _ | lemma | orthogonal_projection_norm_le | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"linear_map.mk_continuous_norm_le"
] | The orthogonal projection has norm `≤ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_orthogonal_projection_singleton {v : E} (w : E) :
(‖v‖ ^ 2 : 𝕜) • (orthogonal_projection (𝕜 ∙ v) w : E) = ⟪v, w⟫ • v | begin
suffices : ↑(orthogonal_projection (𝕜 ∙ v) ((‖v‖ ^ 2 : 𝕜) • w)) = ⟪v, w⟫ • v,
{ simpa using this },
apply eq_orthogonal_projection_of_mem_of_inner_eq_zero,
{ rw submodule.mem_span_singleton,
use ⟪v, w⟫ },
{ intros x hx,
obtain ⟨c, rfl⟩ := submodule.mem_span_singleton.mp hx,
have hv : ↑‖v‖ ... | lemma | smul_orthogonal_projection_singleton | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"eq_orthogonal_projection_of_mem_of_inner_eq_zero",
"inner_smul_left",
"inner_smul_right",
"inner_sub_left",
"map_div₀",
"mul_comm",
"orthogonal_projection",
"submodule.mem_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_projection_singleton {v : E} (w : E) :
(orthogonal_projection (𝕜 ∙ v) w : E) = (⟪v, w⟫ / ‖v‖ ^ 2) • v | begin
by_cases hv : v = 0,
{ rw [hv, eq_orthogonal_projection_of_eq_submodule (submodule.span_zero_singleton 𝕜)],
{ simp },
{ apply_instance } },
have hv' : ‖v‖ ≠ 0 := ne_of_gt (norm_pos_iff.mpr hv),
have key : ((‖v‖ ^ 2 : 𝕜)⁻¹ * ‖v‖ ^ 2) • ↑(orthogonal_projection (𝕜 ∙ v) w)
= ((‖v‖ ^ 2... | lemma | orthogonal_projection_singleton | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"eq_orthogonal_projection_of_eq_submodule",
"orthogonal_projection",
"smul_orthogonal_projection_singleton",
"submodule.span_zero_singleton"
] | Formula for orthogonal projection onto a single vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_unit_singleton {v : E} (hv : ‖v‖ = 1) (w : E) :
(orthogonal_projection (𝕜 ∙ v) w : E) = ⟪v, w⟫ • v | by { rw ← smul_orthogonal_projection_singleton 𝕜 w, simp [hv] } | lemma | orthogonal_projection_unit_singleton | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"orthogonal_projection",
"smul_orthogonal_projection_singleton"
] | Formula for orthogonal projection onto a single unit vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_linear_equiv : E ≃ₗ[𝕜] E | linear_equiv.of_involutive
(bit0 (K.subtype.comp (orthogonal_projection K).to_linear_map) - linear_map.id)
(λ x, by simp [bit0]) | def | reflection_linear_equiv | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"linear_equiv.of_involutive",
"linear_map.id",
"orthogonal_projection"
] | Auxiliary definition for `reflection`: the reflection as a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection : E ≃ₗᵢ[𝕜] E | { norm_map' := begin
intros x,
let w : K := orthogonal_projection K x,
let v := x - w,
have : ⟪v, w⟫ = 0 := orthogonal_projection_inner_eq_zero x w w.2,
convert norm_sub_eq_norm_add this using 2,
{ rw [linear_equiv.coe_mk, reflection_linear_equiv,
linear_equiv.to_fun_eq_coe, linear_equiv... | def | reflection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"continuous_linear_map.coe_coe",
"continuous_linear_map.to_linear_map_eq_coe",
"linear_equiv.coe_mk",
"linear_equiv.coe_of_involutive",
"linear_equiv.to_fun_eq_coe",
"linear_map.add_apply",
"linear_map.comp_apply",
"linear_map.id_apply",
"linear_map.sub_apply",
"norm_sub_eq_norm_add",
"orthogona... | Reflection in a complete subspace of an inner product space. The word "reflection" is
sometimes understood to mean specifically reflection in a codimension-one subspace, and sometimes
more generally to cover operations such as reflection in a point. The definition here, of
reflection in a subspace, is a more general ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_apply (p : E) : reflection K p = bit0 ↑(orthogonal_projection K p) - p | rfl | lemma | reflection_apply | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"orthogonal_projection",
"reflection"
] | The result of reflecting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_symm : (reflection K).symm = reflection K | rfl | lemma | reflection_symm | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection"
] | Reflection is its own inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_inv : (reflection K)⁻¹ = reflection K | rfl | lemma | reflection_inv | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection"
] | Reflection is its own inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_reflection (p : E) : reflection K (reflection K p) = p | (reflection K).left_inv p | lemma | reflection_reflection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection"
] | Reflecting twice in the same subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_involutive : function.involutive (reflection K) | reflection_reflection K | lemma | reflection_involutive | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection",
"reflection_reflection"
] | Reflection is involutive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_trans_reflection :
(reflection K).trans (reflection K) = linear_isometry_equiv.refl 𝕜 E | linear_isometry_equiv.ext $ reflection_involutive K | lemma | reflection_trans_reflection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"linear_isometry_equiv.ext",
"linear_isometry_equiv.refl",
"reflection",
"reflection_involutive"
] | Reflection is involutive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_mul_reflection : reflection K * reflection K = 1 | reflection_trans_reflection _ | lemma | reflection_mul_reflection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection",
"reflection_trans_reflection"
] | Reflection is involutive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_eq_self_iff (x : E) : reflection K x = x ↔ x ∈ K | begin
rw [←orthogonal_projection_eq_self_iff, reflection_apply, sub_eq_iff_eq_add', ← two_smul 𝕜,
← two_smul' 𝕜],
refine (smul_right_injective E _).eq_iff,
exact two_ne_zero
end | lemma | reflection_eq_self_iff | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection",
"reflection_apply",
"smul_right_injective",
"two_ne_zero",
"two_smul",
"two_smul'"
] | A point is its own reflection if and only if it is in the subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_mem_subspace_eq_self {x : E} (hx : x ∈ K) : reflection K x = x | (reflection_eq_self_iff x).mpr hx | lemma | reflection_mem_subspace_eq_self | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"reflection",
"reflection_eq_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflection_map_apply {E E' : Type*}
[normed_add_comm_group E] [normed_add_comm_group E']
[inner_product_space 𝕜 E] [inner_product_space 𝕜 E']
(f : E ≃ₗᵢ[𝕜] E') (K : submodule 𝕜 E) [complete_space K] (x : E') :
reflection (K.map (f.to_linear_equiv : E →ₗ[𝕜] E')) x = f (reflection K (f.symm x)) | by simp [bit0, reflection_apply, orthogonal_projection_map_apply f K x] | lemma | reflection_map_apply | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_product_space",
"normed_add_comm_group",
"orthogonal_projection_map_apply",
"reflection",
"reflection_apply",
"submodule"
] | Reflection in the `submodule.map` of a subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_map {E E' : Type*}
[normed_add_comm_group E] [normed_add_comm_group E']
[inner_product_space 𝕜 E] [inner_product_space 𝕜 E']
(f : E ≃ₗᵢ[𝕜] E') (K : submodule 𝕜 E) [complete_space K] :
reflection (K.map (f.to_linear_equiv : E →ₗ[𝕜] E')) = f.symm.trans ((reflection K).trans f) | linear_isometry_equiv.ext $ reflection_map_apply f K | lemma | reflection_map | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_product_space",
"linear_isometry_equiv.ext",
"normed_add_comm_group",
"reflection",
"reflection_map_apply",
"submodule"
] | Reflection in the `submodule.map` of a subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_bot : reflection (⊥ : submodule 𝕜 E) = linear_isometry_equiv.neg 𝕜 | by ext; simp [reflection_apply] | lemma | reflection_bot | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"linear_isometry_equiv.neg",
"reflection",
"reflection_apply",
"submodule"
] | Reflection through the trivial subspace {0} is just negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.sup_orthogonal_inf_of_complete_space {K₁ K₂ : submodule 𝕜 E} (h : K₁ ≤ K₂)
[complete_space K₁] : K₁ ⊔ (K₁ᗮ ⊓ K₂) = K₂ | begin
ext x,
rw submodule.mem_sup,
let v : K₁ := orthogonal_projection K₁ x,
have hvm : x - v ∈ K₁ᗮ := sub_orthogonal_projection_mem_orthogonal x,
split,
{ rintro ⟨y, hy, z, hz, rfl⟩,
exact K₂.add_mem (h hy) hz.2 },
{ exact λ hx, ⟨v, v.prop, x - v, ⟨hvm, K₂.sub_mem hx (h v.prop)⟩, add_sub_cancel'_righ... | lemma | submodule.sup_orthogonal_inf_of_complete_space | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"orthogonal_projection",
"sub_orthogonal_projection_mem_orthogonal",
"submodule",
"submodule.mem_sup"
] | If `K₁` is complete and contained in `K₂`, `K₁` and `K₁ᗮ ⊓ K₂` span `K₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.sup_orthogonal_of_complete_space [complete_space K] : K ⊔ Kᗮ = ⊤ | begin
convert submodule.sup_orthogonal_inf_of_complete_space (le_top : K ≤ ⊤),
simp
end | lemma | submodule.sup_orthogonal_of_complete_space | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"le_top",
"submodule.sup_orthogonal_inf_of_complete_space"
] | If `K` is complete, `K` and `Kᗮ` span the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.exists_sum_mem_mem_orthogonal [complete_space K] (v : E) :
∃ (y ∈ K) (z ∈ Kᗮ), v = y + z | begin
have h_mem : v ∈ K ⊔ Kᗮ := by simp [submodule.sup_orthogonal_of_complete_space],
obtain ⟨y, hy, z, hz, hyz⟩ := submodule.mem_sup.mp h_mem,
exact ⟨y, hy, z, hz, hyz.symm⟩
end | lemma | submodule.exists_sum_mem_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"submodule.sup_orthogonal_of_complete_space"
] | If `K` is complete, any `v` in `E` can be expressed as a sum of elements of `K` and `Kᗮ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.orthogonal_orthogonal [complete_space K] : Kᗮᗮ = K | begin
ext v,
split,
{ obtain ⟨y, hy, z, hz, rfl⟩ := K.exists_sum_mem_mem_orthogonal v,
intros hv,
have hz' : z = 0,
{ have hyz : ⟪z, y⟫ = 0 := by simp [hz y hy, inner_eq_zero_symm],
simpa [inner_add_right, hyz] using hv z hz },
simp [hy, hz'] },
{ intros hv w hw,
rw inner_eq_zero_symm,... | lemma | submodule.orthogonal_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_add_right",
"inner_eq_zero_symm"
] | If `K` is complete, then the orthogonal complement of its orthogonal complement is itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.orthogonal_orthogonal_eq_closure [complete_space E] :
Kᗮᗮ = K.topological_closure | begin
refine le_antisymm _ _,
{ convert submodule.orthogonal_orthogonal_monotone K.le_topological_closure,
haveI : complete_space K.topological_closure :=
K.is_closed_topological_closure.complete_space_coe,
rw K.topological_closure.orthogonal_orthogonal },
{ exact K.topological_closure_minimal K.le_... | lemma | submodule.orthogonal_orthogonal_eq_closure | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"submodule.orthogonal_orthogonal_monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.is_compl_orthogonal_of_complete_space [complete_space K] : is_compl K Kᗮ | ⟨K.orthogonal_disjoint, codisjoint_iff.2 submodule.sup_orthogonal_of_complete_space⟩ | lemma | submodule.is_compl_orthogonal_of_complete_space | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"is_compl"
] | If `K` is complete, `K` and `Kᗮ` are complements of each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.orthogonal_eq_bot_iff [complete_space (K : set E)] :
Kᗮ = ⊥ ↔ K = ⊤ | begin
refine ⟨_, λ h, by rw [h, submodule.top_orthogonal_eq_bot] ⟩,
intro h,
have : K ⊔ Kᗮ = ⊤ := submodule.sup_orthogonal_of_complete_space,
rwa [h, sup_comm, bot_sup_eq] at this,
end | lemma | submodule.orthogonal_eq_bot_iff | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"bot_sup_eq",
"complete_space",
"submodule.sup_orthogonal_of_complete_space",
"submodule.top_orthogonal_eq_bot",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_orthogonal_projection_of_mem_orthogonal
[complete_space K] {u v : E} (hv : v ∈ K) (hvo : u - v ∈ Kᗮ) :
(orthogonal_projection K u : E) = v | eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hv (λ w, inner_eq_zero_symm.mp ∘ (hvo w)) | lemma | eq_orthogonal_projection_of_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero",
"orthogonal_projection"
] | A point in `K` with the orthogonality property (here characterized in terms of `Kᗮ`) must be the
orthogonal projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_orthogonal_projection_of_mem_orthogonal'
[complete_space K] {u v z : E} (hv : v ∈ K) (hz : z ∈ Kᗮ) (hu : u = v + z) :
(orthogonal_projection K u : E) = v | eq_orthogonal_projection_of_mem_orthogonal hv (by simpa [hu]) | lemma | eq_orthogonal_projection_of_mem_orthogonal' | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_orthogonal_projection_of_mem_orthogonal",
"orthogonal_projection"
] | A point in `K` with the orthogonality property (here characterized in terms of `Kᗮ`) must be the
orthogonal projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero
[complete_space K] {v : E} (hv : v ∈ Kᗮ) :
orthogonal_projection K v = 0 | by { ext, convert eq_orthogonal_projection_of_mem_orthogonal _ _; simp [hv] } | lemma | orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_orthogonal_projection_of_mem_orthogonal",
"orthogonal_projection"
] | The orthogonal projection onto `K` of an element of `Kᗮ` is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.is_ortho.orthogonal_projection_comp_subtypeL {U V : submodule 𝕜 E}
[complete_space U] (h : U ⟂ V) :
orthogonal_projection U ∘L V.subtypeL = 0 | continuous_linear_map.ext $ λ v,
orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero $ h.symm v.prop | lemma | submodule.is_ortho.orthogonal_projection_comp_subtypeL | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"continuous_linear_map.ext",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero",
"submodule"
] | The projection into `U` from an orthogonal submodule `V` is the zero map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_comp_subtypeL_eq_zero_iff {U V : submodule 𝕜 E}
[complete_space U] :
orthogonal_projection U ∘L V.subtypeL = 0 ↔ U ⟂ V | ⟨λ h u hu v hv, begin
convert orthogonal_projection_inner_eq_zero v u hu using 2,
have : orthogonal_projection U v = 0 := fun_like.congr_fun h ⟨_, hv⟩,
rw [this, submodule.coe_zero, sub_zero]
end, submodule.is_ortho.orthogonal_projection_comp_subtypeL⟩ | lemma | orthogonal_projection_comp_subtypeL_eq_zero_iff | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"fun_like.congr_fun",
"orthogonal_projection",
"orthogonal_projection_inner_eq_zero",
"submodule",
"submodule.coe_zero"
] | The projection into `U` from `V` is the zero map if and only if `U` and `V` are orthogonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_eq_linear_proj [complete_space K] (x : E) :
orthogonal_projection K x =
K.linear_proj_of_is_compl _ submodule.is_compl_orthogonal_of_complete_space x | begin
have : is_compl K Kᗮ := submodule.is_compl_orthogonal_of_complete_space,
nth_rewrite 0 [← submodule.linear_proj_add_linear_proj_of_is_compl_eq_self this x],
rw [map_add, orthogonal_projection_mem_subspace_eq_self,
orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero (submodule.coe_mem _),
... | lemma | orthogonal_projection_eq_linear_proj | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"is_compl",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_eq_self",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero",
"submodule.coe_mem",
"submodule.is_compl_orthogonal_of_complete_space",
"submodule.linear_proj_add_linear_proj_of_is_compl_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_projection_coe_linear_map_eq_linear_proj [complete_space K] :
(orthogonal_projection K : E →ₗ[𝕜] K) =
K.linear_proj_of_is_compl _ submodule.is_compl_orthogonal_of_complete_space | linear_map.ext $ orthogonal_projection_eq_linear_proj | lemma | orthogonal_projection_coe_linear_map_eq_linear_proj | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"linear_map.ext",
"orthogonal_projection",
"orthogonal_projection_eq_linear_proj",
"submodule.is_compl_orthogonal_of_complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflection_mem_subspace_orthogonal_complement_eq_neg
[complete_space K] {v : E} (hv : v ∈ Kᗮ) :
reflection K v = - v | by simp [reflection_apply, orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero hv] | lemma | reflection_mem_subspace_orthogonal_complement_eq_neg | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero",
"reflection",
"reflection_apply"
] | The reflection in `K` of an element of `Kᗮ` is its negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero
[complete_space E] {v : E} (hv : v ∈ K) :
orthogonal_projection Kᗮ v = 0 | orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero (K.le_orthogonal_orthogonal hv) | lemma | orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero"
] | The orthogonal projection onto `Kᗮ` of an element of `K` is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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