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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|---|---|---|---|---|---|---|---|---|---|---|
is_open_ball {x : E} {r : ℝ} : is_open (ball x r) | metric.is_open_ball.preimage to_euclidean.continuous | lemma | euclidean.is_open_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_self {x : E} {r : ℝ} (hr : 0 < r) : x ∈ ball x r | metric.mem_ball_self hr | lemma | euclidean.mem_ball_self | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"metric.mem_ball_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_image (x : E) (r : ℝ) :
closed_ball x r = to_euclidean.symm '' metric.closed_ball (to_euclidean x) r | by rw [to_euclidean.image_symm_eq_preimage, closed_ball_eq_preimage] | lemma | euclidean.closed_ball_eq_image | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"metric.closed_ball",
"to_euclidean"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_closed_ball {x : E} {r : ℝ} : is_compact (closed_ball x r) | begin
rw closed_ball_eq_image,
exact (is_compact_closed_ball _ _).image to_euclidean.symm.continuous
end | lemma | euclidean.is_compact_closed_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_closed_ball {x : E} {r : ℝ} : is_closed (closed_ball x r) | is_compact_closed_ball.is_closed | lemma | euclidean.is_closed_closed_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_ball (x : E) {r : ℝ} (h : r ≠ 0) : closure (ball x r) = closed_ball x r | by rw [ball_eq_preimage, ← to_euclidean.preimage_closure, closure_ball (to_euclidean x) h,
closed_ball_eq_preimage] | lemma | euclidean.closure_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"closure",
"closure_ball",
"to_euclidean"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_lt_subset_ball {R : ℝ} {s : set E} {x : E}
(hR : 0 < R) (hs : is_closed s) (h : s ⊆ ball x R) :
∃ r ∈ Ioo 0 R, s ⊆ ball x r | begin
rw [ball_eq_preimage, ← image_subset_iff] at h,
rcases exists_pos_lt_subset_ball hR (to_euclidean.is_closed_image.2 hs) h with ⟨r, hr, hsr⟩,
exact ⟨r, hr, image_subset_iff.1 hsr⟩
end | lemma | euclidean.exists_pos_lt_subset_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"exists_pos_lt_subset_ball",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_closed_ball {x : E} :
(𝓝 x).has_basis (λ r : ℝ, 0 < r) (closed_ball x) | begin
rw [to_euclidean.to_homeomorph.nhds_eq_comap x],
exact metric.nhds_basis_closed_ball.comap _
end | lemma | euclidean.nhds_basis_closed_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : closed_ball x r ∈ 𝓝 x | nhds_basis_closed_ball.mem_of_mem hr | lemma | euclidean.closed_ball_mem_nhds | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_ball {x : E} :
(𝓝 x).has_basis (λ r : ℝ, 0 < r) (ball x) | begin
rw [to_euclidean.to_homeomorph.nhds_eq_comap x],
exact metric.nhds_basis_ball.comap _
end | lemma | euclidean.nhds_basis_ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : ball x r ∈ 𝓝 x | nhds_basis_ball.mem_of_mem hr | lemma | euclidean.ball_mem_nhds | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.euclidean_dist (hf : cont_diff ℝ n f) (hg : cont_diff ℝ n g)
(h : ∀ x, f x ≠ g x) :
cont_diff ℝ n (λ x, euclidean.dist (f x) (g x)) | begin
simp only [euclidean.dist],
apply @cont_diff.dist ℝ,
exacts [(@to_euclidean G _ _ _ _ _ _ _).cont_diff.comp hf,
(@to_euclidean G _ _ _ _ _ _ _).cont_diff.comp hg, λ x, to_euclidean.injective.ne (h x)]
end | lemma | cont_diff.euclidean_dist | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"cont_diff",
"cont_diff.comp",
"cont_diff.dist",
"euclidean.dist",
"to_euclidean"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt (f : ι → E) : ι → E | | n := f n - ∑ i : Iio n, orthogonal_projection (𝕜 ∙ gram_schmidt i) (f n)
using_well_founded { dec_tac := `[exact mem_Iio.1 i.2] } | def | gram_schmidt | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"orthogonal_projection"
] | The Gram-Schmidt process takes a set of vectors as input
and outputs a set of orthogonal vectors which have the same span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_def (f : ι → E) (n : ι):
gram_schmidt 𝕜 f n = f n - ∑ i in Iio n,
orthogonal_projection (𝕜 ∙ gram_schmidt 𝕜 f i) (f n) | by { rw [←sum_attach, attach_eq_univ, gram_schmidt], refl } | lemma | gram_schmidt_def | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"orthogonal_projection"
] | This lemma uses `∑ i in` instead of `∑ i :`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_def' (f : ι → E) (n : ι):
f n = gram_schmidt 𝕜 f n + ∑ i in Iio n,
orthogonal_projection (𝕜 ∙ gram_schmidt 𝕜 f i) (f n) | by rw [gram_schmidt_def, sub_add_cancel] | lemma | gram_schmidt_def' | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_def",
"orthogonal_projection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_def'' (f : ι → E) (n : ι):
f n = gram_schmidt 𝕜 f n
+ ∑ i in Iio n, (⟪gram_schmidt 𝕜 f i, f n⟫ / ‖gram_schmidt 𝕜 f i‖ ^ 2) • gram_schmidt 𝕜 f i | begin
convert gram_schmidt_def' 𝕜 f n,
ext i,
rw orthogonal_projection_singleton,
end | lemma | gram_schmidt_def'' | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_def'",
"orthogonal_projection_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_zero {ι : Type*} [linear_order ι] [locally_finite_order ι]
[order_bot ι] [is_well_order ι (<)] (f : ι → E) : gram_schmidt 𝕜 f ⊥ = f ⊥ | by rw [gram_schmidt_def, Iio_eq_Ico, finset.Ico_self, finset.sum_empty, sub_zero] | lemma | gram_schmidt_zero | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"finset.Ico_self",
"gram_schmidt",
"gram_schmidt_def",
"is_well_order",
"locally_finite_order",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
⟪gram_schmidt 𝕜 f a, gram_schmidt 𝕜 f b⟫ = 0 | begin
suffices : ∀ a b : ι, a < b → ⟪gram_schmidt 𝕜 f a, gram_schmidt 𝕜 f b⟫ = 0,
{ cases h₀.lt_or_lt with ha hb,
{ exact this _ _ ha, },
{ rw inner_eq_zero_symm,
exact this _ _ hb, }, },
clear h₀ a b,
intros a b h₀,
revert a,
apply well_founded.induction (@is_well_founded.wf ι (<) _) b,
i... | theorem | gram_schmidt_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"div_eq_zero_iff",
"div_mul_cancel",
"finset.mem_range",
"gram_schmidt",
"gram_schmidt_def",
"ih",
"inner_eq_zero_symm",
"inner_self_eq_norm_sq_to_K",
"inner_self_eq_zero",
"inner_self_ne_zero",
"inner_smul_right",
"inner_sub_right",
"inner_sum",
"inner_zero_left",
"mul_eq_zero",
"orth... | **Gram-Schmidt Orthogonalisation**:
`gram_schmidt` produces an orthogonal system of vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_pairwise_orthogonal (f : ι → E) :
pairwise (λ a b, ⟪gram_schmidt 𝕜 f a, gram_schmidt 𝕜 f b⟫ = 0) | λ a b, gram_schmidt_orthogonal 𝕜 f | theorem | gram_schmidt_pairwise_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_orthogonal",
"pairwise"
] | This is another version of `gram_schmidt_orthogonal` using `pairwise` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
⟪gram_schmidt 𝕜 v j, v i⟫ = 0 | begin
rw gram_schmidt_def'' 𝕜 v,
simp only [inner_add_right, inner_sum, inner_smul_right],
set b : ι → E := gram_schmidt 𝕜 v,
convert zero_add (0:𝕜),
{ exact gram_schmidt_orthogonal 𝕜 v hij.ne' },
apply finset.sum_eq_zero,
rintros k hki',
have hki : k < i := by simpa using hki',
have : ⟪b j, b k⟫ ... | lemma | gram_schmidt_inv_triangular | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_def''",
"gram_schmidt_orthogonal",
"inner_add_right",
"inner_smul_right",
"inner_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_gram_schmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
f i ∈ span 𝕜 (gram_schmidt 𝕜 f '' Iic j) | begin
rw [gram_schmidt_def' 𝕜 f i],
simp_rw orthogonal_projection_singleton,
exact submodule.add_mem _ (subset_span $ mem_image_of_mem _ hij)
(submodule.sum_mem _ $ λ k hk, smul_mem (span 𝕜 (gram_schmidt 𝕜 f '' Iic j)) _ $
subset_span $ mem_image_of_mem (gram_schmidt 𝕜 f) $ (finset.mem_Iio.1 hk).le.tr... | lemma | mem_span_gram_schmidt | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_def'",
"orthogonal_projection_singleton",
"submodule.add_mem",
"submodule.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_mem_span (f : ι → E) :
∀ {j i}, i ≤ j → gram_schmidt 𝕜 f i ∈ span 𝕜 (f '' Iic j) | | j := λ i hij,
begin
rw [gram_schmidt_def 𝕜 f i],
simp_rw orthogonal_projection_singleton,
refine submodule.sub_mem _ (subset_span (mem_image_of_mem _ hij))
(submodule.sum_mem _ $ λ k hk, _),
let hkj : k < j := (finset.mem_Iio.1 hk).trans_le hij,
exact smul_mem _ _ (span_mono (image_subset f $ Iic_subse... | lemma | gram_schmidt_mem_span | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_def",
"le_rfl",
"orthogonal_projection_singleton",
"submodule.sub_mem",
"submodule.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_gram_schmidt_Iic (f : ι → E) (c : ι) :
span 𝕜 (gram_schmidt 𝕜 f '' Iic c) = span 𝕜 (f '' Iic c) | span_eq_span (set.image_subset_iff.2 $ λ i, gram_schmidt_mem_span _ _) $
set.image_subset_iff.2 $ λ i, mem_span_gram_schmidt _ _ | lemma | span_gram_schmidt_Iic | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_mem_span",
"mem_span_gram_schmidt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_gram_schmidt_Iio (f : ι → E) (c : ι) :
span 𝕜 (gram_schmidt 𝕜 f '' Iio c) = span 𝕜 (f '' Iio c) | span_eq_span
(set.image_subset_iff.2 $ λ i hi, span_mono (image_subset _ $ Iic_subset_Iio.2 hi) $
gram_schmidt_mem_span _ _ le_rfl) $
set.image_subset_iff.2 $ λ i hi, span_mono (image_subset _ $ Iic_subset_Iio.2 hi) $
mem_span_gram_schmidt _ _ le_rfl | lemma | span_gram_schmidt_Iio | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_mem_span",
"le_rfl",
"mem_span_gram_schmidt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_gram_schmidt (f : ι → E) : span 𝕜 (range (gram_schmidt 𝕜 f)) = span 𝕜 (range f) | span_eq_span (range_subset_iff.2 $ λ i, span_mono (image_subset_range _ _) $
gram_schmidt_mem_span _ _ le_rfl) $
range_subset_iff.2 $ λ i, span_mono (image_subset_range _ _) $ mem_span_gram_schmidt _ _ le_rfl | lemma | span_gram_schmidt | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_mem_span",
"le_rfl",
"mem_span_gram_schmidt"
] | `gram_schmidt` preserves span of vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_of_orthogonal {f : ι → E} (hf : pairwise (λ i j, ⟪f i, f j⟫ = 0)) :
gram_schmidt 𝕜 f = f | begin
ext i,
rw gram_schmidt_def,
transitivity f i - 0,
{ congr,
apply finset.sum_eq_zero,
intros j hj,
rw coe_eq_zero,
suffices : span 𝕜 (f '' set.Iic j) ⟂ 𝕜 ∙ f i,
{ apply orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero,
rw mem_orthogonal_singleton_iff_inner_left,
... | lemma | gram_schmidt_of_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_def",
"gram_schmidt_mem_span",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero",
"pairwise",
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_ne_zero_coe
{f : ι → E} (n : ι) (h₀ : linear_independent 𝕜 (f ∘ (coe : set.Iic n → ι))) :
gram_schmidt 𝕜 f n ≠ 0 | begin
by_contra h,
have h₁ : f n ∈ span 𝕜 (f '' Iio n),
{ rw [← span_gram_schmidt_Iio 𝕜 f n, gram_schmidt_def' _ f, h, zero_add],
apply submodule.sum_mem _ _,
simp_intros a ha only [finset.mem_Ico],
simp only [set.mem_image, set.mem_Iio, orthogonal_projection_singleton],
apply submodule.smul_mem... | lemma | gram_schmidt_ne_zero_coe | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"by_contra",
"finset.mem_Ico",
"finset.mem_Iio",
"gram_schmidt",
"gram_schmidt_def'",
"linear_independent",
"linear_independent.not_mem_span_image",
"lt_self_iff_false",
"orthogonal_projection_singleton",
"set.Iic",
"set.mem_Iio",
"set.mem_image",
"span_gram_schmidt_Iio",
"submodule.smul_m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_ne_zero {f : ι → E} (n : ι) (h₀ : linear_independent 𝕜 f) :
gram_schmidt 𝕜 f n ≠ 0 | gram_schmidt_ne_zero_coe _ (linear_independent.comp h₀ _ subtype.coe_injective) | lemma | gram_schmidt_ne_zero | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_ne_zero_coe",
"linear_independent",
"linear_independent.comp",
"subtype.coe_injective"
] | If the input vectors of `gram_schmidt` are linearly independent,
then the output vectors are non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_triangular {i j : ι} (hij : i < j) (b : basis ι 𝕜 E) :
b.repr (gram_schmidt 𝕜 b i) j = 0 | begin
have : gram_schmidt 𝕜 b i ∈ span 𝕜 (gram_schmidt 𝕜 b '' set.Iio j),
from subset_span ((set.mem_image _ _ _).2 ⟨i, hij, rfl⟩),
have : gram_schmidt 𝕜 b i ∈ span 𝕜 (b '' set.Iio j),
by rwa [← span_gram_schmidt_Iio 𝕜 b j],
have : ↑(((b.repr) (gram_schmidt 𝕜 b i)).support) ⊆ set.Iio j,
from ba... | lemma | gram_schmidt_triangular | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"basis",
"basis.repr_support_subset_of_mem_span",
"finsupp.mem_supported",
"finsupp.mem_supported'",
"gram_schmidt",
"set.Iio",
"set.mem_image",
"set.not_mem_Iio_self",
"span_gram_schmidt_Iio"
] | `gram_schmidt` produces a triangular matrix of vectors when given a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_linear_independent {f : ι → E} (h₀ : linear_independent 𝕜 f) :
linear_independent 𝕜 (gram_schmidt 𝕜 f) | linear_independent_of_ne_zero_of_inner_eq_zero
(λ i, gram_schmidt_ne_zero _ h₀) (λ i j, gram_schmidt_orthogonal 𝕜 f) | lemma | gram_schmidt_linear_independent | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_ne_zero",
"gram_schmidt_orthogonal",
"linear_independent",
"linear_independent_of_ne_zero_of_inner_eq_zero"
] | `gram_schmidt` produces linearly independent vectors when given linearly independent vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_basis (b : basis ι 𝕜 E) : basis ι 𝕜 E | basis.mk
(gram_schmidt_linear_independent b.linear_independent)
((span_gram_schmidt 𝕜 b).trans b.span_eq).ge | def | gram_schmidt_basis | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"basis",
"basis.mk",
"gram_schmidt_linear_independent",
"span_gram_schmidt"
] | When given a basis, `gram_schmidt` produces a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_gram_schmidt_basis (b : basis ι 𝕜 E) :
(gram_schmidt_basis b : ι → E) = gram_schmidt 𝕜 b | basis.coe_mk _ _ | lemma | coe_gram_schmidt_basis | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"basis",
"basis.coe_mk",
"gram_schmidt",
"gram_schmidt_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_normed (f : ι → E) (n : ι) : E | (‖gram_schmidt 𝕜 f n‖ : 𝕜)⁻¹ • (gram_schmidt 𝕜 f n) | def | gram_schmidt_normed | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt"
] | the normalized `gram_schmidt`
(i.e each vector in `gram_schmidt_normed` has unit length.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_normed_unit_length_coe
{f : ι → E} (n : ι) (h₀ : linear_independent 𝕜 (f ∘ (coe : set.Iic n → ι))) :
‖gram_schmidt_normed 𝕜 f n‖ = 1 | by simp only [gram_schmidt_ne_zero_coe n h₀,
gram_schmidt_normed, norm_smul_inv_norm, ne.def, not_false_iff] | lemma | gram_schmidt_normed_unit_length_coe | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_ne_zero_coe",
"gram_schmidt_normed",
"linear_independent",
"norm_smul_inv_norm",
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_normed_unit_length {f : ι → E} (n : ι) (h₀ : linear_independent 𝕜 f) :
‖gram_schmidt_normed 𝕜 f n‖ = 1 | gram_schmidt_normed_unit_length_coe _ (linear_independent.comp h₀ _ subtype.coe_injective) | lemma | gram_schmidt_normed_unit_length | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed_unit_length_coe",
"linear_independent",
"linear_independent.comp",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_normed_unit_length' {f : ι → E} {n : ι} (hn : gram_schmidt_normed 𝕜 f n ≠ 0) :
‖gram_schmidt_normed 𝕜 f n‖ = 1 | begin
rw gram_schmidt_normed at *,
rw [norm_smul_inv_norm],
simpa using hn,
end | lemma | gram_schmidt_normed_unit_length' | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed",
"norm_smul_inv_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthonormal {f : ι → E} (h₀ : linear_independent 𝕜 f) :
orthonormal 𝕜 (gram_schmidt_normed 𝕜 f) | begin
unfold orthonormal,
split,
{ simp only [gram_schmidt_normed_unit_length, h₀, eq_self_iff_true, implies_true_iff], },
{ intros i j hij,
simp only [gram_schmidt_normed, inner_smul_left, inner_smul_right, is_R_or_C.conj_inv,
is_R_or_C.conj_of_real, mul_eq_zero, inv_eq_zero, is_R_or_C.of_real_eq_zer... | theorem | gram_schmidt_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed",
"gram_schmidt_normed_unit_length",
"gram_schmidt_orthogonal",
"inner_smul_left",
"inner_smul_right",
"inv_eq_zero",
"is_R_or_C.conj_inv",
"is_R_or_C.conj_of_real",
"is_R_or_C.of_real_eq_zero",
"linear_independent",
"mul_eq_zero",
"norm_eq_zero",
"orthonormal"
] | **Gram-Schmidt Orthonormalization**:
`gram_schmidt_normed` applied to a linearly independent set of vectors produces an orthornormal
system of vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_orthonormal' (f : ι → E) :
orthonormal 𝕜 (λ i : {i | gram_schmidt_normed 𝕜 f i ≠ 0}, gram_schmidt_normed 𝕜 f i) | begin
refine ⟨λ i, gram_schmidt_normed_unit_length' i.prop, _⟩,
rintros i j (hij : ¬ _),
rw subtype.ext_iff at hij,
simp [gram_schmidt_normed, inner_smul_left, inner_smul_right, gram_schmidt_orthogonal 𝕜 f hij],
end | lemma | gram_schmidt_orthonormal' | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed",
"gram_schmidt_normed_unit_length'",
"gram_schmidt_orthogonal",
"inner_smul_left",
"inner_smul_right",
"orthonormal",
"subtype.ext_iff"
] | **Gram-Schmidt Orthonormalization**:
`gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
become zero in the process. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_gram_schmidt_normed (f : ι → E) (s : set ι) :
span 𝕜 (gram_schmidt_normed 𝕜 f '' s) = span 𝕜 (gram_schmidt 𝕜 f '' s) | begin
refine span_eq_span (set.image_subset_iff.2 $ λ i hi, smul_mem _ _ $ subset_span $
mem_image_of_mem _ hi)
(set.image_subset_iff.2 $ λ i hi, span_mono (image_subset _ $ singleton_subset_set_iff.2 hi) _),
simp only [coe_singleton, set.image_singleton],
by_cases h : gram_schmidt 𝕜 f i = 0,
{ simp [h... | lemma | span_gram_schmidt_normed | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_normed",
"set.image_singleton",
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_gram_schmidt_normed_range (f : ι → E) :
span 𝕜 (range (gram_schmidt_normed 𝕜 f)) = span 𝕜 (range (gram_schmidt 𝕜 f)) | by simpa only [image_univ.symm] using span_gram_schmidt_normed f univ | lemma | span_gram_schmidt_normed_range | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt",
"gram_schmidt_normed",
"span_gram_schmidt_normed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthonormal_basis : orthonormal_basis ι 𝕜 E | ((gram_schmidt_orthonormal' f).exists_orthonormal_basis_extension_of_card_eq h).some | def | gram_schmidt_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_orthonormal'",
"orthonormal_basis"
] | Given an indexed family `f : ι → E` of vectors in an inner product space `E`, for which the
size of the index set is the dimension of `E`, produce an orthonormal basis for `E` which agrees
with the orthonormal set produced by the Gram-Schmidt orthonormalization process on the elements of
`ι` for which this process give... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_orthonormal_basis_apply {f : ι → E} {i : ι}
(hi : gram_schmidt_normed 𝕜 f i ≠ 0) :
gram_schmidt_orthonormal_basis h f i = gram_schmidt_normed 𝕜 f i | ((gram_schmidt_orthonormal' f).exists_orthonormal_basis_extension_of_card_eq h).some_spec i hi | lemma | gram_schmidt_orthonormal_basis_apply | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed",
"gram_schmidt_orthonormal'",
"gram_schmidt_orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthonormal_basis_apply_of_orthogonal {f : ι → E}
(hf : pairwise (λ i j, ⟪f i, f j⟫ = 0)) {i : ι} (hi : f i ≠ 0) :
gram_schmidt_orthonormal_basis h f i = (‖f i‖⁻¹ : 𝕜) • f i | begin
have H : gram_schmidt_normed 𝕜 f i = (‖f i‖⁻¹ : 𝕜) • f i,
{ rw [gram_schmidt_normed, gram_schmidt_of_orthogonal 𝕜 hf] },
rw [gram_schmidt_orthonormal_basis_apply h, H],
simpa [H] using hi,
end | lemma | gram_schmidt_orthonormal_basis_apply_of_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed",
"gram_schmidt_of_orthogonal",
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_apply",
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_gram_schmidt_orthonormal_basis_eq_zero {f : ι → E} {i : ι}
(hi : gram_schmidt_normed 𝕜 f i = 0) (j : ι) :
⟪gram_schmidt_orthonormal_basis h f i, f j⟫ = 0 | begin
rw ←mem_orthogonal_singleton_iff_inner_right,
suffices : span 𝕜 (gram_schmidt_normed 𝕜 f '' Iic j) ⟂ 𝕜 ∙ gram_schmidt_orthonormal_basis h f i,
{ apply this,
rw span_gram_schmidt_normed,
exact mem_span_gram_schmidt 𝕜 f le_rfl },
rw is_ortho_span,
rintros u ⟨k, hk, rfl⟩ v (rfl : v = _),
by_c... | lemma | inner_gram_schmidt_orthonormal_basis_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_normed",
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_apply",
"inner_zero_left",
"le_rfl",
"mem_span_gram_schmidt",
"span_gram_schmidt_normed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthonormal_basis_inv_triangular {i j : ι} (hij : i < j) :
⟪gram_schmidt_orthonormal_basis h f j, f i⟫ = 0 | begin
by_cases hi : gram_schmidt_normed 𝕜 f j = 0,
{ rw inner_gram_schmidt_orthonormal_basis_eq_zero h hi },
{ simp [gram_schmidt_orthonormal_basis_apply h hi, gram_schmidt_normed, inner_smul_left,
gram_schmidt_inv_triangular 𝕜 f hij] }
end | lemma | gram_schmidt_orthonormal_basis_inv_triangular | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_inv_triangular",
"gram_schmidt_normed",
"gram_schmidt_orthonormal_basis_apply",
"inner_gram_schmidt_orthonormal_basis_eq_zero",
"inner_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthonormal_basis_inv_triangular' {i j : ι} (hij : i < j) :
(gram_schmidt_orthonormal_basis h f).repr (f i) j = 0 | by simpa [orthonormal_basis.repr_apply_apply]
using gram_schmidt_orthonormal_basis_inv_triangular h f hij | lemma | gram_schmidt_orthonormal_basis_inv_triangular' | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_inv_triangular",
"orthonormal_basis.repr_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gram_schmidt_orthonormal_basis_inv_block_triangular :
((gram_schmidt_orthonormal_basis h f).to_basis.to_matrix f).block_triangular id | λ i j, gram_schmidt_orthonormal_basis_inv_triangular' h f | lemma | gram_schmidt_orthonormal_basis_inv_block_triangular | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_inv_triangular'"
] | Given an indexed family `f : ι → E` of vectors in an inner product space `E`, for which the
size of the index set is the dimension of `E`, the matrix of coefficients of `f` with respect to the
orthonormal basis `gram_schmidt_orthonormal_basis` constructed from `f` is upper-triangular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gram_schmidt_orthonormal_basis_det :
(gram_schmidt_orthonormal_basis h f).to_basis.det f =
∏ i, ⟪gram_schmidt_orthonormal_basis h f i, f i⟫ | begin
convert matrix.det_of_upper_triangular (gram_schmidt_orthonormal_basis_inv_block_triangular h f),
ext i,
exact ((gram_schmidt_orthonormal_basis h f).repr_apply_apply (f i) i).symm,
end | lemma | gram_schmidt_orthonormal_basis_det | analysis.inner_product_space | src/analysis/inner_product_space/gram_schmidt_ortho.lean | [
"analysis.inner_product_space.pi_L2",
"linear_algebra.matrix.block"
] | [
"gram_schmidt_orthonormal_basis",
"gram_schmidt_orthonormal_basis_inv_block_triangular",
"matrix.det_of_upper_triangular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_inner (f g : lp G 2) : summable (λ i, ⟪f i, g i⟫) | begin
-- Apply the Direct Comparison Test, comparing with ∑' i, ‖f i‖ * ‖g i‖ (summable by Hölder)
refine summable_of_norm_bounded (λ i, ‖f i‖ * ‖g i‖) (lp.summable_mul _ f g) _,
{ rw real.is_conjugate_exponent_iff; norm_num },
intros i,
-- Then apply Cauchy-Schwarz pointwise
exact norm_inner_le_norm _ _,
e... | lemma | lp.summable_inner | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"lp",
"lp.summable_mul",
"norm_inner_le_norm",
"real.is_conjugate_exponent_iff",
"summable",
"summable_of_norm_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_eq_tsum (f g : lp G 2) : ⟪f, g⟫ = ∑' i, ⟪f i, g i⟫ | rfl | lemma | lp.inner_eq_tsum | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_inner (f g : lp G 2) : has_sum (λ i, ⟪f i, g i⟫) ⟪f, g⟫ | (summable_inner f g).has_sum | lemma | lp.has_sum_inner | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"has_sum",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_single_left (i : ι) (a : G i) (f : lp G 2) : ⟪lp.single 2 i a, f⟫ = ⟪a, f i⟫ | begin
refine (has_sum_inner (lp.single 2 i a) f).unique _,
convert has_sum_ite_eq i ⟪a, f i⟫,
ext j,
rw lp.single_apply,
split_ifs,
{ subst h },
{ simp }
end | lemma | lp.inner_single_left | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"has_sum_ite_eq",
"lp",
"lp.single",
"lp.single_apply",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_single_right (i : ι) (a : G i) (f : lp G 2) : ⟪f, lp.single 2 i a⟫ = ⟪f i, a⟫ | by simpa [inner_conj_symm] using congr_arg conj (@inner_single_left _ 𝕜 _ _ _ _ i a f) | lemma | lp.inner_single_right | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"inner_conj_symm",
"lp",
"lp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_lp (f : lp G 2) : summable (λ i, V i (f i)) | begin
rw hV.summable_iff_norm_sq_summable,
convert (lp.mem_ℓp f).summable _,
{ norm_cast },
{ norm_num }
end | lemma | orthogonal_family.summable_of_lp | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"lp",
"lp.mem_ℓp",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry : lp G 2 →ₗᵢ[𝕜] E | { to_fun := λ f, ∑' i, V i (f i),
map_add' := λ f g, by simp only [tsum_add (hV.summable_of_lp f) (hV.summable_of_lp g),
lp.coe_fn_add, pi.add_apply, linear_isometry.map_add],
map_smul' := λ c f, by simpa only [linear_isometry.map_smul, pi.smul_apply, lp.coe_fn_smul]
using tsum_const_smul c (hV.summable_of_... | def | orthogonal_family.linear_isometry | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"linear_isometry",
"linear_isometry.map_add",
"linear_isometry.map_smul",
"lp",
"lp.coe_fn_add",
"lp.coe_fn_smul",
"lp.has_sum_norm",
"pi.smul_apply",
"real.rpow_left_inj_on",
"tendsto_nhds_unique",
"tsum_add",
"tsum_const_smul"
] | A mutually orthogonal family of subspaces of `E` induce a linear isometry from `lp 2` of the
subspaces into `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_apply (f : lp G 2) :
hV.linear_isometry f = ∑' i, V i (f i) | rfl | lemma | orthogonal_family.linear_isometry_apply | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_linear_isometry (f : lp G 2) :
has_sum (λ i, V i (f i)) (hV.linear_isometry f) | (hV.summable_of_lp f).has_sum | lemma | orthogonal_family.has_sum_linear_isometry | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"has_sum",
"lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_apply_single {i : ι} (x : G i) :
hV.linear_isometry (lp.single 2 i x) = V i x | begin
rw [hV.linear_isometry_apply, ← tsum_ite_eq i (V i x)],
congr,
ext j,
rw [lp.single_apply],
split_ifs,
{ subst h },
{ simp }
end | lemma | orthogonal_family.linear_isometry_apply_single | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"lp.single",
"lp.single_apply",
"tsum_ite_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_apply_dfinsupp_sum_single (W₀ : Π₀ (i : ι), G i) :
hV.linear_isometry (W₀.sum (lp.single 2)) = W₀.sum (λ i, V i) | begin
have : hV.linear_isometry (∑ i in W₀.support, lp.single 2 i (W₀ i))
= ∑ i in W₀.support, hV.linear_isometry (lp.single 2 i (W₀ i)),
{ exact hV.linear_isometry.to_linear_map.map_sum },
simp [dfinsupp.sum, this] {contextual := tt},
end | lemma | orthogonal_family.linear_isometry_apply_dfinsupp_sum_single | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"lp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_linear_isometry [Π i, complete_space (G i)] :
hV.linear_isometry.to_linear_map.range = (⨆ i, (V i).to_linear_map.range).topological_closure | begin
refine le_antisymm _ _,
{ rintros x ⟨f, rfl⟩,
refine mem_closure_of_tendsto (hV.has_sum_linear_isometry f) (eventually_of_forall _),
intros s,
rw set_like.mem_coe,
refine sum_mem _,
intros i hi,
refine mem_supr_of_mem i _,
exact linear_map.mem_range_self _ (f i) },
{ apply topolo... | lemma | orthogonal_family.range_linear_isometry | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"linear_map.mem_range_self",
"lp.single",
"mem_closure_of_tendsto",
"set_like.mem_coe",
"supr_le"
] | The canonical linear isometry from the `lp 2` of a mutually orthogonal family of subspaces of
`E` into E, has range the closure of the span of the subspaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum : Prop | of_surjective ::
(orthogonal_family : orthogonal_family 𝕜 G V)
(surjective_isometry : function.surjective (orthogonal_family.linear_isometry)) | structure | is_hilbert_sum | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"orthogonal_family",
"orthogonal_family.linear_isometry"
] | Given a family of Hilbert spaces `G : ι → Type*`, a Hilbert sum of `G` consists of a Hilbert
space `E` and an orthogonal family `V : Π i, G i →ₗᵢ[𝕜] E` such that the induced isometry
`Φ : lp G 2 → E` is surjective.
Keeping in mind that `lp G 2` is "the" external Hilbert sum of `G : ι → Type*`, this is analogous
to `d... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.mk [Π i, complete_space $ G i]
(hVortho : orthogonal_family 𝕜 G V)
(hVtotal : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) :
is_hilbert_sum 𝕜 G V | { orthogonal_family := hVortho,
surjective_isometry :=
begin
rw [←linear_isometry.coe_to_linear_map],
exact linear_map.range_eq_top.mp (eq_top_iff.mpr $
hVtotal.trans_eq hVortho.range_linear_isometry.symm)
end } | lemma | is_hilbert_sum.mk | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"is_hilbert_sum",
"orthogonal_family"
] | If `V : Π i, G i →ₗᵢ[𝕜] E` is an orthogonal family such that the supremum of the ranges of
`V i` is dense, then `(E, V)` is a Hilbert sum of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.mk_internal [Π i, complete_space $ F i]
(hFortho : orthogonal_family 𝕜 (λ i, F i) (λ i, (F i).subtypeₗᵢ))
(hFtotal : ⊤ ≤ (⨆ i, (F i)).topological_closure) :
is_hilbert_sum 𝕜 (λ i, F i) (λ i, (F i).subtypeₗᵢ) | is_hilbert_sum.mk hFortho (by simpa [subtypeₗᵢ_to_linear_map, range_subtype] using hFtotal) | lemma | is_hilbert_sum.mk_internal | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"is_hilbert_sum",
"is_hilbert_sum.mk",
"orthogonal_family"
] | This is `orthogonal_family.is_hilbert_sum` in the case of actual inclusions from subspaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.linear_isometry_equiv (hV : is_hilbert_sum 𝕜 G V) :
E ≃ₗᵢ[𝕜] lp G 2 | linear_isometry_equiv.symm $
linear_isometry_equiv.of_surjective
hV.orthogonal_family.linear_isometry hV.surjective_isometry | def | is_hilbert_sum.linear_isometry_equiv | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum",
"linear_isometry_equiv.of_surjective",
"linear_isometry_equiv.symm",
"lp"
] | *A* Hilbert sum `(E, V)` of `G` is canonically isomorphic to *the* Hilbert sum of `G`,
i.e `lp G 2`.
Note that this goes in the opposite direction from `orthogonal_family.linear_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.linear_isometry_equiv_symm_apply
(hV : is_hilbert_sum 𝕜 G V) (w : lp G 2) :
hV.linear_isometry_equiv.symm w = ∑' i, V i (w i) | by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.linear_isometry_apply] | lemma | is_hilbert_sum.linear_isometry_equiv_symm_apply | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum",
"is_hilbert_sum.linear_isometry_equiv",
"lp",
"orthogonal_family.linear_isometry_apply"
] | In the canonical isometric isomorphism between a Hilbert sum `E` of `G` and `lp G 2`,
a vector `w : lp G 2` is the image of the infinite sum of the associated elements in `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.has_sum_linear_isometry_equiv_symm
(hV : is_hilbert_sum 𝕜 G V) (w : lp G 2) :
has_sum (λ i, V i (w i)) (hV.linear_isometry_equiv.symm w) | by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.has_sum_linear_isometry] | lemma | is_hilbert_sum.has_sum_linear_isometry_equiv_symm | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"has_sum",
"is_hilbert_sum",
"is_hilbert_sum.linear_isometry_equiv",
"lp",
"orthogonal_family.has_sum_linear_isometry"
] | In the canonical isometric isomorphism between a Hilbert sum `E` of `G` and `lp G 2`,
a vector `w : lp G 2` is the image of the infinite sum of the associated elements in `E`, and this
sum indeed converges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.linear_isometry_equiv_symm_apply_single
(hV : is_hilbert_sum 𝕜 G V) {i : ι} (x : G i) :
hV.linear_isometry_equiv.symm (lp.single 2 i x) = V i x | by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.linear_isometry_apply_single] | lemma | is_hilbert_sum.linear_isometry_equiv_symm_apply_single | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum",
"is_hilbert_sum.linear_isometry_equiv",
"lp.single",
"orthogonal_family.linear_isometry_apply_single"
] | In the canonical isometric isomorphism between a Hilbert sum `E` of `G : ι → Type*` and
`lp G 2`, an "elementary basis vector" in `lp G 2` supported at `i : ι` is the image of the
associated element in `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.linear_isometry_equiv_symm_apply_dfinsupp_sum_single
(hV : is_hilbert_sum 𝕜 G V) (W₀ : Π₀ (i : ι), G i) :
hV.linear_isometry_equiv.symm (W₀.sum (lp.single 2)) = (W₀.sum (λ i, V i)) | by simp [is_hilbert_sum.linear_isometry_equiv,
orthogonal_family.linear_isometry_apply_dfinsupp_sum_single] | lemma | is_hilbert_sum.linear_isometry_equiv_symm_apply_dfinsupp_sum_single | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum",
"is_hilbert_sum.linear_isometry_equiv",
"lp.single",
"orthogonal_family.linear_isometry_apply_dfinsupp_sum_single"
] | In the canonical isometric isomorphism between a Hilbert sum `E` of `G : ι → Type*` and
`lp G 2`, a finitely-supported vector in `lp G 2` is the image of the associated finite sum of
elements of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_hilbert_sum.linear_isometry_equiv_apply_dfinsupp_sum_single
(hV : is_hilbert_sum 𝕜 G V) (W₀ : Π₀ (i : ι), G i) :
(hV.linear_isometry_equiv (W₀.sum (λ i, V i)) : Π i, G i) = W₀ | begin
rw ← hV.linear_isometry_equiv_symm_apply_dfinsupp_sum_single,
rw linear_isometry_equiv.apply_symm_apply,
ext i,
simp [dfinsupp.sum, lp.single_apply] {contextual := tt},
end | lemma | is_hilbert_sum.linear_isometry_equiv_apply_dfinsupp_sum_single | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum",
"linear_isometry_equiv.apply_symm_apply",
"lp.single_apply"
] | In the canonical isometric isomorphism between a Hilbert sum `E` of `G : ι → Type*` and
`lp G 2`, a finitely-supported vector in `lp G 2` is the image of the associated finite sum of
elements of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal.is_hilbert_sum {v : ι → E} (hv : orthonormal 𝕜 v)
(hsp : ⊤ ≤ (span 𝕜 (set.range v)).topological_closure) :
is_hilbert_sum 𝕜 (λ i : ι, 𝕜) (λ i, linear_isometry.to_span_singleton 𝕜 E (hv.1 i)) | is_hilbert_sum.mk hv.orthogonal_family
begin
convert hsp,
simp [← linear_map.span_singleton_eq_range, ← submodule.span_Union],
end | lemma | orthonormal.is_hilbert_sum | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum",
"is_hilbert_sum.mk",
"linear_isometry.to_span_singleton",
"linear_map.span_singleton_eq_range",
"orthonormal",
"set.range",
"submodule.span_Union"
] | Given a total orthonormal family `v : ι → E`, `E` is a Hilbert sum of `λ i : ι, 𝕜` relative to
the family of linear isometries `λ i, λ k, k • v i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.is_hilbert_sum_orthogonal (K : submodule 𝕜 E) [hK : complete_space K] :
is_hilbert_sum 𝕜 (λ b, ↥(cond b K Kᗮ)) (λ b, (cond b K Kᗮ).subtypeₗᵢ) | begin
haveI : Π b, complete_space ↥(cond b K Kᗮ),
{ intro b,
cases b;
exact orthogonal.complete_space K <|> assumption },
refine is_hilbert_sum.mk_internal _ K.orthogonal_family_self _,
refine le_trans _ (submodule.le_topological_closure _),
rw [supr_bool_eq, cond, cond],
refine codisjoint.top_le _,... | lemma | submodule.is_hilbert_sum_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"codisjoint.top_le",
"complete_space",
"is_hilbert_sum",
"is_hilbert_sum.mk_internal",
"submodule",
"submodule.le_topological_closure",
"supr_bool_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hilbert_basis | of_repr :: (repr : E ≃ₗᵢ[𝕜] ℓ²(ι, 𝕜)) | structure | hilbert_basis | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [] | A Hilbert basis on `ι` for an inner product space `E` is an identification of `E` with the `lp`
space `ℓ²(ι, 𝕜)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_symm_single (b : hilbert_basis ι 𝕜 E) (i : ι) :
b.repr.symm (lp.single 2 i (1:𝕜)) = b i | rfl | lemma | hilbert_basis.repr_symm_single | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"lp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_self (b : hilbert_basis ι 𝕜 E) (i : ι) :
b.repr (b i) = lp.single 2 i (1:𝕜) | by rw [← b.repr_symm_single, linear_isometry_equiv.apply_symm_apply] | lemma | hilbert_basis.repr_self | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"linear_isometry_equiv.apply_symm_apply",
"lp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_apply_apply (b : hilbert_basis ι 𝕜 E) (v : E) (i : ι) :
b.repr v i = ⟪b i, v⟫ | begin
rw [← b.repr.inner_map_map (b i) v, b.repr_self, lp.inner_single_left],
simp,
end | lemma | hilbert_basis.repr_apply_apply | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"lp.inner_single_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthonormal (b : hilbert_basis ι 𝕜 E) : orthonormal 𝕜 b | begin
rw orthonormal_iff_ite,
intros i j,
rw [← b.repr.inner_map_map (b i) (b j), b.repr_self, b.repr_self, lp.inner_single_left,
lp.single_apply],
simp,
end | lemma | hilbert_basis.orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"lp.inner_single_left",
"lp.single_apply",
"orthonormal",
"orthonormal_iff_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_repr_symm (b : hilbert_basis ι 𝕜 E) (f : ℓ²(ι, 𝕜)) :
has_sum (λ i, f i • b i) (b.repr.symm f) | begin
suffices H : (λ (i : ι), f i • b i) =
(λ (b_1 : ι), (b.repr.symm.to_continuous_linear_equiv) ((λ (i : ι), lp.single 2 i (f i)) b_1)),
{ rw H,
have : has_sum (λ (i : ι), lp.single 2 i (f i)) f := lp.has_sum_single ennreal.two_ne_top f,
exact (↑(b.repr.symm.to_continuous_linear_equiv) : ℓ²(ι, 𝕜) →L... | lemma | hilbert_basis.has_sum_repr_symm | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"ennreal.two_ne_top",
"has_sum",
"hilbert_basis",
"linear_isometry_equiv.coe_to_continuous_linear_equiv",
"linear_isometry_equiv.map_smul",
"lp",
"lp.has_sum_single",
"lp.single",
"lp.single_smul",
"mul_one",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_repr (b : hilbert_basis ι 𝕜 E) (x : E) :
has_sum (λ i, b.repr x i • b i) x | by simpa using b.has_sum_repr_symm (b.repr x) | lemma | hilbert_basis.has_sum_repr | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"has_sum",
"hilbert_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_span (b : hilbert_basis ι 𝕜 E) :
(span 𝕜 (set.range b)).topological_closure = ⊤ | begin
classical,
rw eq_top_iff,
rintros x -,
refine mem_closure_of_tendsto (b.has_sum_repr x) (eventually_of_forall _),
intros s,
simp only [set_like.mem_coe],
refine sum_mem _,
rintros i -,
refine smul_mem _ _ _,
exact subset_span ⟨i, rfl⟩
end | lemma | hilbert_basis.dense_span | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"eq_top_iff",
"hilbert_basis",
"mem_closure_of_tendsto",
"set.range",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_inner_mul_inner (b : hilbert_basis ι 𝕜 E) (x y : E) :
has_sum (λ i, ⟪x, b i⟫ * ⟪b i, y⟫) ⟪x, y⟫ | begin
convert (b.has_sum_repr y).mapL (innerSL _ x),
ext i,
rw [innerSL_apply, b.repr_apply_apply, inner_smul_right, mul_comm]
end | lemma | hilbert_basis.has_sum_inner_mul_inner | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"has_sum",
"hilbert_basis",
"innerSL",
"innerSL_apply",
"inner_smul_right",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_inner_mul_inner (b : hilbert_basis ι 𝕜 E) (x y : E) :
summable (λ i, ⟪x, b i⟫ * ⟪b i, y⟫) | (b.has_sum_inner_mul_inner x y).summable | lemma | hilbert_basis.summable_inner_mul_inner | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_inner_mul_inner (b : hilbert_basis ι 𝕜 E) (x y : E) :
∑' i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫ | (b.has_sum_inner_mul_inner x y).tsum_eq | lemma | hilbert_basis.tsum_inner_mul_inner | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_orthonormal_basis [fintype ι] (b : hilbert_basis ι 𝕜 E) :
orthonormal_basis ι 𝕜 E | orthonormal_basis.mk b.orthonormal
begin
refine eq.ge _,
have := (span 𝕜 (finset.univ.image b : set E)).closed_of_finite_dimensional,
simpa only [finset.coe_image, finset.coe_univ, set.image_univ, hilbert_basis.dense_span] using
this.submodule_topological_closure_eq.symm
end | def | hilbert_basis.to_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"eq.ge",
"finset.coe_image",
"finset.coe_univ",
"fintype",
"hilbert_basis",
"hilbert_basis.dense_span",
"orthonormal_basis",
"orthonormal_basis.mk",
"set.image_univ"
] | A finite Hilbert basis is an orthonormal basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_orthonormal_basis [fintype ι] (b : hilbert_basis ι 𝕜 E) :
(b.to_orthonormal_basis : ι → E) = b | orthonormal_basis.coe_mk _ _ | lemma | hilbert_basis.coe_to_orthonormal_basis | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"fintype",
"hilbert_basis",
"orthonormal_basis.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_orthogonal_projection {U : submodule 𝕜 E}
[complete_space U] (b : hilbert_basis ι 𝕜 U) (x : E) :
has_sum (λ i, ⟪(b i : E), x⟫ • b i) (orthogonal_projection U x) | by simpa only [b.repr_apply_apply, inner_orthogonal_projection_eq_of_mem_left]
using b.has_sum_repr (orthogonal_projection U x) | lemma | hilbert_basis.has_sum_orthogonal_projection | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"complete_space",
"has_sum",
"hilbert_basis",
"inner_orthogonal_projection_eq_of_mem_left",
"orthogonal_projection",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_spans_dense (b : hilbert_basis ι 𝕜 E) :
(⨆ J : finset ι, span 𝕜 (J.image b : set E)).topological_closure = ⊤ | eq_top_iff.mpr $ b.dense_span.ge.trans
begin
simp_rw [← submodule.span_Union],
exact topological_closure_mono (span_mono $ set.range_subset_iff.mpr $
λ i, set.mem_Union_of_mem {i} $ finset.mem_coe.mpr $ finset.mem_image_of_mem _ $
finset.mem_singleton_self i)
end | lemma | hilbert_basis.finite_spans_dense | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"finset",
"finset.mem_image_of_mem",
"finset.mem_singleton_self",
"hilbert_basis",
"set.mem_Union_of_mem",
"submodule.span_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (hsp : ⊤ ≤ (span 𝕜 (set.range v)).topological_closure) :
hilbert_basis ι 𝕜 E | hilbert_basis.of_repr $
(hv.is_hilbert_sum hsp).linear_isometry_equiv | def | hilbert_basis.mk | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"linear_isometry_equiv",
"set.range"
] | An orthonormal family of vectors whose span is dense in the whole module is a Hilbert basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.orthonormal.linear_isometry_equiv_symm_apply_single_one (h i) :
(hv.is_hilbert_sum h).linear_isometry_equiv.symm (lp.single 2 i 1) = v i | by rw [is_hilbert_sum.linear_isometry_equiv_symm_apply_single,
linear_isometry.to_span_singleton_apply, one_smul] | lemma | orthonormal.linear_isometry_equiv_symm_apply_single_one | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"is_hilbert_sum.linear_isometry_equiv_symm_apply_single",
"linear_isometry.to_span_singleton_apply",
"linear_isometry_equiv.symm",
"lp.single",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (hsp : ⊤ ≤ (span 𝕜 (set.range v)).topological_closure) :
⇑(hilbert_basis.mk hv hsp) = v | by apply (funext $ orthonormal.linear_isometry_equiv_symm_apply_single_one hv hsp) | lemma | hilbert_basis.coe_mk | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis.mk",
"orthonormal.linear_isometry_equiv_symm_apply_single_one",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_orthogonal_eq_bot (hsp : (span 𝕜 (set.range v))ᗮ = ⊥) : hilbert_basis ι 𝕜 E | hilbert_basis.mk hv
(by rw [← orthogonal_orthogonal_eq_closure, ← eq_top_iff, orthogonal_eq_top_iff, hsp]) | def | hilbert_basis.mk_of_orthogonal_eq_bot | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"eq_top_iff",
"hilbert_basis",
"hilbert_basis.mk",
"mk_of_orthogonal_eq_bot",
"set.range"
] | An orthonormal family of vectors whose span has trivial orthogonal complement is a Hilbert
basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_orthogonal_eq_bot_mk (hsp : (span 𝕜 (set.range v))ᗮ = ⊥) :
⇑(hilbert_basis.mk_of_orthogonal_eq_bot hv hsp) = v | hilbert_basis.coe_mk hv _ | lemma | hilbert_basis.coe_of_orthogonal_eq_bot_mk | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"coe_of_orthogonal_eq_bot_mk",
"hilbert_basis.coe_mk",
"hilbert_basis.mk_of_orthogonal_eq_bot",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.orthonormal_basis.to_hilbert_basis [fintype ι] (b : orthonormal_basis ι 𝕜 E) :
hilbert_basis ι 𝕜 E | hilbert_basis.mk b.orthonormal $
by simpa only [← orthonormal_basis.coe_to_basis, b.to_basis.span_eq, eq_top_iff]
using @subset_closure E _ _ | def | orthonormal_basis.to_hilbert_basis | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"eq_top_iff",
"fintype",
"hilbert_basis",
"hilbert_basis.mk",
"orthonormal_basis",
"orthonormal_basis.coe_to_basis",
"subset_closure"
] | An orthonormal basis is an Hilbert basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.orthonormal_basis.coe_to_hilbert_basis [fintype ι]
(b : orthonormal_basis ι 𝕜 E) : (b.to_hilbert_basis : ι → E) = b | hilbert_basis.coe_mk _ _ | lemma | orthonormal_basis.coe_to_hilbert_basis | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"fintype",
"hilbert_basis.coe_mk",
"orthonormal_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.orthonormal.exists_hilbert_basis_extension
{s : set E} (hs : orthonormal 𝕜 (coe : s → E)) :
∃ (w : set E) (b : hilbert_basis w 𝕜 E), s ⊆ w ∧ ⇑b = (coe : w → E) | let ⟨w, hws, hw_ortho, hw_max⟩ := exists_maximal_orthonormal hs in
⟨ w,
hilbert_basis.mk_of_orthogonal_eq_bot hw_ortho
(by simpa [maximal_orthonormal_iff_orthogonal_complement_eq_bot hw_ortho] using hw_max),
hws,
hilbert_basis.coe_of_orthogonal_eq_bot_mk _ _ ⟩ | lemma | orthonormal.exists_hilbert_basis_extension | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"exists_maximal_orthonormal",
"hilbert_basis",
"hilbert_basis.coe_of_orthogonal_eq_bot_mk",
"hilbert_basis.mk_of_orthogonal_eq_bot",
"maximal_orthonormal_iff_orthogonal_complement_eq_bot",
"orthonormal"
] | A Hilbert space admits a Hilbert basis extending a given orthonormal subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.exists_hilbert_basis :
∃ (w : set E) (b : hilbert_basis w 𝕜 E), ⇑b = (coe : w → E) | let ⟨w, hw, hw', hw''⟩ := (orthonormal_empty 𝕜 E).exists_hilbert_basis_extension in ⟨w, hw, hw''⟩ | lemma | exists_hilbert_basis | analysis.inner_product_space | src/analysis/inner_product_space/l2_space.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.lp_space",
"analysis.inner_product_space.pi_L2"
] | [
"hilbert_basis",
"orthonormal_empty"
] | A Hilbert space admits a Hilbert basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_below (coercive : is_coercive B) :
∃ C, 0 < C ∧ ∀ v, C * ‖v‖ ≤ ‖B♯ v‖ | begin
rcases coercive with ⟨C, C_ge_0, coercivity⟩,
refine ⟨C, C_ge_0, _⟩,
intro v,
by_cases h : 0 < ‖v‖,
{ refine (mul_le_mul_right h).mp _,
calc C * ‖v‖ * ‖v‖
≤ B v v : coercivity v
... = ⟪B♯ v, v⟫_ℝ : (continuous_linear_map_of_bilin_apply ℝ B v v).symm
... ≤ ‖B♯ v‖ * ‖v‖ : real_inner_le... | lemma | is_coercive.bounded_below | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"is_coercive",
"mul_le_mul_right",
"real_inner_le_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz (coercive : is_coercive B) :
∃ C : ℝ≥0, 0 < C ∧ antilipschitz_with C B♯ | begin
rcases coercive.bounded_below with ⟨C, C_pos, below_bound⟩,
refine ⟨(C⁻¹).to_nnreal, real.to_nnreal_pos.mpr (inv_pos.mpr C_pos), _⟩,
refine continuous_linear_map.antilipschitz_of_bound B♯ _,
simp_rw [real.coe_to_nnreal',
max_eq_left_of_lt (inv_pos.mpr C_pos),
←inv_mul_le_iff (inv_pos.mpr C_pos)],
... | lemma | is_coercive.antilipschitz | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"antilipschitz_with",
"continuous_linear_map.antilipschitz_of_bound",
"is_coercive",
"real.coe_to_nnreal'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_bot (coercive : is_coercive B) : ker B♯ = ⊥ | begin
rw [linear_map_class.ker_eq_bot],
rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩,
exact antilipschitz.injective,
end | lemma | is_coercive.ker_eq_bot | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"is_coercive",
"linear_map_class.ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_range (coercive : is_coercive B) : is_closed (range B♯ : set V) | begin
rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩,
exact antilipschitz.is_closed_range B♯.uniform_continuous,
end | lemma | is_coercive.closed_range | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"is_closed",
"is_coercive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_top (coercive : is_coercive B) : range B♯ = ⊤ | begin
haveI := coercive.closed_range.complete_space_coe,
rw ← (range B♯).orthogonal_orthogonal,
rw submodule.eq_top_iff',
intros v w mem_w_orthogonal,
rcases coercive with ⟨C, C_pos, coercivity⟩,
obtain rfl : w = 0,
{ rw [←norm_eq_zero, ←mul_self_eq_zero, ←mul_right_inj' C_pos.ne', mul_zero, ←mul_assoc],
... | lemma | is_coercive.range_eq_top | analysis.inner_product_space | src/analysis/inner_product_space/lax_milgram.lean | [
"analysis.inner_product_space.projection",
"analysis.inner_product_space.dual",
"analysis.normed_space.banach",
"analysis.normed_space.operator_norm",
"topology.metric_space.antilipschitz"
] | [
"inner_zero_left",
"is_coercive",
"mul_zero",
"submodule.eq_top_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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