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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orthogonal_family.independent {V : ι → submodule 𝕜 E}
(hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) :
complete_lattice.independent V | begin
classical,
apply complete_lattice.independent_of_dfinsupp_lsum_injective,
rw [← @linear_map.ker_eq_bot _ _ _ _ _ _ (direct_sum.add_comm_group (λ i, V i)),
submodule.eq_bot_iff],
intros v hv,
rw linear_map.mem_ker at hv,
ext i,
suffices : ⟪(v i : E), v i⟫ = 0,
{ simpa only [inner_self_eq_zero] ... | lemma | orthogonal_family.independent | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"complete_lattice.independent",
"complete_lattice.independent_of_dfinsupp_lsum_injective",
"dfinsupp.lsum",
"dfinsupp.sum_add_hom_apply",
"inner_self_eq_zero",
"inner_zero_right",
"linear_map.ker_eq_bot",
"linear_map.mem_ker",
"orthogonal_family",
"submodule",
"submodule.eq_bot_iff"
] | An orthogonal family forms an independent family of subspaces; that is, any collection of
elements each from a different subspace in the family is linearly independent. In particular, the
pairwise intersections of elements of the family are 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.is_internal.collected_basis_orthonormal {V : ι → submodule 𝕜 E}
(hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ))
(hV_sum : direct_sum.is_internal (λ i, V i))
{α : ι → Type*}
{v_family : Π i, basis (α i) 𝕜 (V i)} (hv_family : ∀ i, orthonormal 𝕜 (v_family i)) :
orthonormal 𝕜 (hV_sum.... | by simpa only [hV_sum.collected_basis_coe] using hV.orthonormal_sigma_orthonormal hv_family | lemma | direct_sum.is_internal.collected_basis_orthonormal | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"basis",
"direct_sum.is_internal",
"orthogonal_family",
"orthonormal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_inner.is_R_or_C_to_real : has_inner ℝ E | { inner := λ x y, re ⟪x, y⟫ } | def | has_inner.is_R_or_C_to_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_inner"
] | A general inner product implies a real inner product. This is not registered as an instance
since it creates problems with the case `𝕜 = ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_space.is_R_or_C_to_real : inner_product_space ℝ E | { norm_sq_eq_inner := norm_sq_eq_inner,
conj_symm := λ x y, inner_re_symm _ _,
add_left := λ x y z, by
{ change re ⟪x + y, z⟫ = re ⟪x, z⟫ + re ⟪y, z⟫,
simp only [inner_add_left, map_add] },
smul_left := λ x y r, by
{ change re ⟪(r : 𝕜) • x, y⟫ = r * re ⟪x, y⟫,
simp only [inner_smul_left, conj_of_real... | def | inner_product_space.is_R_or_C_to_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_inner.is_R_or_C_to_real",
"inner_add_left",
"inner_product_space",
"inner_re_symm",
"inner_smul_left",
"normed_space.restrict_scalars"
] | A general inner product space structure implies a real inner product structure. This is not
registered as an instance since it creates problems with the case `𝕜 = ℝ`, but in can be used in a
proof to obtain a real inner product space structure from a given `𝕜`-inner product space
structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real_inner_eq_re_inner (x y : E) :
@has_inner.inner ℝ E (has_inner.is_R_or_C_to_real 𝕜 E) x y = re ⟪x, y⟫ | rfl | lemma | real_inner_eq_re_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_inner.is_R_or_C_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_inner_I_smul_self (x : E) :
@has_inner.inner ℝ E (has_inner.is_R_or_C_to_real 𝕜 E) x ((I : 𝕜) • x) = 0 | by simp [real_inner_eq_re_inner, inner_smul_right] | lemma | real_inner_I_smul_self | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"has_inner.is_R_or_C_to_real",
"inner_smul_right",
"real_inner_eq_re_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_product_space.complex_to_real
[normed_add_comm_group G] [inner_product_space ℂ G] : inner_product_space ℝ G | inner_product_space.is_R_or_C_to_real ℂ G | instance | inner_product_space.complex_to_real | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"inner_product_space",
"inner_product_space.is_R_or_C_to_real",
"normed_add_comm_group"
] | A complex inner product implies a real inner product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.inner (w z : ℂ) : ⟪w, z⟫_ℝ = (conj w * z).re | rfl | lemma | complex.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_map_complex [normed_add_comm_group G] [inner_product_space ℝ G]
(f : G ≃ₗᵢ[ℝ] ℂ) (x y : G) :
⟪x, y⟫_ℝ = (conj (f x) * f y).re | by rw [← complex.inner, f.inner_map_map] | lemma | inner_map_complex | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"complex.inner",
"inner_product_space",
"normed_add_comm_group"
] | The inner product on an inner product space of dimension 2 can be evaluated in terms
of a complex-number representation of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_inner : continuous (λ p : E × E, ⟪p.1, p.2⟫) | begin
letI : inner_product_space ℝ E := inner_product_space.is_R_or_C_to_real 𝕜 E,
exact is_bounded_bilinear_map_inner.continuous
end | lemma | continuous_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous",
"inner_product_space",
"inner_product_space.is_R_or_C_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inner {f g : α → E} {l : filter α} {x y : E} (hf : tendsto f l (𝓝 x))
(hg : tendsto g l (𝓝 y)) :
tendsto (λ t, ⟪f t, g t⟫) l (𝓝 ⟪x, y⟫) | (continuous_inner.tendsto _).comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.inner (hf : continuous_within_at f s x)
(hg : continuous_within_at g s x) :
continuous_within_at (λ t, ⟪f t, g t⟫) s x | hf.inner hg | lemma | continuous_within_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.inner (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (λ t, ⟪f t, g t⟫) x | hf.inner hg | lemma | continuous_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.inner (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λ t, ⟪f t, g t⟫) s | λ x hx, (hf x hx).inner (hg x hx) | lemma | continuous_on.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.inner (hf : continuous f) (hg : continuous g) : continuous (λ t, ⟪f t, g t⟫) | continuous_iff_continuous_at.2 $ λ x, hf.continuous_at.inner hg.continuous_at | lemma | continuous.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.re_apply_inner_self (T : E →L[𝕜] E) (x : E) : ℝ | re ⟪T x, x⟫ | def | continuous_linear_map.re_apply_inner_self | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | Extract a real bilinear form from an operator `T`, by taking the pairing `λ x, re ⟪T x, x⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.re_apply_inner_self_apply (T : E →L[𝕜] E) (x : E) :
T.re_apply_inner_self x = re ⟪T x, x⟫ | rfl | lemma | continuous_linear_map.re_apply_inner_self_apply | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.re_apply_inner_self_continuous (T : E →L[𝕜] E) :
continuous T.re_apply_inner_self | re_clm.continuous.comp $ T.continuous.inner continuous_id | lemma | continuous_linear_map.re_apply_inner_self_continuous | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.re_apply_inner_self_smul (T : E →L[𝕜] E) (x : E) {c : 𝕜} :
T.re_apply_inner_self (c • x) = ‖c‖ ^ 2 * T.re_apply_inner_self x | by simp only [continuous_linear_map.map_smul, continuous_linear_map.re_apply_inner_self_apply,
inner_smul_left, inner_smul_right, ← mul_assoc, mul_conj, norm_sq_eq_def', ← smul_re,
algebra.smul_def (‖c‖ ^ 2) ⟪T x, x⟫, algebra_map_eq_of_real] | lemma | continuous_linear_map.re_apply_inner_self_smul | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"algebra.smul_def",
"continuous_linear_map.map_smul",
"continuous_linear_map.re_apply_inner_self_apply",
"inner_smul_left",
"inner_smul_right",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_coe (a b : E) :
inner (a : completion E) (b : completion E) = (inner a b : 𝕜) | (dense_inducing_coe.prod dense_inducing_coe).extend_eq
(continuous_inner : continuous (uncurry inner : E × E → 𝕜)) (a, b) | lemma | uniform_space.completion.inner_coe | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous",
"continuous_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inner :
continuous (uncurry inner : completion E × completion E → 𝕜) | begin
let inner' : E →+ E →+ 𝕜 :=
{ to_fun := λ x, (innerₛₗ 𝕜 x).to_add_monoid_hom,
map_zero' := by ext x; exact inner_zero_left _,
map_add' := λ x y, by ext z; exact inner_add_left _ _ _ },
have : continuous (λ p : E × E, inner' p.1 p.2) := continuous_inner,
rw [completion.has_inner, uncurry_curry _]... | lemma | uniform_space.completion.continuous_inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous",
"continuous_inner",
"extend",
"inner_add_left",
"inner_zero_left",
"innerₛₗ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.inner {α : Type*} [topological_space α]
{f g : α → completion E} (hf : continuous f) (hg : continuous g) :
continuous (λ x : α, inner (f x) (g x) : α → 𝕜) | uniform_space.completion.continuous_inner.comp (hf.prod_mk hg : _) | lemma | uniform_space.completion.continuous.inner | analysis.inner_product_space | src/analysis/inner_product_space/basic.lean | [
"algebra.direct_sum.module",
"analysis.complex.basic",
"analysis.convex.uniform",
"analysis.normed_space.completion",
"analysis.normed_space.bounded_linear_maps",
"linear_algebra.bilinear_form"
] | [
"continuous",
"continuous.inner",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_inner_clm (p : E × E) : E × E →L[ℝ] 𝕜 | is_bounded_bilinear_map_inner.deriv p | def | fderiv_inner_clm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [] | Derivative of the inner product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_inner_clm_apply (p x : E × E) :
fderiv_inner_clm 𝕜 p x = ⟪p.1, x.2⟫ + ⟪x.1, p.2⟫ | rfl | lemma | fderiv_inner_clm_apply | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"fderiv_inner_clm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_inner {n} : cont_diff ℝ n (λ p : E × E, ⟪p.1, p.2⟫) | is_bounded_bilinear_map_inner.cont_diff | lemma | cont_diff_inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_inner {p : E × E} {n} :
cont_diff_at ℝ n (λ p : E × E, ⟪p.1, p.2⟫) p | cont_diff_inner.cont_diff_at | lemma | cont_diff_at_inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_inner : differentiable ℝ (λ p : E × E, ⟪p.1, p.2⟫) | is_bounded_bilinear_map_inner.differentiable_at | lemma | differentiable_inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.inner (hf : cont_diff_within_at ℝ n f s x)
(hg : cont_diff_within_at ℝ n g s x) :
cont_diff_within_at ℝ n (λ x, ⟪f x, g x⟫) s x | cont_diff_at_inner.comp_cont_diff_within_at x (hf.prod hg) | lemma | cont_diff_within_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.inner (hf : cont_diff_at ℝ n f x)
(hg : cont_diff_at ℝ n g x) :
cont_diff_at ℝ n (λ x, ⟪f x, g x⟫) x | hf.inner 𝕜 hg | lemma | cont_diff_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.inner (hf : cont_diff_on ℝ n f s) (hg : cont_diff_on ℝ n g s) :
cont_diff_on ℝ n (λ x, ⟪f x, g x⟫) s | λ x hx, (hf x hx).inner 𝕜 (hg x hx) | lemma | cont_diff_on.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.inner (hf : cont_diff ℝ n f) (hg : cont_diff ℝ n g) :
cont_diff ℝ n (λ x, ⟪f x, g x⟫) | cont_diff_inner.comp (hf.prod hg) | lemma | cont_diff.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.inner (hf : has_fderiv_within_at f f' s x)
(hg : has_fderiv_within_at g g' s x) :
has_fderiv_within_at (λ t, ⟪f t, g t⟫) ((fderiv_inner_clm 𝕜 (f x, g x)).comp $ f'.prod g') s x | (is_bounded_bilinear_map_inner.has_fderiv_at (f x, g x)).comp_has_fderiv_within_at x (hf.prod hg) | lemma | has_fderiv_within_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"fderiv_inner_clm",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.inner (hf : has_strict_fderiv_at f f' x)
(hg : has_strict_fderiv_at g g' x) :
has_strict_fderiv_at (λ t, ⟪f t, g t⟫) ((fderiv_inner_clm 𝕜 (f x, g x)).comp $ f'.prod g') x | (is_bounded_bilinear_map_inner.has_strict_fderiv_at (f x, g x)).comp x (hf.prod hg) | lemma | has_strict_fderiv_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"fderiv_inner_clm",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.inner (hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :
has_fderiv_at (λ t, ⟪f t, g t⟫) ((fderiv_inner_clm 𝕜 (f x, g x)).comp $ f'.prod g') x | (is_bounded_bilinear_map_inner.has_fderiv_at (f x, g x)).comp x (hf.prod hg) | lemma | has_fderiv_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"fderiv_inner_clm",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.inner {f g : ℝ → E} {f' g' : E} {s : set ℝ} {x : ℝ}
(hf : has_deriv_within_at f f' s x) (hg : has_deriv_within_at g g' s x) :
has_deriv_within_at (λ t, ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) s x | by simpa using (hf.has_fderiv_within_at.inner 𝕜 hg.has_fderiv_within_at).has_deriv_within_at | lemma | has_deriv_within_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.inner {f g : ℝ → E} {f' g' : E} {x : ℝ} :
has_deriv_at f f' x → has_deriv_at g g' x →
has_deriv_at (λ t, ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x | by simpa only [← has_deriv_within_at_univ] using has_deriv_within_at.inner 𝕜 | lemma | has_deriv_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"has_deriv_at",
"has_deriv_within_at.inner",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.inner (hf : differentiable_within_at ℝ f s x)
(hg : differentiable_within_at ℝ g s x) :
differentiable_within_at ℝ (λ x, ⟪f x, g x⟫) s x | ((differentiable_inner _).has_fderiv_at.comp_has_fderiv_within_at x
(hf.prod hg).has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_inner",
"differentiable_within_at",
"has_fderiv_at.comp_has_fderiv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.inner (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) :
differentiable_at ℝ (λ x, ⟪f x, g x⟫) x | (differentiable_inner _).comp x (hf.prod hg) | lemma | differentiable_at.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at",
"differentiable_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.inner (hf : differentiable_on ℝ f s) (hg : differentiable_on ℝ g s) :
differentiable_on ℝ (λ x, ⟪f x, g x⟫) s | λ x hx, (hf x hx).inner 𝕜 (hg x hx) | lemma | differentiable_on.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.inner (hf : differentiable ℝ f) (hg : differentiable ℝ g) :
differentiable ℝ (λ x, ⟪f x, g x⟫) | λ x, (hf x).inner 𝕜 (hg x) | lemma | differentiable.inner | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_inner_apply (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) (y : G) :
fderiv ℝ (λ t, ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ | by { rw [(hf.has_fderiv_at.inner 𝕜 hg.has_fderiv_at).fderiv], refl } | lemma | fderiv_inner_apply | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_inner_apply {f g : ℝ → E} {x : ℝ} (hf : differentiable_at ℝ f x)
(hg : differentiable_at ℝ g x) :
deriv (λ t, ⟪f t, g t⟫) x = ⟪f x, deriv g x⟫ + ⟪deriv f x, g x⟫ | (hf.has_deriv_at.inner 𝕜 hg.has_deriv_at).deriv | lemma | deriv_inner_apply | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_norm_sq : cont_diff ℝ n (λ x : E, ‖x‖ ^ 2) | begin
simp only [sq, ← @inner_self_eq_norm_mul_norm 𝕜],
exact (re_clm : 𝕜 →L[ℝ] ℝ).cont_diff.comp (cont_diff_id.inner 𝕜 cont_diff_id)
end | lemma | cont_diff_norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff",
"cont_diff.comp",
"cont_diff_id",
"inner_self_eq_norm_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.norm_sq (hf : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, ‖f x‖ ^ 2) | (cont_diff_norm_sq 𝕜).comp hf | lemma | cont_diff.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff",
"cont_diff_norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.norm_sq (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ y, ‖f y‖ ^ 2) s x | (cont_diff_norm_sq 𝕜).cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at.comp_cont_diff_within_at",
"cont_diff_norm_sq",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.norm_sq (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ y, ‖f y‖ ^ 2) x | hf.norm_sq 𝕜 | lemma | cont_diff_at.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_norm {x : E} (hx : x ≠ 0) : cont_diff_at ℝ n norm x | have ‖id x‖ ^ 2 ≠ 0, from pow_ne_zero _ (norm_pos_iff.2 hx).ne',
by simpa only [id, sqrt_sq, norm_nonneg] using (cont_diff_at_id.norm_sq 𝕜).sqrt this | lemma | cont_diff_at_norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.norm (hf : cont_diff_at ℝ n f x) (h0 : f x ≠ 0) :
cont_diff_at ℝ n (λ y, ‖f y‖) x | (cont_diff_at_norm 𝕜 h0).comp x hf | lemma | cont_diff_at.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at",
"cont_diff_at_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.dist (hf : cont_diff_at ℝ n f x) (hg : cont_diff_at ℝ n g x)
(hne : f x ≠ g x) :
cont_diff_at ℝ n (λ y, dist (f y) (g y)) x | by { simp only [dist_eq_norm], exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) } | lemma | cont_diff_at.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.norm (hf : cont_diff_within_at ℝ n f s x) (h0 : f x ≠ 0) :
cont_diff_within_at ℝ n (λ y, ‖f y‖) s x | (cont_diff_at_norm 𝕜 h0).comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at_norm",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.dist (hf : cont_diff_within_at ℝ n f s x)
(hg : cont_diff_within_at ℝ n g s x) (hne : f x ≠ g x) :
cont_diff_within_at ℝ n (λ y, dist (f y) (g y)) s x | by { simp only [dist_eq_norm], exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) } | lemma | cont_diff_within_at.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.norm_sq (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ y, ‖f y‖ ^ 2) s | (λ x hx, (hf x hx).norm_sq 𝕜) | lemma | cont_diff_on.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.norm (hf : cont_diff_on ℝ n f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
cont_diff_on ℝ n (λ y, ‖f y‖) s | λ x hx, (hf x hx).norm 𝕜 (h0 x hx) | lemma | cont_diff_on.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.dist (hf : cont_diff_on ℝ n f s)
(hg : cont_diff_on ℝ n g s) (hne : ∀ x ∈ s, f x ≠ g x) :
cont_diff_on ℝ n (λ y, dist (f y) (g y)) s | λ x hx, (hf x hx).dist 𝕜 (hg x hx) (hne x hx) | lemma | cont_diff_on.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.norm (hf : cont_diff ℝ n f) (h0 : ∀ x, f x ≠ 0) :
cont_diff ℝ n (λ y, ‖f y‖) | cont_diff_iff_cont_diff_at.2 $ λ x, hf.cont_diff_at.norm 𝕜 (h0 x) | lemma | cont_diff.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.dist (hf : cont_diff ℝ n f) (hg : cont_diff ℝ n g)
(hne : ∀ x, f x ≠ g x) :
cont_diff ℝ n (λ y, dist (f y) (g y)) | cont_diff_iff_cont_diff_at.2 $
λ x, hf.cont_diff_at.dist 𝕜 hg.cont_diff_at (hne x) | lemma | cont_diff.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_norm_sq (x : F) :
has_strict_fderiv_at (λ x, ‖x‖ ^ 2) (bit0 (innerSL ℝ x)) x | begin
simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ],
convert (has_strict_fderiv_at_id x).inner ℝ (has_strict_fderiv_at_id x),
ext y,
simp [bit0, real_inner_comm],
end | lemma | has_strict_fderiv_at_norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at_id",
"innerSL",
"inner_self_eq_norm_mul_norm",
"real_inner_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.norm_sq (hf : differentiable_at ℝ f x) :
differentiable_at ℝ (λ y, ‖f y‖ ^ 2) x | ((cont_diff_at_id.norm_sq 𝕜).differentiable_at le_rfl).comp x hf | lemma | differentiable_at.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.norm (hf : differentiable_at ℝ f x) (h0 : f x ≠ 0) :
differentiable_at ℝ (λ y, ‖f y‖) x | ((cont_diff_at_norm 𝕜 h0).differentiable_at le_rfl).comp x hf | lemma | differentiable_at.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at_norm",
"differentiable_at",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.dist (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x)
(hne : f x ≠ g x) :
differentiable_at ℝ (λ y, dist (f y) (g y)) x | by { simp only [dist_eq_norm], exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) } | lemma | differentiable_at.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.norm_sq (hf : differentiable ℝ f) : differentiable ℝ (λ y, ‖f y‖ ^ 2) | λ x, (hf x).norm_sq 𝕜 | lemma | differentiable.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.norm (hf : differentiable ℝ f) (h0 : ∀ x, f x ≠ 0) :
differentiable ℝ (λ y, ‖f y‖) | λ x, (hf x).norm 𝕜 (h0 x) | lemma | differentiable.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.dist (hf : differentiable ℝ f) (hg : differentiable ℝ g)
(hne : ∀ x, f x ≠ g x) :
differentiable ℝ (λ y, dist (f y) (g y)) | λ x, (hf x).dist 𝕜 (hg x) (hne x) | lemma | differentiable.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.norm_sq (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ y, ‖f y‖ ^ 2) s x | ((cont_diff_at_id.norm_sq 𝕜).differentiable_at le_rfl).comp_differentiable_within_at x hf | lemma | differentiable_within_at.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at",
"differentiable_within_at",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.norm (hf : differentiable_within_at ℝ f s x) (h0 : f x ≠ 0) :
differentiable_within_at ℝ (λ y, ‖f y‖) s x | ((cont_diff_at_id.norm 𝕜 h0).differentiable_at le_rfl).comp_differentiable_within_at x hf | lemma | differentiable_within_at.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at",
"differentiable_within_at",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.dist (hf : differentiable_within_at ℝ f s x)
(hg : differentiable_within_at ℝ g s x) (hne : f x ≠ g x) :
differentiable_within_at ℝ (λ y, dist (f y) (g y)) s x | by { simp only [dist_eq_norm], exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne) } | lemma | differentiable_within_at.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.norm_sq (hf : differentiable_on ℝ f s) :
differentiable_on ℝ (λ y, ‖f y‖ ^ 2) s | λ x hx, (hf x hx).norm_sq 𝕜 | lemma | differentiable_on.norm_sq | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.norm (hf : differentiable_on ℝ f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
differentiable_on ℝ (λ y, ‖f y‖) s | λ x hx, (hf x hx).norm 𝕜 (h0 x hx) | lemma | differentiable_on.norm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.dist (hf : differentiable_on ℝ f s) (hg : differentiable_on ℝ g s)
(hne : ∀ x ∈ s, f x ≠ g x) :
differentiable_on ℝ (λ y, dist (f y) (g y)) s | λ x hx, (hf x hx).dist 𝕜 (hg x hx) (hne x hx) | lemma | differentiable_on.dist | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_euclidean :
differentiable_within_at 𝕜 f t y ↔ ∀ i, differentiable_within_at 𝕜 (λ x, f x i) t y | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_differentiable_within_at_iff, differentiable_within_at_pi],
refl
end | lemma | differentiable_within_at_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_within_at",
"differentiable_within_at_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_euclidean :
differentiable_at 𝕜 f y ↔ ∀ i, differentiable_at 𝕜 (λ x, f x i) y | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_differentiable_at_iff, differentiable_at_pi],
refl
end | lemma | differentiable_at_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_at",
"differentiable_at_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_euclidean :
differentiable_on 𝕜 f t ↔ ∀ i, differentiable_on 𝕜 (λ x, f x i) t | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_differentiable_on_iff, differentiable_on_pi],
refl
end | lemma | differentiable_on_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable_on",
"differentiable_on_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_euclidean :
differentiable 𝕜 f ↔ ∀ i, differentiable 𝕜 (λ x, f x i) | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_differentiable_iff, differentiable_pi],
refl
end | lemma | differentiable_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"differentiable",
"differentiable_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_euclidean :
has_strict_fderiv_at f f' y ↔ ∀ i, has_strict_fderiv_at (λ x, f x i)
(euclidean_space.proj i ∘L f') y | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_has_strict_fderiv_at_iff, has_strict_fderiv_at_pi'],
refl
end | lemma | has_strict_fderiv_at_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"euclidean_space.equiv",
"euclidean_space.proj",
"has_strict_fderiv_at",
"has_strict_fderiv_at_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_euclidean :
has_fderiv_within_at f f' t y ↔ ∀ i, has_fderiv_within_at (λ x, f x i)
(euclidean_space.proj i ∘L f') t y | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_has_fderiv_within_at_iff, has_fderiv_within_at_pi'],
refl
end | lemma | has_fderiv_within_at_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"euclidean_space.equiv",
"euclidean_space.proj",
"has_fderiv_within_at",
"has_fderiv_within_at_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_euclidean {n : ℕ∞} :
cont_diff_within_at 𝕜 n f t y ↔ ∀ i, cont_diff_within_at 𝕜 n (λ x, f x i) t y | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_cont_diff_within_at_iff, cont_diff_within_at_pi],
refl
end | lemma | cont_diff_within_at_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_within_at",
"cont_diff_within_at_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_euclidean {n : ℕ∞} :
cont_diff_at 𝕜 n f y ↔ ∀ i, cont_diff_at 𝕜 n (λ x, f x i) y | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_cont_diff_at_iff, cont_diff_at_pi],
refl
end | lemma | cont_diff_at_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_at",
"cont_diff_at_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_euclidean {n : ℕ∞} :
cont_diff_on 𝕜 n f t ↔ ∀ i, cont_diff_on 𝕜 n (λ x, f x i) t | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_cont_diff_on_iff, cont_diff_on_pi],
refl
end | lemma | cont_diff_on_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff_on",
"cont_diff_on_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_euclidean {n : ℕ∞} :
cont_diff 𝕜 n f ↔ ∀ i, cont_diff 𝕜 n (λ x, f x i) | begin
rw [← (euclidean_space.equiv ι 𝕜).comp_cont_diff_iff, cont_diff_pi],
refl
end | lemma | cont_diff_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff",
"cont_diff_pi",
"euclidean_space.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_homeomorph_unit_ball :
cont_diff ℝ n $ λ (x : E), (homeomorph_unit_ball x : E) | begin
suffices : cont_diff ℝ n (λ x, (1 + ‖x‖^2).sqrt⁻¹), { exact this.smul cont_diff_id, },
have h : ∀ (x : E), 0 < 1 + ‖x‖ ^ 2 := λ x, by positivity,
refine cont_diff.inv _ (λ x, real.sqrt_ne_zero'.mpr (h x)),
exact (cont_diff_const.add $ cont_diff_norm_sq ℝ).sqrt (λ x, (h x).ne.symm),
end | lemma | cont_diff_homeomorph_unit_ball | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"cont_diff",
"cont_diff.inv",
"cont_diff_id",
"cont_diff_norm_sq",
"homeomorph_unit_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_homeomorph_unit_ball_symm
{f : E → E} (h : ∀ y (hy : y ∈ ball (0 : E) 1), f y = homeomorph_unit_ball.symm ⟨y, hy⟩) :
cont_diff_on ℝ n f $ ball 0 1 | begin
intros y hy,
apply cont_diff_at.cont_diff_within_at,
have hf : f =ᶠ[𝓝 y] λ y, (1 - ‖(y : E)‖^2).sqrt⁻¹ • (y : E),
{ rw eventually_eq_iff_exists_mem,
refine ⟨ball (0 : E) 1, mem_nhds_iff.mpr ⟨ball (0 : E) 1, set.subset.refl _, is_open_ball, hy⟩,
λ z hz, _⟩,
rw h z hz,
refl, },
refine c... | lemma | cont_diff_on_homeomorph_unit_ball_symm | analysis.inner_product_space | src/analysis/inner_product_space/calculus.lean | [
"analysis.inner_product_space.pi_L2",
"analysis.special_functions.sqrt"
] | [
"abs_norm",
"cont_diff_at",
"cont_diff_at.comp",
"cont_diff_at.congr_of_eventually_eq",
"cont_diff_at.cont_diff_within_at",
"cont_diff_at.inv",
"cont_diff_at_id",
"cont_diff_norm_sq",
"cont_diff_on",
"one_pow",
"set.subset.refl",
"sq_lt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_iff (f : E →L[ℝ] F) :
is_conformal_map f ↔ ∃ (c : ℝ), 0 < c ∧ ∀ (u v : E), ⟪f u, f v⟫ = c * ⟪u, v⟫ | begin
split,
{ rintros ⟨c₁, hc₁, li, rfl⟩,
refine ⟨c₁ * c₁, mul_self_pos.2 hc₁, λ u v, _⟩,
simp only [real_inner_smul_left, real_inner_smul_right, mul_assoc, coe_smul',
coe_to_continuous_linear_map, pi.smul_apply, inner_map_map] },
{ rintros ⟨c₁, hc₁, huv⟩,
obtain ⟨c, hc, rfl⟩ : ∃ c : ℝ, 0 < c ∧... | lemma | is_conformal_map_iff | analysis.inner_product_space | src/analysis/inner_product_space/conformal_linear_map.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.inner_product_space.basic"
] | [
"continuous_linear_map.coe_coe",
"inv_mul_cancel_left₀",
"is_conformal_map",
"linear_map.smul_apply",
"mul_assoc",
"pi.smul_apply",
"real.mul_self_sqrt",
"real_inner_smul_left",
"real_inner_smul_right",
"smul_inv_smul₀"
] | A map between two inner product spaces is a conformal map if and only if it preserves inner
products up to a scalar factor, i.e., there exists a positive `c : ℝ` such that `⟪f u, f v⟫ = c *
⟪u, v⟫` for all `u`, `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_dual_map : E →ₗᵢ⋆[𝕜] normed_space.dual 𝕜 E | { norm_map' := innerSL_apply_norm _,
..innerSL 𝕜 } | def | inner_product_space.to_dual_map | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"innerSL",
"innerSL_apply_norm",
"normed_space.dual"
] | An element `x` of an inner product space `E` induces an element of the dual space `dual 𝕜 E`,
the map `λ y, ⟪x, y⟫`; moreover this operation is a conjugate-linear isometric embedding of `E`
into `dual 𝕜 E`.
If `E` is complete, this operation is surjective, hence a conjugate-linear isometric equivalence;
see `to_dual`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_dual_map_apply {x y : E} : to_dual_map 𝕜 E x y = ⟪x, y⟫ | rfl | lemma | inner_product_space.to_dual_map_apply | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
innerSL_norm [nontrivial E] : ‖(innerSL 𝕜 : E →L⋆[𝕜] E →L[𝕜] 𝕜)‖ = 1 | show ‖(to_dual_map 𝕜 E).to_continuous_linear_map‖ = 1,
from linear_isometry.norm_to_continuous_linear_map _ | lemma | inner_product_space.innerSL_norm | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"innerSL",
"linear_isometry.norm_to_continuous_linear_map",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_inner_left_basis {ι : Type*} {x y : E} (b : basis ι 𝕜 E)
(h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y | begin
apply (to_dual_map 𝕜 E).map_eq_iff.mp,
refine (function.injective.eq_iff continuous_linear_map.coe_injective).mp (basis.ext b _),
intro i,
simp only [to_dual_map_apply, continuous_linear_map.coe_coe],
rw [←inner_conj_symm],
nth_rewrite_rhs 0 [←inner_conj_symm],
exact congr_arg conj (h i)
end | lemma | inner_product_space.ext_inner_left_basis | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"basis",
"basis.ext",
"continuous_linear_map.coe_coe",
"continuous_linear_map.coe_injective",
"function.injective.eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_inner_right_basis {ι : Type*} {x y : E} (b : basis ι 𝕜 E)
(h : ∀ i : ι, ⟪x, b i⟫ = ⟪y, b i⟫) : x = y | begin
refine ext_inner_left_basis b (λ i, _),
rw [←inner_conj_symm],
nth_rewrite_rhs 0 [←inner_conj_symm],
exact congr_arg conj (h i)
end | lemma | inner_product_space.ext_inner_right_basis | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual : E ≃ₗᵢ⋆[𝕜] normed_space.dual 𝕜 E | linear_isometry_equiv.of_surjective (to_dual_map 𝕜 E)
begin
intros ℓ,
set Y := linear_map.ker ℓ with hY,
by_cases htriv : Y = ⊤,
{ have hℓ : ℓ = 0,
{ have h' := linear_map.ker_eq_top.mp htriv,
rw [←coe_zero] at h',
apply coe_injective,
exact h' },
exact ⟨0, by simp [hℓ]⟩ },
{ rw [← ... | def | inner_product_space.to_dual | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"algebra.id.smul_eq_mul",
"continuous_linear_map.map_smul",
"inner_smul_left",
"inner_smul_right",
"inner_sub_right",
"linear_isometry_equiv.of_surjective",
"linear_map.ker",
"linear_map.mem_ker",
"mul_comm",
"normed_space.dual",
"submodule.ne_bot_iff",
"submodule.orthogonal_eq_bot_iff"
] | Fréchet-Riesz representation: any `ℓ` in the dual of a Hilbert space `E` is of the form
`λ u, ⟪y, u⟫` for some `y : E`, i.e. `to_dual_map` is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_dual_apply {x y : E} : to_dual 𝕜 E x y = ⟪x, y⟫ | rfl | lemma | inner_product_space.to_dual_apply | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_symm_apply {x : E} {y : normed_space.dual 𝕜 E} :
⟪(to_dual 𝕜 E).symm y, x⟫ = y x | begin
rw ← to_dual_apply,
simp only [linear_isometry_equiv.apply_symm_apply],
end | lemma | inner_product_space.to_dual_symm_apply | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"linear_isometry_equiv.apply_symm_apply",
"normed_space.dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_of_bilin (B : E →L⋆[𝕜] E →L[𝕜] 𝕜) : E →L[𝕜] E | comp (to_dual 𝕜 E).symm.to_continuous_linear_equiv.to_continuous_linear_map B | def | inner_product_space.continuous_linear_map_of_bilin | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [] | Maps a bounded sesquilinear form to its continuous linear map,
given by interpreting the form as a map `B : E →L⋆[𝕜] normed_space.dual 𝕜 E`
and dualizing the result using `to_dual`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map_of_bilin_apply (v w : E) : ⟪(B♯ v), w⟫ = B v w | by simp [continuous_linear_map_of_bilin] | lemma | inner_product_space.continuous_linear_map_of_bilin_apply | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_continuous_linear_map_of_bilin {v f : E}
(is_lax_milgram : (∀ w, ⟪f, w⟫ = B v w)) :
f = B♯ v | begin
refine ext_inner_right 𝕜 _,
intro w,
rw continuous_linear_map_of_bilin_apply,
exact is_lax_milgram w,
end | lemma | inner_product_space.unique_continuous_linear_map_of_bilin | analysis.inner_product_space | src/analysis/inner_product_space/dual.lean | [
"analysis.inner_product_space.projection",
"analysis.normed_space.dual",
"analysis.normed_space.star.basic"
] | [
"ext_inner_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_euclidean : E ≃L[ℝ] euclidean_space ℝ (fin $ finrank ℝ E) | continuous_linear_equiv.of_finrank_eq finrank_euclidean_space_fin.symm | def | to_euclidean | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"continuous_linear_equiv.of_finrank_eq",
"euclidean_space"
] | If `E` is a finite dimensional space over `ℝ`, then `to_euclidean` is a continuous `ℝ`-linear
equivalence between `E` and the Euclidean space of the same dimension. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist (x y : E) : ℝ | dist (to_euclidean x) (to_euclidean y) | def | 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"
] | [
"to_euclidean"
] | If `x` and `y` are two points in a finite dimensional space over `ℝ`, then `euclidean.dist x y`
is the distance between these points in the metric defined by some inner product space structure on
`E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_ball (x : E) (r : ℝ) : set E | {y | dist y x ≤ r} | def | euclidean.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"
] | [] | Closed ball w.r.t. the euclidean distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball (x : E) (r : ℝ) : set E | {y | dist y x < r} | def | euclidean.ball | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [] | Open ball w.r.t. the euclidean distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_eq_preimage (x : E) (r : ℝ) :
ball x r = to_euclidean ⁻¹' (metric.ball (to_euclidean x) r) | rfl | lemma | euclidean.ball_eq_preimage | analysis.inner_product_space | src/analysis/inner_product_space/euclidean_dist.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.pi_L2"
] | [
"metric.ball",
"to_euclidean"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_preimage (x : E) (r : ℝ) :
closed_ball x r = to_euclidean ⁻¹' (metric.closed_ball (to_euclidean x) r) | rfl | lemma | euclidean.closed_ball_eq_preimage | 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 | |
ball_subset_closed_ball {x : E} {r : ℝ} : ball x r ⊆ closed_ball x r | λ y (hy : _ < _), le_of_lt hy | lemma | euclidean.ball_subset_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 |
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