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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|---|---|---|---|---|---|---|---|---|---|---|
norm_coe (a : ℝ) : ‖(a : ℍ)‖ = ‖a‖ | by rw [norm_eq_sqrt_real_inner, inner_self, norm_sq_coe, real.sqrt_sq_eq_abs, real.norm_eq_abs] | lemma | quaternion.norm_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"norm_eq_sqrt_real_inner",
"real.norm_eq_abs",
"real.sqrt_sq_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_coe (a : ℝ) : ‖(a : ℍ)‖₊ = ‖a‖₊ | subtype.ext $ norm_coe a | lemma | quaternion.nnnorm_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_star (a : ℍ) : ‖star a‖ = ‖a‖ | by simp_rw [norm_eq_sqrt_real_inner, inner_self, norm_sq_star] | lemma | quaternion.norm_star | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"norm_eq_sqrt_real_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_star (a : ℍ) : ‖star a‖₊ = ‖a‖₊ | subtype.ext $ norm_star a | lemma | quaternion.nnnorm_star | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"nnnorm_star",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_re (z : ℂ) : (z : ℍ).re = z.re | rfl | lemma | quaternion.coe_complex_re | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_im_i (z : ℂ) : (z : ℍ).im_i = z.im | rfl | lemma | quaternion.coe_complex_im_i | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_im_j (z : ℂ) : (z : ℍ).im_j = 0 | rfl | lemma | quaternion.coe_complex_im_j | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_im_k (z : ℂ) : (z : ℍ).im_k = 0 | rfl | lemma | quaternion.coe_complex_im_k | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_add (z w : ℂ) : ↑(z + w) = (z + w : ℍ) | by ext; simp | lemma | quaternion.coe_complex_add | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_mul (z w : ℂ) : ↑(z * w) = (z * w : ℍ) | by ext; simp | lemma | quaternion.coe_complex_mul | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_zero : ((0 : ℂ) : ℍ) = 0 | rfl | lemma | quaternion.coe_complex_zero | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_one : ((1 : ℂ) : ℍ) = 1 | rfl | lemma | quaternion.coe_complex_one | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_real_complex_mul (r : ℝ) (z : ℂ) : (r • z : ℍ) = ↑r * ↑z | by ext; simp | lemma | quaternion.coe_real_complex_mul | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_complex_coe (r : ℝ) : ((r : ℂ) : ℍ) = r | rfl | lemma | quaternion.coe_complex_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_complex : ℂ →ₐ[ℝ] ℍ | { to_fun := coe,
map_one' := rfl,
map_zero' := rfl,
map_add' := coe_complex_add,
map_mul' := coe_complex_mul,
commutes' := λ x, rfl } | def | quaternion.of_complex | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | Coercion `ℂ →ₐ[ℝ] ℍ` as an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_complex : ⇑of_complex = coe | rfl | lemma | quaternion.coe_of_complex | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pi_Lp_equiv_symm_equiv_tuple (x : ℍ) :
‖(pi_Lp.equiv 2 (λ _ : fin 4, _)).symm (equiv_tuple ℝ x)‖ = ‖x‖ | begin
rw [norm_eq_sqrt_real_inner, norm_eq_sqrt_real_inner, inner_self, norm_sq_def', pi_Lp.inner_apply,
fin.sum_univ_four],
simp_rw [is_R_or_C.inner_apply, star_ring_end_apply, star_trivial, ←sq],
refl,
end | lemma | quaternion.norm_pi_Lp_equiv_symm_equiv_tuple | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"is_R_or_C.inner_apply",
"norm_eq_sqrt_real_inner",
"pi_Lp.equiv",
"pi_Lp.inner_apply",
"star_ring_end_apply"
] | The norm of the components as a euclidean vector equals the norm of the quaternion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv_tuple : ℍ ≃ₗᵢ[ℝ] euclidean_space ℝ (fin 4) | { to_fun := λ a, (pi_Lp.equiv _ (λ _ : fin 4, _)).symm ![a.1, a.2, a.3, a.4],
inv_fun := λ a, ⟨a 0, a 1, a 2, a 3⟩,
norm_map' := norm_pi_Lp_equiv_symm_equiv_tuple,
..(quaternion_algebra.linear_equiv_tuple (-1 : ℝ) (-1 : ℝ)).trans
(pi_Lp.linear_equiv 2 ℝ (λ _ : fin 4, ℝ)).symm } | def | quaternion.linear_isometry_equiv_tuple | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"euclidean_space",
"inv_fun",
"pi_Lp.equiv",
"pi_Lp.linear_equiv",
"quaternion_algebra.linear_equiv_tuple"
] | `quaternion_algebra.linear_equiv_tuple` as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_coe : continuous (coe : ℝ → ℍ) | continuous_algebra_map ℝ ℍ | lemma | quaternion.continuous_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous",
"continuous_algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_norm_sq : continuous (norm_sq : ℍ → ℝ) | by simpa [←norm_sq_eq_norm_sq]
using (continuous_norm.mul continuous_norm : continuous (λ q : ℍ, ‖q‖ * ‖q‖)) | lemma | quaternion.continuous_norm_sq | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_re : continuous (λ q : ℍ, q.re) | (continuous_apply 0).comp linear_isometry_equiv_tuple.continuous | lemma | quaternion.continuous_re | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous",
"continuous_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_im_i : continuous (λ q : ℍ, q.im_i) | (continuous_apply 1).comp linear_isometry_equiv_tuple.continuous | lemma | quaternion.continuous_im_i | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous",
"continuous_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_im_j : continuous (λ q : ℍ, q.im_j) | (continuous_apply 2).comp linear_isometry_equiv_tuple.continuous | lemma | quaternion.continuous_im_j | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous",
"continuous_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_im_k : continuous (λ q : ℍ, q.im_k) | (continuous_apply 3).comp linear_isometry_equiv_tuple.continuous | lemma | quaternion.continuous_im_k | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous",
"continuous_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_im : continuous (λ q : ℍ, q.im) | by simpa only [←sub_self_re] using continuous_id.sub (continuous_coe.comp continuous_re) | lemma | quaternion.continuous_im | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_coe {f : α → ℝ} {r : ℝ} :
has_sum (λ a, (f a : ℍ)) (↑r : ℍ) ↔ has_sum f r | ⟨λ h, by simpa only using
h.map (show ℍ →ₗ[ℝ] ℝ, from quaternion_algebra.re_lm _ _) continuous_re,
λ h, by simpa only using h.map (algebra_map ℝ ℍ) (continuous_algebra_map _ _)⟩ | lemma | quaternion.has_sum_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"algebra_map",
"continuous_algebra_map",
"has_sum",
"quaternion_algebra.re_lm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_coe {f : α → ℝ} : summable (λ a, (f a : ℍ)) ↔ summable f | by simpa only using summable.map_iff_of_left_inverse (algebra_map ℝ ℍ)
(show ℍ →ₗ[ℝ] ℝ, from quaternion_algebra.re_lm _ _)
(continuous_algebra_map _ _) continuous_re coe_re | lemma | quaternion.summable_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"algebra_map",
"continuous_algebra_map",
"quaternion_algebra.re_lm",
"summable",
"summable.map_iff_of_left_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe (f : α → ℝ) : ∑' a, (f a : ℍ) = ↑(∑' a, f a) | begin
by_cases hf : summable f,
{ exact (has_sum_coe.mpr hf.has_sum).tsum_eq, },
{ simp [tsum_eq_zero_of_not_summable hf,
tsum_eq_zero_of_not_summable (summable_coe.not.mpr hf)] },
end | lemma | quaternion.tsum_coe | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
schwartz_map | (to_fun : E → F)
(smooth' : cont_diff ℝ ⊤ to_fun)
(decay' : ∀ (k n : ℕ), ∃ (C : ℝ), ∀ x, ‖x‖^k * ‖iterated_fderiv ℝ n to_fun x‖ ≤ C) | structure | schwartz_map | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"cont_diff"
] | A function is a Schwartz function if it is smooth and all derivatives decay faster than
any power of `‖x‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_like : fun_like 𝓢(E, F) E (λ _, F) | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr' } | instance | schwartz_map.fun_like | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fun_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decay (f : 𝓢(E, F)) (k n : ℕ) : ∃ (C : ℝ) (hC : 0 < C),
∀ x, ‖x‖^k * ‖iterated_fderiv ℝ n f x‖ ≤ C | begin
rcases f.decay' k n with ⟨C, hC⟩,
exact ⟨max C 1, by positivity, λ x, (hC x).trans (le_max_left _ _)⟩,
end | lemma | schwartz_map.decay | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | All derivatives of a Schwartz function are rapidly decaying. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smooth (f : 𝓢(E, F)) (n : ℕ∞) : cont_diff ℝ n f | f.smooth'.of_le le_top | lemma | schwartz_map.smooth | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"cont_diff",
"le_top",
"smooth"
] | Every Schwartz function is smooth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (f : 𝓢(E, F)) : continuous f | (f.smooth 0).continuous | lemma | schwartz_map.continuous | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"continuous"
] | Every Schwartz function is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable (f : 𝓢(E, F)) : differentiable ℝ f | (f.smooth 1).differentiable rfl.le | lemma | schwartz_map.differentiable | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"differentiable"
] | Every Schwartz function is differentiable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at (f : 𝓢(E, F)) {x : E} : differentiable_at ℝ f x | f.differentiable.differentiable_at | lemma | schwartz_map.differentiable_at | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"differentiable_at"
] | Every Schwartz function is differentiable at any point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : 𝓢(E, F)} (h : ∀ x, (f : E → F) x = g x) : f = g | fun_like.ext f g h | lemma | schwartz_map.ext | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_cocompact_zpow_neg_nat (k : ℕ) :
asymptotics.is_O (filter.cocompact E) f (λ x, ‖x‖ ^ (-k : ℤ)) | begin
obtain ⟨d, hd, hd'⟩ := f.decay k 0,
simp_rw norm_iterated_fderiv_zero at hd',
simp_rw [asymptotics.is_O, asymptotics.is_O_with],
refine ⟨d, filter.eventually.filter_mono filter.cocompact_le_cofinite _⟩,
refine (filter.eventually_cofinite_ne 0).mp (filter.eventually_of_forall (λ x hx, _)),
rwa [real.no... | lemma | schwartz_map.is_O_cocompact_zpow_neg_nat | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"asymptotics.is_O",
"asymptotics.is_O_with",
"filter.cocompact",
"filter.cocompact_le_cofinite",
"filter.eventually.filter_mono",
"filter.eventually_cofinite_ne",
"filter.eventually_of_forall",
"le_div_iff'",
"norm_iterated_fderiv_zero",
"real.norm_of_nonneg",
"zpow_neg",
"zpow_nonneg",
"zpo... | Auxiliary lemma, used in proving the more general result `is_O_cocompact_zpow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_cocompact_rpow [proper_space E] (s : ℝ) :
asymptotics.is_O (filter.cocompact E) f (λ x, ‖x‖ ^ s) | begin
let k := ⌈-s⌉₊,
have hk : -(k : ℝ) ≤ s, from neg_le.mp (nat.le_ceil (-s)),
refine (is_O_cocompact_zpow_neg_nat f k).trans _,
refine (_ : asymptotics.is_O filter.at_top
(λ x:ℝ, x ^ (-k : ℤ)) (λ x:ℝ, x ^ s)).comp_tendsto tendsto_norm_cocompact_at_top,
simp_rw [asymptotics.is_O, asymptotics.is_O_with],... | lemma | schwartz_map.is_O_cocompact_rpow | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"asymptotics.is_O",
"asymptotics.is_O_with",
"filter.at_top",
"filter.cocompact",
"filter.eventually_ge_at_top",
"filter.eventually_of_mem",
"int.cast_coe_nat",
"int.cast_neg",
"nat.le_ceil",
"one_mul",
"proper_space",
"real.norm_of_nonneg",
"real.rpow_le_rpow_of_exponent_le",
"real.rpow_n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_cocompact_zpow [proper_space E] (k : ℤ) :
asymptotics.is_O (filter.cocompact E) f (λ x, ‖x‖ ^ k) | by simpa only [real.rpow_int_cast] using is_O_cocompact_rpow f k | lemma | schwartz_map.is_O_cocompact_zpow | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"asymptotics.is_O",
"filter.cocompact",
"proper_space",
"real.rpow_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_nonempty (k n : ℕ) (f : 𝓢(E, F)) :
∃ (c : ℝ), c ∈ {c : ℝ | 0 ≤ c ∧ ∀ (x : E), ‖x‖^k * ‖iterated_fderiv ℝ n f x‖ ≤ c} | let ⟨M, hMp, hMb⟩ := f.decay k n in ⟨M, le_of_lt hMp, hMb⟩ | lemma | schwartz_map.bounds_nonempty | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_bdd_below (k n : ℕ) (f : 𝓢(E, F)) :
bdd_below {c | 0 ≤ c ∧ ∀ x, ‖x‖^k * ‖iterated_fderiv ℝ n f x‖ ≤ c} | ⟨0, λ _ ⟨hn, _⟩, hn⟩ | lemma | schwartz_map.bounds_bdd_below | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decay_add_le_aux (k n : ℕ) (f g : 𝓢(E, F)) (x : E) :
‖x‖^k * ‖iterated_fderiv ℝ n (f+g) x‖ ≤
‖x‖^k * ‖iterated_fderiv ℝ n f x‖
+ ‖x‖^k * ‖iterated_fderiv ℝ n g x‖ | begin
rw ←mul_add,
refine mul_le_mul_of_nonneg_left _ (by positivity),
convert norm_add_le _ _,
exact iterated_fderiv_add_apply (f.smooth _) (g.smooth _),
end | lemma | schwartz_map.decay_add_le_aux | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"iterated_fderiv_add_apply",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decay_neg_aux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
‖x‖ ^ k * ‖iterated_fderiv ℝ n (-f) x‖ = ‖x‖ ^ k * ‖iterated_fderiv ℝ n f x‖ | begin
nth_rewrite 3 ←norm_neg,
congr,
exact iterated_fderiv_neg_apply,
end | lemma | schwartz_map.decay_neg_aux | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"iterated_fderiv_neg_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decay_smul_aux (k n : ℕ) (f : 𝓢(E, F)) (c : 𝕜) (x : E) :
‖x‖ ^ k * ‖iterated_fderiv ℝ n (c • f) x‖ =
‖c‖ * ‖x‖ ^ k * ‖iterated_fderiv ℝ n f x‖ | by rw [mul_comm (‖c‖), mul_assoc, iterated_fderiv_const_smul_apply (f.smooth _), norm_smul] | lemma | schwartz_map.decay_smul_aux | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"iterated_fderiv_const_smul_apply",
"mul_assoc",
"mul_comm",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_aux (k n : ℕ) (f : 𝓢(E, F)) : ℝ | Inf {c | 0 ≤ c ∧ ∀ x, ‖x‖^k * ‖iterated_fderiv ℝ n f x‖ ≤ c} | def | schwartz_map.seminorm_aux | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | Helper definition for the seminorms of the Schwartz space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_aux_nonneg (k n : ℕ) (f : 𝓢(E, F)) : 0 ≤ f.seminorm_aux k n | le_cInf (bounds_nonempty k n f) (λ _ ⟨hx, _⟩, hx) | lemma | schwartz_map.seminorm_aux_nonneg | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"le_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_seminorm_aux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
‖x‖ ^ k * ‖iterated_fderiv ℝ n ⇑f x‖ ≤ f.seminorm_aux k n | le_cInf (bounds_nonempty k n f) (λ y ⟨_, h⟩, h x) | lemma | schwartz_map.le_seminorm_aux | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"le_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_aux_le_bound (k n : ℕ) (f : 𝓢(E, F)) {M : ℝ} (hMp: 0 ≤ M)
(hM : ∀ x, ‖x‖^k * ‖iterated_fderiv ℝ n f x‖ ≤ M) :
f.seminorm_aux k n ≤ M | cInf_le (bounds_bdd_below k n f) ⟨hMp, hM⟩ | lemma | schwartz_map.seminorm_aux_le_bound | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"cInf_le"
] | If one controls the norm of every `A x`, then one controls the norm of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_apply {f : 𝓢(E, F)} {c : 𝕜} {x : E} : (c • f) x = c • (f x) | rfl | lemma | schwartz_map.smul_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_aux_smul_le (k n : ℕ) (c : 𝕜) (f : 𝓢(E, F)) :
(c • f).seminorm_aux k n ≤ ‖c‖ * f.seminorm_aux k n | begin
refine (c • f).seminorm_aux_le_bound k n (mul_nonneg (norm_nonneg _) (seminorm_aux_nonneg _ _ _))
(λ x, (decay_smul_aux k n f c x).le.trans _),
rw mul_assoc,
exact mul_le_mul_of_nonneg_left (f.le_seminorm_aux k n x) (norm_nonneg _),
end | lemma | schwartz_map.seminorm_aux_smul_le | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"mul_assoc",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul : has_smul ℕ 𝓢(E, F) | ⟨λ c f, { to_fun := c • f,
smooth' := (f.smooth _).const_smul c,
decay' := begin
have : c • (f : E → F) = (c : ℝ) • f,
{ ext x, simp only [pi.smul_apply, ← nsmul_eq_smul_cast] },
simp only [this],
exact ((c : ℝ) • f).decay',
end}⟩ | instance | schwartz_map.has_nsmul | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"has_smul",
"nsmul_eq_smul_cast",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zsmul : has_smul ℤ 𝓢(E, F) | ⟨λ c f, { to_fun := c • f,
smooth' := (f.smooth _).const_smul c,
decay' := begin
have : c • (f : E → F) = (c : ℝ) • f,
{ ext x, simp only [pi.smul_apply, ← zsmul_eq_smul_cast] },
simp only [this],
exact ((c : ℝ) • f).decay',
end}⟩ | instance | schwartz_map.has_zsmul | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"has_smul",
"pi.smul_apply",
"zsmul_eq_smul_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ↑(0 : 𝓢(E, F)) = (0 : E → F) | rfl | lemma | schwartz_map.coe_zero | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_zero : coe_fn (0 : 𝓢(E, F)) = (0 : E → F) | rfl | lemma | schwartz_map.coe_fn_zero | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply {x : E} : (0 : 𝓢(E, F)) x = 0 | rfl | lemma | schwartz_map.zero_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_aux_zero (k n : ℕ) :
(0 : 𝓢(E, F)).seminorm_aux k n = 0 | le_antisymm (seminorm_aux_le_bound k n _ rfl.le (λ _, by simp [pi.zero_def]))
(seminorm_aux_nonneg _ _ _) | lemma | schwartz_map.seminorm_aux_zero | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply {f g : 𝓢(E, F)} {x : E} : (f + g) x = f x + g x | rfl | lemma | schwartz_map.add_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_aux_add_le (k n : ℕ) (f g : 𝓢(E, F)) :
(f + g).seminorm_aux k n ≤ f.seminorm_aux k n + g.seminorm_aux k n | (f + g).seminorm_aux_le_bound k n
(add_nonneg (seminorm_aux_nonneg _ _ _) (seminorm_aux_nonneg _ _ _)) $
λ x, (decay_add_le_aux k n f g x).trans $
add_le_add (f.le_seminorm_aux k n x) (g.le_seminorm_aux k n x) | lemma | schwartz_map.seminorm_aux_add_le | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply {f g : 𝓢(E, F)} {x : E} : (f - g) x = f x - g x | rfl | lemma | schwartz_map.sub_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_hom : 𝓢(E, F) →+ (E → F) | { to_fun := λ f, f, map_zero' := coe_zero, map_add' := λ _ _, rfl } | def | schwartz_map.coe_hom | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | Coercion as an additive homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe_hom : (coe_hom E F : 𝓢(E, F) → (E → F)) = coe_fn | rfl | lemma | schwartz_map.coe_coe_hom | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_hom_injective : function.injective (coe_hom E F) | by { rw coe_coe_hom, exact fun_like.coe_injective } | lemma | schwartz_map.coe_hom_injective | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm (k n : ℕ) : seminorm 𝕜 𝓢(E, F) | seminorm.of_smul_le (seminorm_aux k n)
(seminorm_aux_zero k n) (seminorm_aux_add_le k n) (seminorm_aux_smul_le k n) | def | schwartz_map.seminorm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"seminorm",
"seminorm.of_smul_le"
] | The seminorms of the Schwartz space given by the best constants in the definition of
`𝓢(E, F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_le_bound (k n : ℕ) (f : 𝓢(E, F)) {M : ℝ} (hMp: 0 ≤ M)
(hM : ∀ x, ‖x‖^k * ‖iterated_fderiv ℝ n f x‖ ≤ M) : seminorm 𝕜 k n f ≤ M | f.seminorm_aux_le_bound k n hMp hM | lemma | schwartz_map.seminorm_le_bound | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"seminorm"
] | If one controls the seminorm for every `x`, then one controls the seminorm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminorm_le_bound' (k n : ℕ) (f : 𝓢(ℝ, F)) {M : ℝ} (hMp: 0 ≤ M)
(hM : ∀ x, |x|^k * ‖iterated_deriv n f x‖ ≤ M) : seminorm 𝕜 k n f ≤ M | begin
refine seminorm_le_bound 𝕜 k n f hMp _,
simpa only [real.norm_eq_abs, norm_iterated_fderiv_eq_norm_iterated_deriv],
end | lemma | schwartz_map.seminorm_le_bound' | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"norm_iterated_fderiv_eq_norm_iterated_deriv",
"real.norm_eq_abs",
"seminorm"
] | If one controls the seminorm for every `x`, then one controls the seminorm.
Variant for functions `𝓢(ℝ, F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_seminorm (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
‖x‖ ^ k * ‖iterated_fderiv ℝ n f x‖ ≤ seminorm 𝕜 k n f | f.le_seminorm_aux k n x | lemma | schwartz_map.le_seminorm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"seminorm"
] | The seminorm controls the Schwartz estimate for any fixed `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_seminorm' (k n : ℕ) (f : 𝓢(ℝ, F)) (x : ℝ) : | |x| ^ k * ‖iterated_deriv n f x‖ ≤ seminorm 𝕜 k n f :=
begin
have := le_seminorm 𝕜 k n f x,
rwa [← real.norm_eq_abs, ← norm_iterated_fderiv_eq_norm_iterated_deriv],
end | lemma | schwartz_map.le_seminorm' | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"norm_iterated_fderiv_eq_norm_iterated_deriv",
"real.norm_eq_abs",
"seminorm"
] | The seminorm controls the Schwartz estimate for any fixed `x`.
Variant for functions `𝓢(ℝ, F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_le_seminorm (f : 𝓢(E, F)) (n : ℕ) (x₀ : E) :
‖iterated_fderiv ℝ n f x₀‖ ≤ (schwartz_map.seminorm 𝕜 0 n) f | begin
have := schwartz_map.le_seminorm 𝕜 0 n f x₀,
rwa [pow_zero, one_mul] at this,
end | lemma | schwartz_map.norm_iterated_fderiv_le_seminorm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"one_mul",
"pow_zero",
"schwartz_map.le_seminorm",
"schwartz_map.seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pow_mul_le_seminorm (f : 𝓢(E, F)) (k : ℕ) (x₀ : E) :
‖x₀‖^k * ‖f x₀‖ ≤ (schwartz_map.seminorm 𝕜 k 0) f | begin
have := schwartz_map.le_seminorm 𝕜 k 0 f x₀,
rwa norm_iterated_fderiv_zero at this,
end | lemma | schwartz_map.norm_pow_mul_le_seminorm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"norm_iterated_fderiv_zero",
"schwartz_map.le_seminorm",
"schwartz_map.seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_seminorm (f : 𝓢(E, F)) (x₀ : E) :
‖f x₀‖ ≤ (schwartz_map.seminorm 𝕜 0 0) f | begin
have := norm_pow_mul_le_seminorm 𝕜 f 0 x₀,
rwa [pow_zero, one_mul] at this,
end | lemma | schwartz_map.norm_le_seminorm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"one_mul",
"pow_zero",
"schwartz_map.seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.schwartz_seminorm_family : seminorm_family 𝕜 𝓢(E, F) (ℕ × ℕ) | λ m, seminorm 𝕜 m.1 m.2 | def | schwartz_seminorm_family | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"seminorm",
"seminorm_family"
] | The family of Schwartz seminorms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
schwartz_seminorm_family_apply (n k : ℕ) :
schwartz_seminorm_family 𝕜 E F (n,k) = schwartz_map.seminorm 𝕜 n k | rfl | lemma | schwartz_map.schwartz_seminorm_family_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"schwartz_map.seminorm",
"schwartz_seminorm_family"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
schwartz_seminorm_family_apply_zero :
schwartz_seminorm_family 𝕜 E F 0 = schwartz_map.seminorm 𝕜 0 0 | rfl | lemma | schwartz_map.schwartz_seminorm_family_apply_zero | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"schwartz_map.seminorm",
"schwartz_seminorm_family"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_le_sup_seminorm_apply {m : ℕ × ℕ} {k n : ℕ} (hk : k ≤ m.1) (hn : n ≤ m.2)
(f : 𝓢(E, F)) (x : E) :
(1 + ‖x‖) ^ k * ‖iterated_fderiv ℝ n f x‖
≤ 2^m.1 * (finset.Iic m).sup (λ m, seminorm 𝕜 m.1 m.2) f | begin
rw [add_comm, add_pow],
simp only [one_pow, mul_one, finset.sum_congr, finset.sum_mul],
norm_cast,
rw ← nat.sum_range_choose m.1,
push_cast,
rw [finset.sum_mul],
have hk' : finset.range (k + 1) ⊆ finset.range (m.1 + 1) :=
by rwa [finset.range_subset, add_le_add_iff_right],
refine le_trans (finse... | lemma | schwartz_map.one_add_le_sup_seminorm_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"add_pow",
"finset.Iic",
"finset.le_sup_of_le",
"finset.range",
"finset.range_subset",
"finset.sum_mul",
"le_rfl",
"mul_assoc",
"mul_comm",
"mul_le_mul",
"mul_one",
"nat.sum_range_choose",
"one_pow",
"seminorm"
] | A more convenient version of `le_sup_seminorm_apply`.
The set `finset.Iic m` is the set of all pairs `(k', n')` with `k' ≤ m.1` and `n' ≤ m.2`.
Note that the constant is far from optimal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.schwartz_with_seminorms : with_seminorms (schwartz_seminorm_family 𝕜 E F) | begin
have A : with_seminorms (schwartz_seminorm_family ℝ E F) := ⟨rfl⟩,
rw seminorm_family.with_seminorms_iff_nhds_eq_infi at ⊢ A,
rw A,
refl
end | lemma | schwartz_with_seminorms | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"schwartz_seminorm_family",
"seminorm_family.with_seminorms_iff_nhds_eq_infi",
"with_seminorms"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.has_temperate_growth (f : E → F) : Prop | cont_diff ℝ ⊤ f ∧ ∀ n : ℕ, ∃ (k : ℕ) (C : ℝ), ∀ x, ‖iterated_fderiv ℝ n f x‖ ≤ C * (1 + ‖x‖)^k | def | function.has_temperate_growth | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"cont_diff"
] | A function is called of temperate growth if it is smooth and all iterated derivatives are
polynomially bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.function.has_temperate_growth.norm_iterated_fderiv_le_uniform_aux {f : E → F}
(hf_temperate : f.has_temperate_growth) (n : ℕ) :
∃ (k : ℕ) (C : ℝ) (hC : 0 ≤ C), ∀ (N : ℕ) (hN : N ≤ n) (x : E),
‖iterated_fderiv ℝ N f x‖ ≤ C * (1 + ‖x‖)^k | begin
choose k C f using hf_temperate.2,
use (finset.range (n+1)).sup k,
let C' := max (0 : ℝ) ((finset.range (n+1)).sup' (by simp) C),
have hC' : 0 ≤ C' := by simp only [le_refl, finset.le_sup'_iff, true_or, le_max_iff],
use [C', hC'],
intros N hN x,
rw ← finset.mem_range_succ_iff at hN,
refine le_tran... | lemma | function.has_temperate_growth.norm_iterated_fderiv_le_uniform_aux | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"finset.le_sup",
"finset.le_sup'_iff",
"finset.mem_range_succ_iff",
"finset.range",
"le_max_iff",
"mul_le_mul",
"pow_le_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_lm (A : (D → E) → (F → G))
(hadd : ∀ (f g : 𝓢(D, E)) x, A (f + g) x = A f x + A g x)
(hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)) x, A (a • f) x = σ a • A f x)
(hsmooth : ∀ (f : 𝓢(D, E)), cont_diff ℝ ⊤ (A f))
(hbound : ∀ (n : ℕ × ℕ), ∃ (s : finset (ℕ × ℕ)) (C : ℝ) (hC : 0 ≤ C), ∀ (f : 𝓢(D, E)) (x : F),
‖x‖ ^ n.fs... | { to_fun := λ f,
{ to_fun := A f,
smooth' := hsmooth f,
decay' :=
begin
intros k n,
rcases hbound ⟨k, n⟩ with ⟨s, C, hC, h⟩,
exact ⟨C * (s.sup (schwartz_seminorm_family 𝕜 D E)) f, h f⟩,
end, },
map_add' := λ f g, ext (hadd f g),
map_smul' := λ a f, ext (hsmul a f), } | def | schwartz_map.mk_lm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"cont_diff",
"finset",
"schwartz_seminorm_family"
] | Create a semilinear map between Schwartz spaces.
Note: This is a helper definition for `mk_clm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_clm [ring_hom_isometric σ] (A : (D → E) → (F → G))
(hadd : ∀ (f g : 𝓢(D, E)) x, A (f + g) x = A f x + A g x)
(hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)) x, A (a • f) x = σ a • A f x)
(hsmooth : ∀ (f : 𝓢(D, E)), cont_diff ℝ ⊤ (A f))
(hbound : ∀ (n : ℕ × ℕ), ∃ (s : finset (ℕ × ℕ)) (C : ℝ) (hC : 0 ≤ C), ∀ (f : 𝓢(D, E... | { cont :=
begin
change continuous (mk_lm A hadd hsmul hsmooth hbound : 𝓢(D, E) →ₛₗ[σ] 𝓢(F, G)),
refine seminorm.continuous_from_bounded (schwartz_with_seminorms 𝕜 D E)
(schwartz_with_seminorms 𝕜' F G) _ (λ n, _),
rcases hbound n with ⟨s, C, hC, h⟩,
refine ⟨s, ⟨C, hC⟩, (λ f, _)⟩,
simp onl... | def | schwartz_map.mk_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"algebra.id.smul_eq_mul",
"cont",
"cont_diff",
"continuous",
"finset",
"nnreal.smul_def",
"ring_hom_isometric",
"schwartz_seminorm_family",
"schwartz_with_seminorms",
"seminorm.comp_apply",
"seminorm.continuous_from_bounded",
"seminorm.smul_apply",
"subtype.coe_mk"
] | Create a continuous semilinear map between Schwartz spaces.
For an example of using this definition, see `fderiv_clm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_clm (m : E) : 𝓢(E, E →L[ℝ] F) →L[𝕜] 𝓢(E, F) | mk_clm (λ f x, f x m)
(λ _ _ _, rfl) (λ _ _ _, rfl) (λ f, cont_diff.clm_apply f.2 cont_diff_const)
(begin
rintro ⟨k, n⟩,
use [{(k, n)}, ‖m‖, norm_nonneg _],
intros f x,
refine le_trans (mul_le_mul_of_nonneg_left (norm_iterated_fderiv_clm_apply_const f.2 le_top)
(by positivity)) _,
rw [← mu... | def | schwartz_map.eval_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"cont_diff.clm_apply",
"cont_diff_const",
"finset.sup_singleton",
"le_top",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"norm_iterated_fderiv_clm_apply_const"
] | The map applying a vector to Hom-valued Schwartz function as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bilin_left_clm (B : E →L[ℝ] F →L[ℝ] G) {g : D → F} (hg : g.has_temperate_growth) :
𝓢(D, E) →L[ℝ] 𝓢(D, G) | -- Todo (after port): generalize to `B : E →L[𝕜] F →L[𝕜] G` and `𝕜`-linear
mk_clm (λ f x, B (f x) (g x))
(λ _ _ _, by simp only [map_add, add_left_inj, pi.add_apply, eq_self_iff_true,
continuous_linear_map.add_apply])
(λ _ _ _, by simp only [pi.smul_apply, continuous_linear_map.coe_smul',
continuous_line... | def | schwartz_map.bilin_left_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"continuous_linear_map.add_apply",
"continuous_linear_map.coe_smul'",
"continuous_linear_map.map_smul",
"continuous_linear_map.norm_iterated_fderiv_le_of_bilinear",
"finset.Iic",
"finset.mem_range_succ_iff",
"finset.mul_sum",
"finset.range",
"finset.sum_mul",
"le_rfl",
"le_top",
"mul_assoc",
... | The map `f ↦ (x ↦ B (f x) (g x))` as a continuous `𝕜`-linear map on Schwartz space,
where `B` is a continuous `𝕜`-linear map and `g` is a function of temperate growth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_clm {g : D → E} (hg : g.has_temperate_growth)
(hg_upper : ∃ (k : ℕ) (C : ℝ), ∀ x, ‖x‖ ≤ C * (1 + ‖g x‖)^k ) :
𝓢(E, F) →L[𝕜] 𝓢(D, F) | mk_clm (λ f x, (f (g x)))
(λ _ _ _, by simp only [add_left_inj, pi.add_apply, eq_self_iff_true])
(λ _ _ _, rfl)
(λ f, f.smooth'.comp hg.1)
(begin
rintros ⟨k, n⟩,
rcases hg.norm_iterated_fderiv_le_uniform_aux n with ⟨l, C, hC, hgrowth⟩,
rcases hg_upper with ⟨kg, Cg, hg_upper'⟩,
have hCg : 1 ≤ 1 +... | def | schwartz_map.comp_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"div_le_iff",
"finset.Iic",
"le_div_iff'",
"le_rfl",
"le_self_pow",
"le_top",
"map_nonneg",
"mul_assoc",
"mul_le_mul",
"mul_le_mul_of_nonneg_right",
"mul_pow",
"nonneg_of_mul_nonneg_left",
"norm_iterated_fderiv_comp_le",
"one_le_pow_of_one_le",
"one_mul",
"pow_add",
"pow_le_pow_of_le... | Composition with a function on the right is a continuous linear map on Schwartz space
provided that the function is temperate and growths polynomially near infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_clm : 𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F) | mk_clm (fderiv ℝ)
(λ f g _, fderiv_add f.differentiable_at g.differentiable_at)
(λ a f _, fderiv_const_smul f.differentiable_at a)
(λ f, (cont_diff_top_iff_fderiv.mp f.smooth').2)
(λ ⟨k, n⟩, ⟨{⟨k, n+1⟩}, 1, zero_le_one, λ f x, by simpa only [schwartz_seminorm_family_apply,
seminorm.comp_apply, finset.sup_si... | def | schwartz_map.fderiv_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fderiv",
"fderiv_add",
"fderiv_const_smul",
"finset.sup_singleton",
"norm_iterated_fderiv_fderiv",
"one_mul",
"one_smul",
"seminorm.comp_apply",
"zero_le_one"
] | The Fréchet derivative on Schwartz space as a continuous `𝕜`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_clm_apply (f : 𝓢(E, F)) (x : E) : fderiv_clm 𝕜 f x = fderiv ℝ f x | rfl | lemma | schwartz_map.fderiv_clm_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fderiv",
"fderiv_clm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_clm : 𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F) | mk_clm (λ f, deriv f)
(λ f g _, deriv_add f.differentiable_at g.differentiable_at)
(λ a f _, deriv_const_smul a f.differentiable_at)
(λ f, (cont_diff_top_iff_deriv.mp f.smooth').2)
(λ ⟨k, n⟩, ⟨{⟨k, n+1⟩}, 1, zero_le_one, λ f x, by simpa only [real.norm_eq_abs,
finset.sup_singleton, schwartz_seminorm_family_... | def | schwartz_map.deriv_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"deriv",
"deriv_add",
"deriv_const_smul",
"finset.sup_singleton",
"iterated_deriv_succ'",
"norm_iterated_fderiv_eq_norm_iterated_deriv",
"one_mul",
"real.norm_eq_abs",
"zero_le_one"
] | The 1-dimensional derivative on Schwartz space as a continuous `𝕜`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_clm_apply (f : 𝓢(ℝ, F)) (x : ℝ) : deriv_clm 𝕜 f x = deriv f x | rfl | lemma | schwartz_map.deriv_clm_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"deriv",
"deriv_clm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pderiv_clm (m : E) : 𝓢(E, F) →L[𝕜] 𝓢(E, F) | (eval_clm m).comp (fderiv_clm 𝕜) | def | schwartz_map.pderiv_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | The partial derivative (or directional derivative) in the direction `m : E` as a
continuous linear map on Schwartz space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pderiv_clm_apply (m : E) (f : 𝓢(E, F)) (x : E) : pderiv_clm 𝕜 m f x = fderiv ℝ f x m | rfl | lemma | schwartz_map.pderiv_clm_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_pderiv {n : ℕ} : (fin n → E) → 𝓢(E, F) →L[𝕜] 𝓢(E, F) | nat.rec_on n
(λ x, continuous_linear_map.id 𝕜 _)
(λ n rec x, (pderiv_clm 𝕜 (x 0)).comp (rec (fin.tail x))) | def | schwartz_map.iterated_pderiv | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"continuous_linear_map.id",
"fin.tail"
] | The iterated partial derivative (or directional derivative) as a continuous linear map on
Schwartz space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_pderiv_zero (m : fin 0 → E) (f : 𝓢(E, F)):
iterated_pderiv 𝕜 m f = f | rfl | lemma | schwartz_map.iterated_pderiv_zero | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_pderiv_one (m : fin 1 → E) (f : 𝓢(E, F)) :
iterated_pderiv 𝕜 m f = pderiv_clm 𝕜 (m 0) f | rfl | lemma | schwartz_map.iterated_pderiv_one | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_pderiv_succ_left {n : ℕ} (m : fin (n + 1) → E) (f : 𝓢(E, F)) :
iterated_pderiv 𝕜 m f = pderiv_clm 𝕜 (m 0) (iterated_pderiv 𝕜 (fin.tail m) f) | rfl | lemma | schwartz_map.iterated_pderiv_succ_left | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fin.tail"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_pderiv_succ_right {n : ℕ} (m : fin (n + 1) → E) (f : 𝓢(E, F)) :
iterated_pderiv 𝕜 m f =
iterated_pderiv 𝕜 (fin.init m) (pderiv_clm 𝕜 (m (fin.last n)) f) | begin
induction n with n IH,
{ rw [iterated_pderiv_zero, iterated_pderiv_one],
refl },
-- The proof is `∂^{n + 2} = ∂ ∂^{n + 1} = ∂ ∂^n ∂ = ∂^{n+1} ∂`
have hmzero : fin.init m 0 = m 0 := by simp only [fin.init_def, fin.cast_succ_zero],
have hmtail : fin.tail m (fin.last n) = m (fin.last n.succ) :=
by si... | lemma | schwartz_map.iterated_pderiv_succ_right | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"fin.cast_succ_zero",
"fin.init",
"fin.init_def",
"fin.last",
"fin.succ_last",
"fin.tail",
"fin.tail_def",
"fin.tail_init_eq_init_tail"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_bounded_continuous_function (f : 𝓢(E, F)) : E →ᵇ F | bounded_continuous_function.of_normed_add_comm_group f (schwartz_map.continuous f)
(schwartz_map.seminorm ℝ 0 0 f) (norm_le_seminorm ℝ f) | def | schwartz_map.to_bounded_continuous_function | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"bounded_continuous_function.of_normed_add_comm_group",
"schwartz_map.continuous",
"schwartz_map.seminorm"
] | Schwartz functions as bounded continuous functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bounded_continuous_function_apply (f : 𝓢(E, F)) (x : E) :
f.to_bounded_continuous_function x = f x | rfl | lemma | schwartz_map.to_bounded_continuous_function_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (f : 𝓢(E, F)) : C(E, F) | f.to_bounded_continuous_function.to_continuous_map | def | schwartz_map.to_continuous_map | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | Schwartz functions as continuous functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bounded_continuous_function_lm : 𝓢(E, F) →ₗ[𝕜] E →ᵇ F | { to_fun := λ f, f.to_bounded_continuous_function,
map_add' := λ f g, by { ext, exact add_apply },
map_smul' := λ a f, by { ext, exact smul_apply } } | def | schwartz_map.to_bounded_continuous_function_lm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | The inclusion map from Schwartz functions to bounded continuous functions as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bounded_continuous_function_lm_apply (f : 𝓢(E, F)) (x : E) :
to_bounded_continuous_function_lm 𝕜 E F f x = f x | rfl | lemma | schwartz_map.to_bounded_continuous_function_lm_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_bounded_continuous_function_clm : 𝓢(E, F) →L[𝕜] E →ᵇ F | { cont :=
begin
change continuous (to_bounded_continuous_function_lm 𝕜 E F),
refine seminorm.continuous_from_bounded (schwartz_with_seminorms 𝕜 E F)
(norm_with_seminorms 𝕜 (E →ᵇ F)) _ (λ i, ⟨{0}, 1, λ f, _⟩),
rw [finset.sup_singleton, one_smul , seminorm.comp_apply, coe_norm_seminorm,
sch... | def | schwartz_map.to_bounded_continuous_function_clm | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [
"bounded_continuous_function.norm_le",
"coe_norm_seminorm",
"cont",
"continuous",
"finset.sup_singleton",
"map_nonneg",
"norm_with_seminorms",
"one_smul",
"schwartz_with_seminorms",
"seminorm.comp_apply",
"seminorm.continuous_from_bounded"
] | The inclusion map from Schwartz functions to bounded continuous functions as a continuous linear
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bounded_continuous_function_clm_apply (f : 𝓢(E, F)) (x : E) :
to_bounded_continuous_function_clm 𝕜 E F f x = f x | rfl | lemma | schwartz_map.to_bounded_continuous_function_clm_apply | analysis | src/analysis/schwartz_space.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.iterated_deriv",
"analysis.locally_convex.with_seminorms",
"topology.algebra.uniform_filter_basis",
"topology.continuous_function.bounded",
"tactic.positivity",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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