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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|---|---|---|---|---|---|---|---|---|---|---|
thickening_ball (hε : 0 < ε) (hδ : 0 < δ) (x : E) :
thickening ε (ball x δ) = ball x (ε + δ) | by rw [←thickening_singleton, thickening_thickening hε hδ, thickening_singleton]; apply_instance | lemma | thickening_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"thickening_thickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_closed_ball (hε : 0 < ε) (hδ : 0 ≤ δ) (x : E) :
thickening ε (closed_ball x δ) = ball x (ε + δ) | by rw [←cthickening_singleton _ hδ, thickening_cthickening hε hδ, thickening_singleton];
apply_instance | lemma | thickening_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"thickening_cthickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (x : E) :
cthickening ε (ball x δ) = closed_ball x (ε + δ) | by rw [←thickening_singleton, cthickening_thickening hε hδ,
cthickening_singleton _ (add_nonneg hε hδ.le)]; apply_instance | lemma | cthickening_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"cthickening_thickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_closed_ball (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (x : E) :
cthickening ε (closed_ball x δ) = closed_ball x (ε + δ) | by rw [←cthickening_singleton _ hδ, cthickening_cthickening hε hδ,
cthickening_singleton _ (add_nonneg hε hδ)]; apply_instance | lemma | cthickening_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"cthickening_cthickening"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_add_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
ball a ε + ball b δ = ball (a + b) (ε + δ) | by rw [ball_add, thickening_ball hε hδ b, metric.vadd_ball, vadd_eq_add] | lemma | ball_add_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"thickening_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_sub_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
ball a ε - ball b δ = ball (a - b) (ε + δ) | by simp_rw [sub_eq_add_neg, neg_ball, ball_add_ball hε hδ] | lemma | ball_sub_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ball_add_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_add_closed_ball (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
ball a ε + closed_ball b δ = ball (a + b) (ε + δ) | by rw [ball_add, thickening_closed_ball hε hδ b, metric.vadd_ball, vadd_eq_add] | lemma | ball_add_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"thickening_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_sub_closed_ball (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
ball a ε - closed_ball b δ = ball (a - b) (ε + δ) | by simp_rw [sub_eq_add_neg, neg_closed_ball, ball_add_closed_ball hε hδ] | lemma | ball_sub_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ball_add_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_add_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E) :
closed_ball a ε + ball b δ = ball (a + b) (ε + δ) | by rw [add_comm, ball_add_closed_ball hδ hε b, add_comm, add_comm δ] | lemma | closed_ball_add_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"ball_add_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_sub_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E) :
closed_ball a ε - ball b δ = ball (a - b) (ε + δ) | by simp_rw [sub_eq_add_neg, neg_ball, closed_ball_add_ball hε hδ] | lemma | closed_ball_sub_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"closed_ball_add_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_add_closed_ball [proper_space E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
closed_ball a ε + closed_ball b δ = closed_ball (a + b) (ε + δ) | by rw [(is_compact_closed_ball _ _).add_closed_ball hδ b, cthickening_closed_ball hδ hε a,
metric.vadd_closed_ball, vadd_eq_add, add_comm, add_comm δ] | lemma | closed_ball_add_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"cthickening_closed_ball",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_sub_closed_ball [proper_space E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
closed_ball a ε - closed_ball b δ = closed_ball (a - b) (ε + δ) | by simp_rw [sub_eq_add_neg, neg_closed_ball, closed_ball_add_closed_ball hε hδ] | lemma | closed_ball_sub_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"closed_ball_add_closed_ball",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
c • closed_ball x r = closed_ball (c • x) (‖c‖ * r) | begin
rcases eq_or_ne c 0 with rfl|hc,
{ simp [hr, zero_smul_set, set.singleton_zero, ← nonempty_closed_ball] },
{ exact smul_closed_ball' hc x r }
end | theorem | smul_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"eq_or_ne",
"smul_closed_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_unit_ball (c : 𝕜) : c • closed_ball (0 : E) (1 : ℝ) = closed_ball (0 : E) (‖c‖) | by rw [smul_closed_ball _ _ zero_le_one, smul_zero, mul_one] | lemma | smul_closed_unit_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"mul_one",
"smul_closed_ball",
"smul_zero",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_unit_ball_of_nonneg {r : ℝ} (hr : 0 ≤ r) :
r • closed_ball 0 1 = closed_ball (0 : E) r | by rw [smul_closed_unit_ball, real.norm_of_nonneg hr] | lemma | smul_closed_unit_ball_of_nonneg | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"real.norm_of_nonneg",
"smul_closed_unit_ball"
] | In a real normed space, the image of the unit closed ball under multiplication by a nonnegative
number `r` is the closed ball of radius `r` with center at the origin. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.sphere_nonempty [nontrivial E] {x : E} {r : ℝ} :
(sphere x r).nonempty ↔ 0 ≤ r | begin
obtain ⟨y, hy⟩ := exists_ne x,
refine ⟨λ h, nonempty_closed_ball.1 (h.mono sphere_subset_closed_ball), λ hr,
⟨r • ‖y - x‖⁻¹ • (y - x) + x, _⟩⟩,
have : ‖y - x‖ ≠ 0, by simpa [sub_eq_zero],
simp [norm_smul, this, real.norm_of_nonneg hr],
end | lemma | normed_space.sphere_nonempty | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"exists_ne",
"nontrivial",
"norm_smul",
"real.norm_of_nonneg"
] | In a nontrivial real normed space, a sphere is nonempty if and only if its radius is
nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_sphere [nontrivial E] (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
c • sphere x r = sphere (c • x) (‖c‖ * r) | begin
rcases eq_or_ne c 0 with rfl|hc,
{ simp [zero_smul_set, set.singleton_zero, hr] },
{ exact smul_sphere' hc x r }
end | lemma | smul_sphere | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"eq_or_ne",
"nontrivial",
"smul_sphere'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affinity_unit_ball {r : ℝ} (hr : 0 < r) (x : E) : x +ᵥ r • ball 0 1 = ball x r | by rw [smul_unit_ball_of_pos hr, vadd_ball_zero] | lemma | affinity_unit_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"smul_unit_ball_of_pos"
] | Any ball `metric.ball x r`, `0 < r` is the image of the unit ball under `λ y, x + r • y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affinity_unit_closed_ball {r : ℝ} (hr : 0 ≤ r) (x : E) :
x +ᵥ r • closed_ball 0 1 = closed_ball x r | by rw [smul_closed_unit_ball, real.norm_of_nonneg hr, vadd_closed_ball_zero] | lemma | affinity_unit_closed_ball | analysis.normed_space | src/analysis/normed_space/pointwise.lean | [
"analysis.normed.group.add_torsor",
"analysis.normed.group.pointwise",
"analysis.normed_space.basic"
] | [
"real.norm_of_nonneg",
"smul_closed_unit_ball",
"vadd_closed_ball_zero"
] | Any closed ball `metric.closed_ball x r`, `0 ≤ r` is the image of the unit closed ball under
`λ y, x + r • y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_coe (r : ℝ) : exp ℝ (r : ℍ[ℝ]) = ↑(exp ℝ r) | (map_exp ℝ (algebra_map ℝ ℍ[ℝ]) (continuous_algebra_map _ _) _).symm | lemma | quaternion.exp_coe | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"algebra_map",
"continuous_algebra_map",
"exp",
"map_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_exp_series_of_imaginary
{q : quaternion ℝ} (hq : q.re = 0) {c s : ℝ}
(hc : has_sum (λ n, (-1)^n * ‖q‖^(2 * n) / (2 * n)!) c)
(hs : has_sum (λ n, (-1)^n * ‖q‖^(2 * n + 1) / (2 * n + 1)!) s) :
has_sum (λ n, exp_series ℝ _ n (λ _, q)) (↑c + (s / ‖q‖) • q) | begin
replace hc := has_sum_coe.mpr hc,
replace hs := (hs.div_const ‖q‖).smul_const q,
obtain rfl | hq0 := eq_or_ne q 0,
{ simp_rw [exp_series_apply_zero, norm_zero, div_zero, zero_smul, add_zero],
simp_rw [norm_zero] at hc,
convert hc,
ext (_ | n) : 1,
{ rw [pow_zero, mul_zero, pow_zero, nat.fa... | lemma | quaternion.has_sum_exp_series_of_imaginary | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"div_div_cancel_left'",
"div_eq_mul_inv",
"div_one",
"div_zero",
"eq_or_ne",
"exp_series",
"exp_series_apply_eq",
"exp_series_apply_zero",
"has_sum",
"has_sum.even_add_odd",
"mul_div_assoc",
"mul_zero",
"nat.cast_one",
"nat.factorial_zero",
"neg_pow",
"one_mul",
"pow_mul",
"pow_suc... | Auxiliary result; if the power series corresponding to `real.cos` and `real.sin` evaluated
at `‖q‖` tend to `c` and `s`, then the exponential series tends to `c + (s / ‖q‖)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_of_re_eq_zero (q : quaternion ℝ) (hq : q.re = 0) :
exp ℝ q = ↑(real.cos ‖q‖) + (real.sin ‖q‖ / ‖q‖) • q | begin
rw [exp_eq_tsum],
refine has_sum.tsum_eq _,
simp_rw ← exp_series_apply_eq,
exact has_sum_exp_series_of_imaginary hq (real.has_sum_cos _) (real.has_sum_sin _),
end | lemma | quaternion.exp_of_re_eq_zero | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"exp",
"exp_eq_tsum",
"exp_series_apply_eq",
"has_sum.tsum_eq",
"quaternion",
"real.cos",
"real.has_sum_cos",
"real.has_sum_sin",
"real.sin"
] | The closed form for the quaternion exponential on imaginary quaternions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_eq (q : quaternion ℝ) :
exp ℝ q = exp ℝ q.re • (↑(real.cos ‖q.im‖) + (real.sin ‖q.im‖ / ‖q.im‖) • q.im) | begin
rw [←exp_of_re_eq_zero q.im q.im_re, ←coe_mul_eq_smul, ←exp_coe, ←exp_add_of_commute, re_add_im],
exact algebra.commutes q.re (_ : ℍ[ℝ]),
end | lemma | quaternion.exp_eq | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"algebra.commutes",
"exp",
"quaternion",
"real.cos",
"real.sin"
] | The closed form for the quaternion exponential on arbitrary quaternions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_exp (q : ℍ[ℝ]) : (exp ℝ q).re = exp ℝ q.re * (real.cos ‖q - q.re‖) | by simp [exp_eq] | lemma | quaternion.re_exp | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"exp",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_exp (q : ℍ[ℝ]) : (exp ℝ q).im = (exp ℝ q.re * (real.sin ‖q.im‖ / ‖q.im‖)) • q.im | by simp [exp_eq, smul_smul] | lemma | quaternion.im_exp | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"exp",
"real.sin",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_exp (q : ℍ[ℝ]) : norm_sq (exp ℝ q) = (exp ℝ q.re)^2 | calc norm_sq (exp ℝ q)
= norm_sq (exp ℝ q.re • (↑(real.cos ‖q.im‖) + (real.sin ‖q.im‖ / ‖q.im‖) • q.im))
: by rw exp_eq
... = (exp ℝ q.re)^2 * norm_sq ((↑(real.cos ‖q.im‖) + (real.sin ‖q.im‖ / ‖q.im‖) • q.im))
: by rw [norm_sq_smul]
... = (exp ℝ q.re)^2 * ((real.cos ‖q.im‖) ^ 2 + (real.sin ‖q.im‖)^2)
: ... | lemma | quaternion.norm_sq_exp | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"div_mul_cancel",
"div_pow",
"eq_or_ne",
"exp",
"mul_one",
"mul_zero",
"pow_ne_zero",
"real.cos",
"real.cos_sq_add_sin_sq",
"real.sin",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp (q : ℍ[ℝ]) : ‖exp ℝ q‖ = ‖exp ℝ q.re‖ | by rw [norm_eq_sqrt_real_inner (exp ℝ q), inner_self, norm_sq_exp, real.sqrt_sq_eq_abs,
real.norm_eq_abs] | lemma | quaternion.norm_exp | analysis.normed_space | src/analysis/normed_space/quaternion_exponential.lean | [
"analysis.quaternion",
"analysis.normed_space.exponential",
"analysis.special_functions.trigonometric.series"
] | [
"exp",
"norm_eq_sqrt_real_inner",
"real.norm_eq_abs",
"real.sqrt_sq_eq_abs"
] | Note that this implies that exponentials of pure imaginary quaternions are unit quaternions
since in that case the RHS is `1` via `exp_zero` and `norm_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_add (h : same_ray ℝ x y) : ‖x + y‖ = ‖x‖ + ‖y‖ | begin
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩,
rw [← add_smul, norm_smul_of_nonneg (add_nonneg ha hb), norm_smul_of_nonneg ha,
norm_smul_of_nonneg hb, add_mul]
end | lemma | same_ray.norm_add | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"add_smul",
"norm_smul_of_nonneg",
"same_ray"
] | If `x` and `y` are on the same ray, then the triangle inequality becomes the equality: the norm
of `x + y` is the sum of the norms of `x` and `y`. The converse is true for a strictly convex
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub (h : same_ray ℝ x y) : ‖x - y‖ = |‖x‖ - ‖y‖| | begin
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩,
wlog hab : b ≤ a,
{ rw same_ray_comm at h, rw [norm_sub_rev, abs_sub_comm],
exact this u b a hb ha h (le_of_not_le hab), },
rw ← sub_nonneg at hab,
rw [← sub_smul, norm_smul_of_nonneg hab, norm_smul_of_nonneg ha,
norm_smul_of_nonneg hb... | lemma | same_ray.norm_sub | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"abs_of_nonneg",
"abs_sub_comm",
"norm_smul_of_nonneg",
"same_ray",
"same_ray_comm",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul_eq (h : same_ray ℝ x y) : ‖x‖ • y = ‖y‖ • x | begin
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩,
simp only [norm_smul_of_nonneg, *, mul_smul, smul_comm (‖u‖)],
apply smul_comm
end | lemma | same_ray.norm_smul_eq | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"norm_smul_of_nonneg",
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inj_on_ray_left (hx : x ≠ 0) : {y | same_ray ℝ x y}.inj_on norm | begin
rintro y hy z hz h,
rcases hy.exists_nonneg_left hx with ⟨r, hr, rfl⟩,
rcases hz.exists_nonneg_left hx with ⟨s, hs, rfl⟩,
rw [norm_smul, norm_smul, mul_left_inj' (norm_ne_zero_iff.2 hx), norm_of_nonneg hr,
norm_of_nonneg hs] at h,
rw h
end | lemma | norm_inj_on_ray_left | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"mul_left_inj'",
"norm_smul",
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inj_on_ray_right (hy : y ≠ 0) : {x | same_ray ℝ x y}.inj_on norm | by simpa only [same_ray_comm] using norm_inj_on_ray_left hy | lemma | norm_inj_on_ray_right | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"norm_inj_on_ray_left",
"same_ray",
"same_ray_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray_iff_norm_smul_eq : same_ray ℝ x y ↔ ‖x‖ • y = ‖y‖ • x | ⟨same_ray.norm_smul_eq, λ h, or_iff_not_imp_left.2 $ λ hx, or_iff_not_imp_left.2 $ λ hy,
⟨‖y‖, ‖x‖, norm_pos_iff.2 hy, norm_pos_iff.2 hx, h.symm⟩⟩ | lemma | same_ray_iff_norm_smul_eq | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray_iff_inv_norm_smul_eq_of_ne (hx : x ≠ 0) (hy : y ≠ 0) :
same_ray ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y | by rw [inv_smul_eq_iff₀, smul_comm, eq_comm, inv_smul_eq_iff₀, same_ray_iff_norm_smul_eq];
rwa norm_ne_zero_iff | lemma | same_ray_iff_inv_norm_smul_eq_of_ne | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"inv_smul_eq_iff₀",
"same_ray",
"same_ray_iff_norm_smul_eq"
] | Two nonzero vectors `x y` in a real normed space are on the same ray if and only if the unit
vectors `‖x‖⁻¹ • x` and `‖y‖⁻¹ • y` are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
same_ray_iff_inv_norm_smul_eq : same_ray ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y | begin
rcases eq_or_ne x 0 with rfl|hx, { simp [same_ray.zero_left] },
rcases eq_or_ne y 0 with rfl|hy, { simp [same_ray.zero_right] },
simp only [same_ray_iff_inv_norm_smul_eq_of_ne hx hy, *, false_or]
end | lemma | same_ray_iff_inv_norm_smul_eq | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"eq_or_ne",
"same_ray",
"same_ray.zero_left",
"same_ray.zero_right",
"same_ray_iff_inv_norm_smul_eq_of_ne"
] | Two vectors `x y` in a real normed space are on the ray if and only if one of them is zero or
the unit vectors `‖x‖⁻¹ • x` and `‖y‖⁻¹ • y` are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
same_ray_iff_of_norm_eq (h : ‖x‖ = ‖y‖) : same_ray ℝ x y ↔ x = y | begin
obtain rfl | hy := eq_or_ne y 0,
{ rw [norm_zero, norm_eq_zero] at h,
exact iff_of_true (same_ray.zero_right _) h },
{ exact ⟨λ hxy, norm_inj_on_ray_right hy hxy same_ray.rfl h, λ hxy, hxy ▸ same_ray.rfl⟩ }
end | lemma | same_ray_iff_of_norm_eq | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"eq_or_ne",
"iff_of_true",
"norm_eq_zero",
"norm_inj_on_ray_right",
"same_ray",
"same_ray.rfl",
"same_ray.zero_right"
] | Two vectors of the same norm are on the same ray if and only if they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_same_ray_iff_of_norm_eq (h : ‖x‖ = ‖y‖) : ¬ same_ray ℝ x y ↔ x ≠ y | (same_ray_iff_of_norm_eq h).not | lemma | not_same_ray_iff_of_norm_eq | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"same_ray",
"same_ray_iff_of_norm_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray.eq_of_norm_eq (h : same_ray ℝ x y) (hn : ‖x‖ = ‖y‖) : x = y | (same_ray_iff_of_norm_eq hn).mp h | lemma | same_ray.eq_of_norm_eq | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"same_ray",
"same_ray_iff_of_norm_eq"
] | If two points on the same ray have the same norm, then they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
same_ray.norm_eq_iff (h : same_ray ℝ x y) : ‖x‖ = ‖y‖ ↔ x = y | ⟨h.eq_of_norm_eq, λ h, h ▸ rfl⟩ | lemma | same_ray.norm_eq_iff | analysis.normed_space | src/analysis/normed_space/ray.lean | [
"linear_algebra.ray",
"analysis.normed_space.basic"
] | [
"same_ray"
] | The norms of two vectors on the same ray are equal if and only if they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
riesz_lemma {F : subspace 𝕜 E} (hFc : is_closed (F : set E))
(hF : ∃ x : E, x ∉ F) {r : ℝ} (hr : r < 1) :
∃ x₀ : E, x₀ ∉ F ∧ ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖ | begin
classical,
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF,
let d := metric.inf_dist x F,
have hFn : (F : set E).nonempty, from ⟨_, F.zero_mem⟩,
have hdp : 0 < d,
from lt_of_le_of_ne metric.inf_dist_nonneg (λ heq, hx
((hFc.mem_iff_inf_dist_zero hFn).2 heq.symm)),
let r' := max r 2⁻¹,
have hr' : r' < 1... | lemma | riesz_lemma | analysis.normed_space | src/analysis/normed_space/riesz_lemma.lean | [
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"by_contradiction",
"is_closed",
"lt_div_iff",
"lt_div_iff'",
"metric.inf_dist",
"metric.inf_dist_le_dist_of_mem",
"metric.inf_dist_lt_iff",
"metric.inf_dist_nonneg",
"mul_le_mul_of_nonneg_right",
"mul_lt_iff_lt_one_right",
"subspace"
] | Riesz's lemma, which usually states that it is possible to find a
vector with norm 1 whose distance to a closed proper subspace is
arbitrarily close to 1. The statement here is in terms of multiples of
norms, since in general the existence of an element of norm exactly 1
is not guaranteed. For a variant giving an eleme... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
riesz_lemma_of_norm_lt
{c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
{F : subspace 𝕜 E} (hFc : is_closed (F : set E)) (hF : ∃ x : E, x ∉ F) :
∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ | begin
have Rpos : 0 < R := (norm_nonneg _).trans_lt hR,
have : ‖c‖ / R < 1, by { rw div_lt_iff Rpos, simpa using hR },
rcases riesz_lemma hFc hF this with ⟨x, xF, hx⟩,
have x0 : x ≠ 0 := λ H, by simpa [H] using xF,
obtain ⟨d, d0, dxlt, ledx, -⟩ :
∃ (d : 𝕜), d ≠ 0 ∧ ‖d • x‖ < R ∧ R / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹... | lemma | riesz_lemma_of_norm_lt | analysis.normed_space | src/analysis/normed_space/riesz_lemma.lean | [
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"div_lt_iff",
"div_nonneg",
"is_closed",
"mul_inv_cancel",
"mul_le_mul_of_nonneg_left",
"norm_smul",
"rescale_to_shell",
"riesz_lemma",
"ring",
"smul_smul",
"smul_sub",
"submodule.smul_mem",
"subspace"
] | A version of Riesz lemma: given a strict closed subspace `F`, one may find an element of norm `≤ R`
which is at distance at least `1` of every element of `F`. Here, `R` is any given constant
strictly larger than the norm of an element of norm `> 1`. For a version without an `R`, see
`riesz_lemma`.
Since we are consid... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.closed_ball_inf_dist_compl_subset_closure {x : F} {s : set F} (hx : x ∈ s) :
closed_ball x (inf_dist x sᶜ) ⊆ closure s | begin
cases eq_or_ne (inf_dist x sᶜ) 0 with h₀ h₀,
{ rw [h₀, closed_ball_zero'],
exact closure_mono (singleton_subset_iff.2 hx) },
{ rw ← closure_ball x h₀,
exact closure_mono ball_inf_dist_compl_subset }
end | lemma | metric.closed_ball_inf_dist_compl_subset_closure | analysis.normed_space | src/analysis/normed_space/riesz_lemma.lean | [
"analysis.normed_space.basic",
"topology.metric_space.hausdorff_distance"
] | [
"closure",
"closure_ball",
"closure_mono",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_radius (𝕜 : Type*) {A : Type*} [normed_field 𝕜] [ring A]
[algebra 𝕜 A] (a : A) : ℝ≥0∞ | ⨆ k ∈ spectrum 𝕜 a, ‖k‖₊ | def | spectral_radius | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra",
"normed_field",
"ring",
"spectrum"
] | The *spectral radius* is the supremum of the `nnnorm` (`‖⬝‖₊`) of elements in the spectrum,
coerced into an element of `ℝ≥0∞`. Note that it is possible for `spectrum 𝕜 a = ∅`. In this
case, `spectral_radius a = 0`. It is also possible that `spectrum 𝕜 a` be unbounded (though
not for Banach algebras, see ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
spectral_radius.of_subsingleton [subsingleton A] (a : A) :
spectral_radius 𝕜 a = 0 | by simp [spectral_radius] | lemma | spectrum.spectral_radius.of_subsingleton | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"spectral_radius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_radius_zero : spectral_radius 𝕜 (0 : A) = 0 | by { nontriviality A, simp [spectral_radius] } | lemma | spectrum.spectral_radius_zero | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"spectral_radius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_resolvent_set_of_spectral_radius_lt {a : A} {k : 𝕜} (h : spectral_radius 𝕜 a < ‖k‖₊) :
k ∈ ρ a | not_not.mp $ λ hn, h.not_le $ le_supr₂ k hn | lemma | spectrum.mem_resolvent_set_of_spectral_radius_lt | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"le_supr₂",
"spectral_radius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_resolvent_set (a : A) : is_open (ρ a) | units.is_open.preimage ((continuous_algebra_map 𝕜 A).sub continuous_const) | lemma | spectrum.is_open_resolvent_set | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"continuous_algebra_map",
"continuous_const",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed (a : A) : is_closed (σ a) | (is_open_resolvent_set a).is_closed_compl | lemma | spectrum.is_closed | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_resolvent_set_of_norm_lt_mul {a : A} {k : 𝕜} (h : ‖a‖ * ‖(1 : A)‖ < ‖k‖) :
k ∈ ρ a | begin
rw [resolvent_set, set.mem_set_of_eq, algebra.algebra_map_eq_smul_one],
nontriviality A,
have hk : k ≠ 0,
from ne_zero_of_norm_ne_zero ((mul_nonneg (norm_nonneg _) (norm_nonneg _)).trans_lt h).ne',
let ku := units.map (↑ₐ).to_monoid_hom (units.mk0 k hk),
rw [←inv_inv (‖(1 : A)‖), mul_inv_lt_iff (inv... | lemma | spectrum.mem_resolvent_set_of_norm_lt_mul | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra.algebra_map_eq_smul_one",
"is_unit",
"mul_inv_lt_iff",
"norm_algebra_map",
"one_ne_zero",
"resolvent_set",
"units.map",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_resolvent_set_of_norm_lt [norm_one_class A] {a : A} {k : 𝕜} (h : ‖a‖ < ‖k‖) :
k ∈ ρ a | mem_resolvent_set_of_norm_lt_mul (by rwa [norm_one, mul_one]) | lemma | spectrum.mem_resolvent_set_of_norm_lt | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"mul_one",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_norm_mul_of_mem {a : A} {k : 𝕜} (hk : k ∈ σ a) :
‖k‖ ≤ ‖a‖ * ‖(1 : A)‖ | le_of_not_lt $ mt mem_resolvent_set_of_norm_lt_mul hk | lemma | spectrum.norm_le_norm_mul_of_mem | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_norm_of_mem [norm_one_class A] {a : A} {k : 𝕜} (hk : k ∈ σ a) :
‖k‖ ≤ ‖a‖ | le_of_not_lt $ mt mem_resolvent_set_of_norm_lt hk | lemma | spectrum.norm_le_norm_of_mem | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_closed_ball_norm_mul (a : A) :
σ a ⊆ metric.closed_ball (0 : 𝕜) (‖a‖ * ‖(1 : A)‖) | λ k hk, by simp [norm_le_norm_mul_of_mem hk] | lemma | spectrum.subset_closed_ball_norm_mul | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_closed_ball_norm [norm_one_class A] (a : A) :
σ a ⊆ metric.closed_ball (0 : 𝕜) (‖a‖) | λ k hk, by simp [norm_le_norm_of_mem hk] | lemma | spectrum.subset_closed_ball_norm | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"metric.closed_ball",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded (a : A) : metric.bounded (σ a) | (metric.bounded_iff_subset_ball 0).mpr ⟨‖a‖ * ‖(1 : A)‖, subset_closed_ball_norm_mul a⟩ | lemma | spectrum.is_bounded | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"metric.bounded",
"metric.bounded_iff_subset_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact [proper_space 𝕜] (a : A) : is_compact (σ a) | metric.is_compact_of_is_closed_bounded (spectrum.is_closed a) (is_bounded a) | theorem | spectrum.is_compact | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"is_compact",
"metric.is_compact_of_is_closed_bounded",
"proper_space",
"spectrum.is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_radius_le_nnnorm [norm_one_class A] (a : A) :
spectral_radius 𝕜 a ≤ ‖a‖₊ | by { refine supr₂_le (λ k hk, _), exact_mod_cast norm_le_norm_of_mem hk } | theorem | spectrum.spectral_radius_le_nnnorm | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"norm_one_class",
"spectral_radius",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nnnorm_eq_spectral_radius_of_nonempty [proper_space 𝕜] {a : A} (ha : (σ a).nonempty) :
∃ k ∈ σ a, (‖k‖₊ : ℝ≥0∞) = spectral_radius 𝕜 a | begin
obtain ⟨k, hk, h⟩ := (spectrum.is_compact a).exists_forall_ge ha continuous_nnnorm.continuous_on,
exact ⟨k, hk, le_antisymm (le_supr₂ k hk) (supr₂_le $ by exact_mod_cast h)⟩,
end | lemma | spectrum.exists_nnnorm_eq_spectral_radius_of_nonempty | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"le_supr₂",
"proper_space",
"spectral_radius",
"spectrum.is_compact",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_radius_lt_of_forall_lt_of_nonempty [proper_space 𝕜] {a : A}
(ha : (σ a).nonempty) {r : ℝ≥0} (hr : ∀ k ∈ σ a, ‖k‖₊ < r) :
spectral_radius 𝕜 a < r | Sup_image.symm.trans_lt $ ((spectrum.is_compact a).Sup_lt_iff_of_continuous ha
(ennreal.continuous_coe.comp continuous_nnnorm).continuous_on (r : ℝ≥0∞)).mpr
(by exact_mod_cast hr) | lemma | spectrum.spectral_radius_lt_of_forall_lt_of_nonempty | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"continuous_on",
"proper_space",
"spectral_radius",
"spectrum.is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_radius_le_pow_nnnorm_pow_one_div (a : A) (n : ℕ) :
spectral_radius 𝕜 a ≤ (‖a ^ (n + 1)‖₊) ^ (1 / (n + 1) : ℝ) * (‖(1 : A)‖₊) ^ (1 / (n + 1) : ℝ) | begin
refine supr₂_le (λ k hk, _),
/- apply easy direction of the spectral mapping theorem for polynomials -/
have pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1)),
by simpa only [one_mul, algebra.algebra_map_eq_smul_one, one_smul, aeval_monomial, one_mul,
eval_monomial] using subset_polynomial_aeval a (monomial... | theorem | spectrum.spectral_radius_le_pow_nnnorm_pow_one_div | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra.algebra_map_eq_smul_one",
"ennreal.coe_mul",
"ennreal.coe_mul_rpow",
"mul_one_div_cancel",
"nat.cast_succ",
"nat.succ_pos'",
"nnnorm_pow",
"nnnorm_pow_le",
"one_mul",
"one_smul",
"pow_mem",
"real.to_nnreal_mono",
"real.to_nnreal_mul",
"spectral_radius",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_radius_le_liminf_pow_nnnorm_pow_one_div (a : A) :
spectral_radius 𝕜 a ≤ at_top.liminf (λ n : ℕ, (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) | begin
refine ennreal.le_of_forall_lt_one_mul_le (λ ε hε, _),
by_cases ε = 0,
{ simp only [h, zero_mul, zero_le'] },
have hε' : ε⁻¹ ≠ ∞,
from λ h', h (by simpa only [inv_inv, inv_top] using congr_arg (λ (x : ℝ≥0∞), x⁻¹) h'),
simp only [ennreal.mul_le_iff_le_inv h (hε.trans_le le_top).ne, mul_comm ε⁻¹,
... | theorem | spectrum.spectral_radius_le_liminf_pow_nnnorm_pow_one_div | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"ennreal.coe_ne_top",
"ennreal.eventually_pow_one_div_le",
"ennreal.infi_mul",
"ennreal.le_of_forall_lt_one_mul_le",
"ennreal.mul_le_iff_le_inv",
"ennreal.supr_mul",
"inv_inv",
"inv_one",
"le_infi",
"le_supr",
"le_top",
"mul_comm",
"mul_le_mul_left'",
"spectral_radius",
"zero_le'",
"ze... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_resolvent {a : A} {k : 𝕜} (hk : k ∈ ρ a) :
has_deriv_at (resolvent a) (-(resolvent a k) ^ 2) k | begin
have H₁ : has_fderiv_at ring.inverse _ (↑ₐk - a) := has_fderiv_at_ring_inverse hk.unit,
have H₂ : has_deriv_at (λ k, ↑ₐk - a) 1 k,
{ simpa using (algebra.linear_map 𝕜 A).has_deriv_at.sub_const a },
simpa [resolvent, sq, hk.unit_spec, ← ring.inverse_unit hk.unit] using H₁.comp_has_deriv_at k H₂,
end | theorem | spectrum.has_deriv_at_resolvent | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra.linear_map",
"has_deriv_at",
"has_deriv_at.sub_const",
"has_fderiv_at",
"has_fderiv_at_ring_inverse",
"resolvent",
"ring.inverse",
"ring.inverse_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_resolvent_le_forall (a : A) :
∀ ε > 0, ∃ R > 0, ∀ z : 𝕜, R ≤ ‖z‖ → ‖resolvent a z‖ ≤ ε | begin
obtain ⟨c, c_pos, hc⟩ := (@normed_ring.inverse_one_sub_norm A _ _).exists_pos,
rw [is_O_with_iff, eventually_iff, metric.mem_nhds_iff] at hc,
rcases hc with ⟨δ, δ_pos, hδ⟩,
simp only [cstar_ring.norm_one, mul_one] at hδ,
intros ε hε,
have ha₁ : 0 < ‖a‖ + 1 := lt_of_le_of_lt (norm_nonneg a) (lt_add_one... | lemma | spectrum.norm_resolvent_le_forall | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra.algebra_map_eq_smul_one",
"cstar_ring.norm_one",
"inv_le_of_inv_le",
"inv_mul_cancel_right₀",
"inv_mul_lt_iff",
"is_unit",
"lt_add_one",
"metric.mem_nhds_iff",
"mul_assoc",
"mul_le_mul",
"mul_le_mul_of_nonneg_right",
"mul_lt_mul_of_pos_left",
"mul_one",
"norm_inv",
"norm_smul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball_inverse_one_sub_smul [complete_space A] (a : A) :
has_fpower_series_on_ball (λ z : 𝕜, ring.inverse (1 - z • a))
(λ n, continuous_multilinear_map.mk_pi_field 𝕜 (fin n) (a ^ n)) 0 (‖a‖₊)⁻¹ | { r_le :=
begin
refine le_of_forall_nnreal_lt (λ r hr, le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) (λ n, _)),
rw [←norm_to_nnreal, norm_mk_pi_field, norm_to_nnreal],
cases n,
{ simp only [le_refl, mul_one, or_true, le_max_iff, pow_zero] },
{ refine le_trans (le_trans (mul_le_mul_right' (nnnorm_... | lemma | spectrum.has_fpower_series_on_ball_inverse_one_sub_smul | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"complete_space",
"continuous_multilinear_map.mk_pi_field",
"has_fpower_series_on_ball",
"has_sum",
"le_max_iff",
"metric.emetric_ball_nnreal",
"mul_comm",
"mul_le_mul_right'",
"mul_one",
"nnnorm_pow_le'",
"nnnorm_smul",
"nnreal.lt_inv_iff_mul_lt",
"normed_ring.inverse_one_sub",
"normed_ri... | In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_one_sub_smul_of_lt_inv_radius {a : A} {z : 𝕜} (h : ↑‖z‖₊ < (spectral_radius 𝕜 a)⁻¹) :
is_unit (1 - z • a) | begin
by_cases hz : z = 0,
{ simp only [hz, is_unit_one, sub_zero, zero_smul] },
{ let u := units.mk0 z hz,
suffices hu : is_unit (u⁻¹ • 1 - a),
{ rwa [is_unit.smul_sub_iff_sub_inv_smul, inv_inv u] at hu },
{ rw [units.smul_def, ←algebra.algebra_map_eq_smul_one, ←mem_resolvent_set_iff],
refine m... | lemma | spectrum.is_unit_one_sub_smul_of_lt_inv_radius | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"inv_inv",
"is_unit",
"is_unit.smul_sub_iff_sub_inv_smul",
"is_unit_one",
"nnnorm_inv",
"spectral_radius",
"units.coe_inv",
"units.coe_mk0",
"units.mk0",
"units.smul_def",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_inverse_one_sub_smul [complete_space A] {a : A} {r : ℝ≥0}
(hr : (r : ℝ≥0∞) < (spectral_radius 𝕜 a)⁻¹) :
differentiable_on 𝕜 (λ z : 𝕜, ring.inverse (1 - z • a)) (metric.closed_ball 0 r) | begin
intros z z_mem,
apply differentiable_at.differentiable_within_at,
have hu : is_unit (1 - z • a),
{ refine is_unit_one_sub_smul_of_lt_inv_radius (lt_of_le_of_lt (coe_mono _) hr),
simpa only [norm_to_nnreal, real.to_nnreal_coe]
using real.to_nnreal_mono (mem_closed_ball_zero_iff.mp z_mem) },
hav... | theorem | spectrum.differentiable_on_inverse_one_sub_smul | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"complete_space",
"differentiable",
"differentiable_at.comp",
"differentiable_at.differentiable_within_at",
"differentiable_at_inverse",
"differentiable_on",
"is_unit",
"metric.closed_ball",
"real.to_nnreal_coe",
"real.to_nnreal_mono",
"ring.inverse",
"spectral_radius"
] | In a Banach algebra `A` over `𝕜`, for `a : A` the function `λ z, (1 - z • a)⁻¹` is
differentiable on any closed ball centered at zero of radius `r < (spectral_radius 𝕜 a)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limsup_pow_nnnorm_pow_one_div_le_spectral_radius (a : A) :
limsup (λ n : ℕ, ↑‖a ^ n‖₊ ^ (1 / n : ℝ)) at_top ≤ spectral_radius ℂ a | begin
refine ennreal.inv_le_inv.mp (le_of_forall_pos_nnreal_lt (λ r r_pos r_lt, _)),
simp_rw [inv_limsup, ←one_div],
let p : formal_multilinear_series ℂ ℂ A :=
λ n, continuous_multilinear_map.mk_pi_field ℂ (fin n) (a ^ n),
suffices h : (r : ℝ≥0∞) ≤ p.radius,
{ convert h,
simp only [p.radius_eq_liminf,... | lemma | spectrum.limsup_pow_nnnorm_pow_one_div_le_spectral_radius | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"continuous_multilinear_map.mk_pi_field",
"ennreal.coe_rpow_def",
"formal_multilinear_series",
"has_fpower_series_on_ball",
"spectral_radius"
] | The `limsup` relationship for the spectral radius used to prove `spectrum.gelfand_formula`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius (a : A) :
tendsto (λ n : ℕ, ((‖a ^ n‖₊ ^ (1 / n : ℝ)) : ℝ≥0∞)) at_top (𝓝 (spectral_radius ℂ a)) | tendsto_of_le_liminf_of_limsup_le (spectral_radius_le_liminf_pow_nnnorm_pow_one_div ℂ a)
(limsup_pow_nnnorm_pow_one_div_le_spectral_radius a) | theorem | spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"spectral_radius",
"tendsto_of_le_liminf_of_limsup_le"
] | **Gelfand's formula**: Given an element `a : A` of a complex Banach algebra, the
`spectral_radius` of `a` is the limit of the sequence `‖a ^ n‖₊ ^ (1 / n)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_norm_pow_one_div_tendsto_nhds_spectral_radius (a : A) :
tendsto (λ n : ℕ, ennreal.of_real (‖a ^ n‖ ^ (1 / n : ℝ))) at_top (𝓝 (spectral_radius ℂ a)) | begin
convert pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius a,
ext1,
rw [←of_real_rpow_of_nonneg (norm_nonneg _) _, ←coe_nnnorm, coe_nnreal_eq],
exact one_div_nonneg.mpr (by exact_mod_cast zero_le _),
end | theorem | spectrum.pow_norm_pow_one_div_tendsto_nhds_spectral_radius | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"ennreal.of_real",
"spectral_radius"
] | **Gelfand's formula**: Given an element `a : A` of a complex Banach algebra, the
`spectral_radius` of `a` is the limit of the sequence `‖a ^ n‖₊ ^ (1 / n)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty : (spectrum ℂ a).nonempty | begin
/- Suppose `σ a = ∅`, then resolvent set is `ℂ`, any `(z • 1 - a)` is a unit, and `resolvent`
is differentiable on `ℂ`. -/
rw set.nonempty_iff_ne_empty,
by_contra h,
have H₀ : resolvent_set ℂ a = set.univ, by rwa [spectrum, set.compl_empty_iff] at h,
have H₁ : differentiable ℂ (λ z : ℂ, resolvent a z)... | theorem | spectrum.nonempty | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"by_contra",
"differentiable",
"differentiable_at",
"em",
"not_is_unit_zero",
"real.norm_of_nonneg",
"resolvent",
"resolvent_set",
"set.compl_empty_iff",
"set.mem_univ",
"set.nonempty_iff_ne_empty",
"spectrum",
"zero_lt_one"
] | In a (nontrivial) complex Banach algebra, every element has nonempty spectrum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nnnorm_eq_spectral_radius : ∃ z ∈ spectrum ℂ a, (‖z‖₊ : ℝ≥0∞) = spectral_radius ℂ a | exists_nnnorm_eq_spectral_radius_of_nonempty (spectrum.nonempty a) | lemma | spectrum.exists_nnnorm_eq_spectral_radius | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"spectral_radius",
"spectrum",
"spectrum.nonempty"
] | In a complex Banach algebra, the spectral radius is always attained by some element of the
spectrum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
spectral_radius_lt_of_forall_lt {r : ℝ≥0} (hr : ∀ z ∈ spectrum ℂ a, ‖z‖₊ < r) :
spectral_radius ℂ a < r | spectral_radius_lt_of_forall_lt_of_nonempty (spectrum.nonempty a) hr | lemma | spectrum.spectral_radius_lt_of_forall_lt | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"spectral_radius",
"spectrum",
"spectrum.nonempty"
] | In a complex Banach algebra, if every element of the spectrum has norm strictly less than
`r : ℝ≥0`, then the spectral radius is also strictly less than `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_polynomial_aeval (p : ℂ[X]) :
spectrum ℂ (aeval a p) = (λ k, eval k p) '' (spectrum ℂ a) | map_polynomial_aeval_of_nonempty a p (spectrum.nonempty a) | lemma | spectrum.map_polynomial_aeval | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"spectrum",
"spectrum.nonempty"
] | The **spectral mapping theorem** for polynomials in a Banach algebra over `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pow (n : ℕ) : spectrum ℂ (a ^ n) = (λ x, x ^ n) '' (spectrum ℂ a) | by simpa only [aeval_X_pow, eval_pow, eval_X] using map_polynomial_aeval a (X ^ n) | lemma | spectrum.map_pow | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"map_pow",
"spectrum"
] | A specialization of the spectral mapping theorem for polynomials in a Banach algebra over `ℂ`
to monic monomials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_eq_of_mem {a : A} {z : ℂ} (h : z ∈ σ a) : algebra_map ℂ A z = a | by rwa [mem_iff, hA, not_not, sub_eq_zero] at h | lemma | spectrum.algebra_map_eq_of_mem | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra_map",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.normed_ring.alg_equiv_complex_of_complete
[complete_space A] : ℂ ≃ₐ[ℂ] A | let nt : nontrivial A := ⟨⟨1, 0, hA.mp ⟨⟨1, 1, mul_one _, mul_one _⟩, rfl⟩⟩⟩ in
{ to_fun := algebra_map ℂ A,
inv_fun := λ a, (@spectrum.nonempty _ _ _ _ nt a).some,
left_inv := λ z, by simpa only [@scalar_eq _ _ _ _ _ nt _] using
(@spectrum.nonempty _ _ _ _ nt $ algebra_map ℂ A z).some_mem,
right_inv := λ a, ... | def | normed_ring.alg_equiv_complex_of_complete | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra.of_id",
"algebra_map",
"complete_space",
"inv_fun",
"mul_one",
"nontrivial",
"spectrum.nonempty"
] | **Gelfand-Mazur theorem**: For a complex Banach division algebra, the natural `algebra_map ℂ A`
is an algebra isomorphism whose inverse is given by selecting the (unique) element of
`spectrum ℂ a`. In addition, `algebra_map_isometry` guarantees this map is an isometry.
Note: because `normed_division_ring` requires the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_mem_exp [is_R_or_C 𝕜] [normed_ring A] [normed_algebra 𝕜 A] [complete_space A]
(a : A) {z : 𝕜} (hz : z ∈ spectrum 𝕜 a) : exp 𝕜 z ∈ spectrum 𝕜 (exp 𝕜 a) | begin
have hexpmul : exp 𝕜 a = exp 𝕜 (a - ↑ₐ z) * ↑ₐ (exp 𝕜 z),
{ rw [algebra_map_exp_comm z, ←exp_add_of_commute (algebra.commutes z (a - ↑ₐz)).symm,
sub_add_cancel] },
let b := ∑' n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐz) ^ n,
have hb : summable (λ n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐz) ^ ... | theorem | spectrum.exp_mem_exp | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"algebra.commutes",
"algebra.smul_mul_assoc",
"algebra_map_exp_comm",
"commute.is_unit_mul_iff",
"complete_space",
"div_eq_mul_inv",
"div_le_div",
"exp",
"exp_eq_tsum",
"exp_series_summable'",
"filter.eventually_cofinite_ne",
"inv_one",
"is_R_or_C",
"is_R_or_C.norm_nat_cast",
"is_unit.su... | For `𝕜 = ℝ` or `𝕜 = ℂ`, `exp 𝕜` maps the spectrum of `a` into the spectrum of `exp 𝕜 a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_linear_map (φ : A →ₐ[𝕜] 𝕜) : A →L[𝕜] 𝕜 | { cont := map_continuous φ, .. φ.to_linear_map } | def | alg_hom.to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"cont"
] | An algebra homomorphism into the base field, as a continuous linear map (since it is
automatically bounded). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_linear_map (φ : A →ₐ[𝕜] 𝕜) :
⇑φ.to_continuous_linear_map = φ | rfl | lemma | alg_hom.coe_to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_apply_le_self_mul_norm_one [alg_hom_class F 𝕜 A 𝕜] (f : F) (a : A) :
‖f a‖ ≤ ‖a‖ * ‖(1 : A)‖ | spectrum.norm_le_norm_mul_of_mem (apply_mem_spectrum f _) | lemma | alg_hom.norm_apply_le_self_mul_norm_one | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"alg_hom_class",
"spectrum.norm_le_norm_mul_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_apply_le_self [norm_one_class A] [alg_hom_class F 𝕜 A 𝕜] (f : F) (a : A) : ‖f a‖ ≤ ‖a‖ | spectrum.norm_le_norm_of_mem (apply_mem_spectrum f _) | lemma | alg_hom.norm_apply_le_self | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"alg_hom_class",
"norm_one_class",
"spectrum.norm_le_norm_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_map_norm [norm_one_class A] (φ : A →ₐ[𝕜] 𝕜) :
‖φ.to_continuous_linear_map‖ = 1 | continuous_linear_map.op_norm_eq_of_bounds zero_le_one
(λ a, (one_mul ‖a‖).symm ▸ spectrum.norm_le_norm_of_mem (apply_mem_spectrum φ _))
(λ _ _ h, by simpa only [coe_to_continuous_linear_map, map_one, norm_one, mul_one] using h 1) | lemma | alg_hom.to_continuous_linear_map_norm | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"continuous_linear_map.op_norm_eq_of_bounds",
"map_one",
"mul_one",
"norm_one_class",
"one_mul",
"spectrum.norm_le_norm_of_mem",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_alg_hom : (character_space 𝕜 A) ≃ (A →ₐ[𝕜] 𝕜) | { to_fun := to_alg_hom,
inv_fun := λ f,
{ val := f.to_continuous_linear_map,
property := by { rw eq_set_map_one_map_mul, exact ⟨map_one f, map_mul f⟩ } },
left_inv := λ f, subtype.ext $ continuous_linear_map.ext $ λ x, rfl,
right_inv := λ f, alg_hom.ext $ λ x, rfl } | def | weak_dual.character_space.equiv_alg_hom | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [
"alg_hom.ext",
"continuous_linear_map.ext",
"inv_fun",
"map_mul",
"subtype.ext"
] | The equivalence between characters and algebra homomorphisms into the base field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_alg_hom_coe (f : character_space 𝕜 A) : ⇑(equiv_alg_hom f) = f | rfl | lemma | weak_dual.character_space.equiv_alg_hom_coe | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_alg_hom_symm_coe (f : A →ₐ[𝕜] 𝕜) : ⇑(equiv_alg_hom.symm f) = f | rfl | lemma | weak_dual.character_space.equiv_alg_hom_symm_coe | analysis.normed_space | src/analysis/normed_space/spectrum.lean | [
"field_theory.is_alg_closed.spectrum",
"analysis.complex.liouville",
"analysis.complex.polynomial",
"analysis.analytic.radius_liminf",
"topology.algebra.module.character_space",
"analysis.normed_space.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_fst_exp_series [field 𝕜] [ring R]
[add_comm_group M] [algebra 𝕜 R]
[module R M] [module Rᵐᵒᵖ M] [smul_comm_class R Rᵐᵒᵖ M]
[module 𝕜 M] [is_scalar_tower 𝕜 R M] [is_scalar_tower 𝕜 Rᵐᵒᵖ M]
[topological_ring R] [topological_add_group M]
[has_continuous_smul R M] [has_continuous_smul Rᵐᵒᵖ M]
(x : t... | by simpa [exp_series_apply_eq] using h | lemma | triv_sq_zero_ext.has_sum_fst_exp_series | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"add_comm_group",
"algebra",
"exp_series",
"exp_series_apply_eq",
"field",
"has_continuous_smul",
"has_sum",
"is_scalar_tower",
"module",
"ring",
"smul_comm_class",
"topological_add_group",
"topological_ring"
] | If `exp R x.fst` converges to `e` then `(exp R x).fst` converges to `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_snd_exp_series_of_smul_comm [field 𝕜] [char_zero 𝕜] [ring R]
[add_comm_group M] [algebra 𝕜 R]
[module R M] [module Rᵐᵒᵖ M] [smul_comm_class R Rᵐᵒᵖ M]
[module 𝕜 M] [is_scalar_tower 𝕜 R M] [is_scalar_tower 𝕜 Rᵐᵒᵖ M]
[topological_ring R] [topological_add_group M]
[has_continuous_smul R M] [has_cont... | begin
simp_rw [exp_series_apply_eq] at *,
conv
{ congr,
funext,
rw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast 𝕜 n, smul_smul, inv_mul_eq_div,
←inv_div, ←smul_assoc], },
apply has_sum.smul_const,
rw [←has_sum_nat_add_iff' 1], swap, apply_instance,
rw [finset.range_one, finset.s... | lemma | triv_sq_zero_ext.has_sum_snd_exp_series_of_smul_comm | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"add_comm_group",
"algebra",
"char_zero",
"div_zero",
"exp_series",
"exp_series_apply_eq",
"field",
"finset.range_one",
"has_continuous_smul",
"has_sum",
"has_sum.smul_const",
"inv_mul_eq_div",
"inv_zero",
"is_scalar_tower",
"module",
"mul_div_cancel_left",
"mul_opposite.op",
"nat.... | If `exp R x.fst` converges to `e` then `(exp R x).snd` converges to `e • x.snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_exp_series_of_smul_comm [field 𝕜] [char_zero 𝕜] [ring R]
[add_comm_group M] [algebra 𝕜 R]
[module R M] [module Rᵐᵒᵖ M] [smul_comm_class R Rᵐᵒᵖ M]
[module 𝕜 M] [is_scalar_tower 𝕜 R M] [is_scalar_tower 𝕜 Rᵐᵒᵖ M]
[topological_ring R] [topological_add_group M]
[has_continuous_smul R M] [has_continuo... | by simpa only [inl_fst_add_inr_snd_eq] using
(has_sum_inl _ $ has_sum_fst_exp_series 𝕜 x h).add
(has_sum_inr _ $ has_sum_snd_exp_series_of_smul_comm 𝕜 x hx h) | lemma | triv_sq_zero_ext.has_sum_exp_series_of_smul_comm | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"add_comm_group",
"algebra",
"char_zero",
"exp_series",
"field",
"has_continuous_smul",
"has_sum",
"is_scalar_tower",
"module",
"mul_opposite.op",
"ring",
"smul_comm_class",
"topological_add_group",
"topological_ring"
] | If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_def_of_smul_comm (x : tsze R M) (hx : mul_opposite.op x.fst • x.snd = x.fst • x.snd) :
exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) | begin
simp_rw [exp, formal_multilinear_series.sum],
refine (has_sum_exp_series_of_smul_comm 𝕜 x hx _).tsum_eq,
exact exp_series_has_sum_exp _,
end | lemma | triv_sq_zero_ext.exp_def_of_smul_comm | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp",
"exp_series_has_sum_exp",
"formal_multilinear_series.sum",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) | begin
rw [exp_def_of_smul_comm, snd_inl, fst_inl, smul_zero, inr_zero, add_zero],
{ rw [snd_inl, fst_inl, smul_zero, smul_zero] }
end | lemma | triv_sq_zero_ext.exp_inl | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m | begin
rw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one],
{ rw [snd_inr, fst_inr, mul_opposite.op_zero, zero_smul, zero_smul] }
end | lemma | triv_sq_zero_ext.exp_inr | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp",
"exp_zero",
"mul_opposite.op_zero",
"one_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) | exp_def_of_smul_comm 𝕜 x (op_smul_eq_smul _ _) | lemma | triv_sq_zero_ext.exp_def | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_exp (x : tsze R M) : fst (exp 𝕜 x) = exp 𝕜 x.fst | by rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | lemma | triv_sq_zero_ext.fst_exp | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd | by rw [exp_def, snd_add, snd_inl, snd_inr, zero_add] | lemma | triv_sq_zero_ext.snd_exp | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_smul_exp_of_invertible (x : tsze R M) [invertible x.fst] :
x = x.fst • exp 𝕜 (⅟x.fst • inr x.snd) | by rw [←inr_smul, exp_inr, smul_add, ←inl_one, ←inl_smul, ←inr_smul, smul_eq_mul, mul_one,
smul_smul, mul_inv_of_self, one_smul, inl_fst_add_inr_snd_eq] | lemma | triv_sq_zero_ext.eq_smul_exp_of_invertible | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp",
"invertible",
"mul_inv_of_self",
"mul_one",
"one_smul",
"smul_add",
"smul_eq_mul",
"smul_smul"
] | Polar form of trivial-square-zero extension. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
x = x.fst • exp 𝕜 (x.fst⁻¹ • inr x.snd) | begin
letI : invertible x.fst := invertible_of_nonzero hx,
exact eq_smul_exp_of_invertible _ _
end | lemma | triv_sq_zero_ext.eq_smul_exp_of_ne_zero | analysis.normed_space | src/analysis/normed_space/triv_sq_zero_ext.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.exponential",
"topology.instances.triv_sq_zero_ext"
] | [
"exp",
"invertible",
"invertible_of_nonzero"
] | More convenient version of `triv_sq_zero_ext.eq_smul_exp_of_invertible` for when `R` is a
field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_sub (t : R) (h : ‖t‖ < 1) : Rˣ | { val := 1 - t,
inv := ∑' n : ℕ, t ^ n,
val_inv := mul_neg_geom_series t h,
inv_val := geom_series_mul_neg t h } | def | units.one_sub | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"geom_series_mul_neg",
"mul_neg_geom_series"
] | In a complete normed ring, a perturbation of `1` by an element `t` of distance less than `1`
from `1` is a unit. Here we construct its `units` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add (x : Rˣ) (t : R) (h : ‖t‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ | units.copy -- to make `coe_add` true definitionally, for convenience
(x * (units.one_sub (-(↑x⁻¹ * t)) begin
nontriviality R using [zero_lt_one],
have hpos : 0 < ‖(↑x⁻¹ : R)‖ := units.norm_pos x⁻¹,
calc ‖-(↑x⁻¹ * t)‖
= ‖↑x⁻¹ * t‖ : by { rw norm_neg }
... ≤ ‖(↑x⁻¹ ... | def | units.add | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"mul_inv_cancel",
"norm_mul_le",
"units.copy",
"units.norm_pos",
"units.one_sub",
"zero_lt_one"
] | In a complete normed ring, a perturbation of a unit `x` by an element `t` of distance less than
`‖x⁻¹‖⁻¹` from `x` is a unit. Here we construct its `units` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_of_nearby (x : Rˣ) (y : R) (h : ‖y - x‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ | units.copy (x.add (y - x : R) h) y (by simp) _ rfl | def | units.unit_of_nearby | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"units.copy"
] | In a complete normed ring, an element `y` of distance less than `‖x⁻¹‖⁻¹` from `x` is a unit.
Here we construct its `units` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open : is_open {x : R | is_unit x} | begin
nontriviality R,
apply metric.is_open_iff.mpr,
rintros x' ⟨x, rfl⟩,
refine ⟨‖(↑x⁻¹ : R)‖⁻¹, _root_.inv_pos.mpr (units.norm_pos x⁻¹), _⟩,
intros y hy,
rw [metric.mem_ball, dist_eq_norm] at hy,
exact (x.unit_of_nearby y hy).is_unit
end | lemma | units.is_open | analysis.normed_space | src/analysis/normed_space/units.lean | [
"topology.algebra.ring.ideal",
"analysis.specific_limits.normed"
] | [
"is_open",
"is_unit",
"metric.mem_ball",
"units.norm_pos"
] | The group of units of a complete normed ring is an open subset of the ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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