| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Filter |
| open scoped Topology Ring |
| namespace BrownLog |
| variable {A : Type*} [NormedRing A] [CompleteSpace A] |
| lemma summable_pow_of_root_tendsto (a : A) |
| (ha : Tendsto (fun n:ℕ=>‖a^n‖^(1/(n:ℝ))) atTop (𝓝 0)) : Summable (fun n:ℕ=>a^n) := by |
| have he : ∀ᶠ n : ℕ in atTop,0<n ∧ ‖a^n‖^(1/(n:ℝ))≤(1/2:ℝ) := by |
| filter_upwards [eventually_gt_atTop 0,ha.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))] with n hn h |
| exact ⟨hn,h.le⟩ |
| apply (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)<1)).of_norm_bounded_eventually_nat |
| filter_upwards [he] with n hn |
| have h := pow_le_pow_left₀ (Real.rpow_nonneg (norm_nonneg _) _) hn.2 n |
| have hi : (‖a^n‖^(1/(n:ℝ)))^n=‖a^n‖ := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul (norm_nonneg _)] |
| rw [one_div_mul_cancel (by exact_mod_cast hn.1.ne'),Real.rpow_one] |
| rwa [hi] at h |
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| omit [CompleteSpace A] in |
| lemma norm_root_smul_tendsto {𝕜 : Type*} [RCLike 𝕜] [NormedAlgebra 𝕜 A] |
| (a : A) (c : 𝕜) (ha : Tendsto (fun n:ℕ=>‖a^n‖^(1/(n:ℝ))) atTop (𝓝 0)) : |
| Tendsto (fun n:ℕ=>‖(c•a)^n‖^(1/(n:ℝ))) atTop (𝓝 0) := by |
| have hc := ha.const_mul ‖c‖ |
| simp only [mul_zero] at hc |
| apply hc.congr' |
| filter_upwards [eventually_gt_atTop 0] with n hn |
| rw [smul_pow,norm_smul,norm_pow c n,Real.mul_rpow (by positivity) (norm_nonneg _)] |
| rw [← Real.rpow_natCast,← Real.rpow_mul (norm_nonneg c)] |
| rw [mul_one_div_cancel (by exact_mod_cast hn.ne'),Real.rpow_one] |
|
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| omit [CompleteSpace A] in |
| lemma inverse_one_sub_eq_tsum (a : A) (ha : Summable (fun n:ℕ=>a^n)) : |
| IsUnit (1-a) ∧ (1-a)⁻¹ʳ=∑' n:ℕ,a^n := by |
| let u : Aˣ := ⟨1-a,∑' n:ℕ,a^n,ha.one_sub_mul_tsum_pow,ha.tsum_pow_mul_one_sub⟩ |
| refine ⟨u.isUnit,?_⟩ |
| change (↑u:A)⁻¹ʳ=(↑u⁻¹:A) |
| exact Ring.inverse_unit u |
| end BrownLog |
| end |
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| end OAI |
| |
