File size: 2,161 Bytes
b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import Mathlib
namespace OAI
noncomputable section
namespace Problem367.StablePowerTransport
variable {𝕜 E V : Type*} [NontriviallyNormedField 𝕜]
[NormedAddCommGroup E] [NormedSpace 𝕜 E]
[NormedAddCommGroup V] [NormedSpace 𝕜 V]
theorem pow_apply_intertwiner
(N : V →L[𝕜] V) (T : E →L[𝕜] E) (R : E →L[𝕜] V)
(hNR : ∀ z, N (R z) = R (T z)) (n : ℕ) (z : E) :
(N ^ n) (R z) = R ((T ^ n) z) := by
induction n with
| zero => simp
| succ n ih =>
rw [pow_succ', pow_succ', mul_apply_eq_comp, mul_apply_eq_comp, ih, hNR]
theorem norm_pow_apply_le_of_stable_intertwiner
(N : V →L[𝕜] V) (T : E →L[𝕜] E) (R : E →L[𝕜] V)
(hNR : ∀ z, N (R z) = R (T z))
{κ : ℝ} (hκ : 0 < κ) (hR : ∀ z, κ * ‖z‖ ≤ ‖R z‖)
(n : ℕ) {x : V} (hx : x ∈ R.range) :
‖(N ^ n) x‖ ≤ (‖R‖ / κ) * ‖T ^ n‖ * ‖x‖ := by
obtain ⟨z, hz⟩ := hx
change R z = x at hz
subst x
rw [pow_apply_intertwiner N T R hNR]
have hzbound : ‖z‖ ≤ ‖R z‖ / κ := by
apply (le_div_iff₀ hκ).mpr
simpa only [mul_comm] using hR z
calc
‖R ((T ^ n) z)‖ ≤ ‖R‖ * ‖(T ^ n) z‖ := R.le_opNorm _
_ ≤ ‖R‖ * (‖T ^ n‖ * ‖z‖) :=
mul_le_mul_of_nonneg_left ((T ^ n).le_opNorm z) (norm_nonneg _)
_ ≤ ‖R‖ * (‖T ^ n‖ * (‖R z‖ / κ)) :=
mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_left hzbound (norm_nonneg _)) (norm_nonneg _)
_ = (‖R‖ / κ) * ‖T ^ n‖ * ‖R z‖ := by ring
theorem norm_pow_apply_le_of_model_bound
(N : V →L[𝕜] V) (T : E →L[𝕜] E) (R : E →L[𝕜] V)
(hNR : ∀ z, N (R z) = R (T z))
{κ : ℝ} (hκ : 0 < κ) (hR : ∀ z, κ * ‖z‖ ≤ ‖R z‖)
(n : ℕ) {C : ℝ} (hC : ‖T ^ n‖ ≤ C) {x : V} (hx : x ∈ R.range) :
‖(N ^ n) x‖ ≤ ((‖R‖ / κ) * C) * ‖x‖ := by
apply (norm_pow_apply_le_of_stable_intertwiner N T R hNR hκ hR n hx).trans
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hC (div_nonneg (norm_nonneg _) hκ.le)) (norm_nonneg _)
end Problem367.StablePowerTransport
end
end OAI
|