| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Metric |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| variable {Q E : Type*} [NormedAddCommGroup Q] [NormedSpace ℝ Q] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| lemma bilinear_lower_of_unit (L : E →L[ℝ] E →L[ℝ] ℝ) {c : ℝ} |
| (h : ∀ v:E, ‖v‖=1 → c ≤ L v v) (v:E) : c*‖v‖^2 ≤ L v v := by |
| by_cases hv : v=0 |
| · simp [hv] |
| have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv |
| let w := ‖v‖⁻¹ • v |
| have hw : ‖w‖=1 := by simp only [w,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hn),inv_mul_cancel₀ hn.ne'] |
| have he : v=‖v‖ • w := by dsimp [w]; rw [smul_smul,mul_inv_cancel₀ hn.ne',one_smul] |
| have H := mul_le_mul_of_nonneg_left (h w hw) (sq_nonneg ‖v‖) |
| calc |
| c*‖v‖^2 = ‖v‖^2*c := mul_comm _ _ |
| _ ≤ ‖v‖^2*L w w := H |
| _ = L (‖v‖ • w) (‖v‖ • w) := by simp only [map_smul,smul_apply,smul_eq_mul]; ring |
| _ = L v v := by rw [←he] |
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| lemma bilinear_diag_abs_le (L : E →L[ℝ] E →L[ℝ] ℝ) (v:E) : |
| |L v v| ≤ ‖L‖*‖v‖^2 := by |
| calc |
| |L v v| = ‖L v v‖ := (Real.norm_eq_abs _).symm |
| _ ≤ ‖L v‖*‖v‖ := (L v).le_opNorm _ |
| _ ≤ (‖L‖*‖v‖)*‖v‖ := mul_le_mul_of_nonneg_right (L.le_opNorm _) (norm_nonneg _) |
| _ = ‖L‖*‖v‖^2 := by ring |
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| lemma bilinear_time_hasDerivAt {H : ℝ×Q → E →L[ℝ] E →L[ℝ] ℝ} |
| {t:ℝ} {q:Q} (hH : DifferentiableAt ℝ H (t,q)) (v:E) : |
| HasDerivAt (fun s => H (s,q) v v) (fderiv ℝ H (t,q) (1,0) v v) t := by |
| have hp : HasDerivAt (fun s : ℝ => (s,q)) (1,0) t := |
| (hasDerivAt_id t).prodMk (hasDerivAt_const t q) |
| have HD : HasDerivAt (fun s : ℝ => H (s,q)) (fderiv ℝ H (t,q) (1,0)) t := |
| HasFDerivAt.comp_hasDerivAt (l := H) (l' := fderiv ℝ H (t,q)) |
| (f := fun s : ℝ => (s,q)) t hH.hasFDerivAt hp |
| have HV := (HD.clm_apply (hasDerivAt_const t v)).clm_apply (hasDerivAt_const t v) |
| simpa only [Function.comp_def,map_zero,add_zero,zero_add] using HV |
| end WeakMTWTransport |
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| end |
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| end OAI |
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