| import Mathlib | |
| import OAI.RepresentationTheory.PartialPermutation.HilbertSchmidt | |
| namespace OAI | |
| namespace PartialPermutation | |
| noncomputable section | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] | |
| [FiniteDimensional ℂ V] | |
| lemma norm_apply_sq_le_hsNormSq (A : Module.End ℂ V) (x : V) : | |
| ‖A x‖^2 ≤ hsNormSq A * ‖x‖^2 := by | |
| let e := stdOrthonormalBasis ℂ V | |
| calc | |
| ‖A x‖^2 = ∑ i, ‖inner ℂ (e i) (A x)‖^2 := (e.sum_sq_norm_inner_right _).symm | |
| _ ≤ ∑ i, ‖LinearMap.adjoint A (e i)‖^2 * ‖x‖^2 := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| rw [← LinearMap.adjoint_inner_left A] | |
| simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg (inner ℂ (LinearMap.adjoint A (e i)) x)) | |
| (norm_inner_le_norm (𝕜 := ℂ) (LinearMap.adjoint A (e i)) x) 2 | |
| _ = hsNormSq A * ‖x‖^2 := by | |
| rw [← Finset.sum_mul, ← hsNormSq_eq_sum e, hsNormSq_adjoint] | |
| lemma opNorm_sq_le_hsNormSq (A : Module.End ℂ V) : | |
| ‖LinearMap.toContinuousLinearMap A‖^2 ≤ hsNormSq A := by | |
| have h : ‖LinearMap.toContinuousLinearMap A‖ ≤ Real.sqrt (hsNormSq A) := by | |
| apply ContinuousLinearMap.opNorm_le_bound _ (Real.sqrt_nonneg _) | |
| intro x | |
| apply (sq_le_sq₀ (norm_nonneg _) (by positivity)).mp | |
| rw [mul_pow, Real.sq_sqrt (hsNormSq_nonneg A)] | |
| exact norm_apply_sq_le_hsNormSq A x | |
| simpa only [Real.sq_sqrt (hsNormSq_nonneg A)] using | |
| pow_le_pow_left₀ (norm_nonneg _) h 2 | |
| end | |
| end PartialPermutation | |
| end OAI | |