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c8720e4 | 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 | 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
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