Sharp quadratic corrector bound on a real Hilbert space

Turn the operator square into a norm identity; Cancel the two block cross terms; Bound the quadratic form and block energy; Apply two-component Cauchy-Schwarz; Take the nonnegative square root and divide by two

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem quadratic_corrector_bound_of_square_identity
    {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
    (K D : H →L[ℝ] H) (hK : IsSelfAdjoint K) (hD : IsSelfAdjoint D)
    (hSquare : (1 : H →L[ℝ] H)+K*K=D*D)
    (c : ℝ) (hc : 0 ≤ c) (hNorm : ‖D‖ ≤ c) (u v : H) :
    |(1/2 : ℝ)*(‖u‖^2-‖v‖^2)-inner ℝ (K u) v| ≤
      (c/2)*(‖u‖^2+‖v‖^2)
 := by
  have hEnergy (x : H) : ‖x‖^2+‖K x‖^2=‖D x‖^2 := by
    have h := congrArg (fun A : H →L[ℝ] H => inner ℝ (A x) x) hSquare
    simp only [ContinuousLinearMap.add_apply, ContinuousLinearMap.one_apply,
      ContinuousLinearMap.mul_apply, inner_add_left] at h
    have hk : inner ℝ (K (K x)) x=inner ℝ (K x) (K x) :=
      hK.isSymmetric (K x) x
    have hd : inner ℝ (D (D x)) x=inner ℝ (D x) (D x) :=
      hD.isSymmetric (D x) x
    rw [hk, hd] at h
    simpa only [real_inner_self_eq_norm_sq] using h
  have hSym : inner ℝ u (K v)=inner ℝ (K u) v :=
    (hK.isSymmetric u v).symm
  have hBlock : ‖u-K v‖^2+‖K u+v‖^2=‖D u‖^2+‖D v‖^2 := by
    rw [norm_sub_sq_real, norm_add_sq_real, hSym]
    nlinarith only [hEnergy u, hEnergy v]
  have hForm : inner ℝ (u-K v) u-inner ℝ (K u+v) v =
      ‖u‖^2-‖v‖^2-2*inner ℝ (K u) v := by
    have hSwap : inner ℝ (K v) u=inner ℝ (K u) v :=
      (hK.isSymmetric v u).trans (real_inner_comm v (K u)).symm
    rw [inner_sub_left, inner_add_left, hSwap, real_inner_self_eq_norm_sq,
      real_inner_self_eq_norm_sq]
    ring
  have hAbs : |‖u‖^2-‖v‖^2-2*inner ℝ (K u) v| ≤
      ‖u-K v‖*‖u‖+‖K u+v‖*‖v‖ := by
    rw [← hForm]
    exact (abs_sub _ _).trans (add_le_add
      (abs_real_inner_le_norm _ _) (abs_real_inner_le_norm _ _))
  have hDx (x : H) : ‖D x‖^2 ≤ c^2*‖x‖^2 := by
    have hb : ‖D x‖ ≤ c*‖x‖ :=
      (D.le_opNorm x).trans (mul_le_mul_of_nonneg_right hNorm (norm_nonneg x))
    have hs := (sq_le_sq₀ (norm_nonneg (D x)) (mul_nonneg hc (norm_nonneg x))).2 hb
    nlinarith only [hs]
  have hBlockBound : ‖u-K v‖^2+‖K u+v‖^2 ≤ c^2*(‖u‖^2+‖v‖^2) := by
    rw [hBlock]
    nlinarith only [hDx u, hDx v]
  have hCS : (‖u-K v‖*‖u‖+‖K u+v‖*‖v‖)^2 ≤
      (‖u‖^2+‖v‖^2)*(‖u-K v‖^2+‖K u+v‖^2) := by
    nlinarith only [sq_nonneg (‖u-K v‖*‖v‖-‖K u+v‖*‖u‖)]
  have hSquared : |‖u‖^2-‖v‖^2-2*inner ℝ (K u) v|^2 ≤
      (c*(‖u‖^2+‖v‖^2))^2 := by
    calc
      _ ≤ (‖u-K v‖*‖u‖+‖K u+v‖*‖v‖)^2 :=
        (sq_le_sq₀ (abs_nonneg _) (by positivity)).2 hAbs
      _ ≤ (‖u‖^2+‖v‖^2)*(‖u-K v‖^2+‖K u+v‖^2) := hCS
      _ ≤ (‖u‖^2+‖v‖^2)*(c^2*(‖u‖^2+‖v‖^2)) :=
        mul_le_mul_of_nonneg_left hBlockBound (by positivity)
      _ = _ := by ring
  have hBound := (sq_le_sq₀ (abs_nonneg
    (‖u‖^2-‖v‖^2-2*inner ℝ (K u) v)) (by positivity)).1 hSquared
  have hHalf : (1/2 : ℝ)*(‖u‖^2-‖v‖^2)-inner ℝ (K u) v =
      (1/2 : ℝ)*(‖u‖^2-‖v‖^2-2*inner ℝ (K u) v) := by ring
  rw [hHalf, abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/2)]
  nlinarith only [hBound]

end
end AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorBound

#print axioms AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorBound.quadratic_corrector_bound_of_square_identity

Exact module and namespace context