theorem extracted_formal_statement_0.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (x : X) (h : (ContinuousLinearMap.adjoint A) (A x) = 0) :
  ⟪A x, A x⟫_ℝ = ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ := sorry

State before:
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
x : X
h : (ContinuousLinearMap.adjoint A) (A x) = 0
⊢ ⟪A x, A x⟫_ℝ = ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ

Tactic: rw [ContinuousLinearMap.adjoint_inner_right]

State after:

Relevant lemmas:
ContinuousLinearMap.adjoint_inner_right : ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜]
  [inst_6 : NormedAddCommGroup E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace 𝕜 E]
  [inst_9 : InnerProductSpace 𝕜 F] [inst_10 : CompleteSpace E] [inst_11 : CompleteSpace F] (A : E →L[𝕜] F) (x : E)
  (y : F), ⟪x, (ContinuousLinearMap.adjoint A) y⟫_𝕜 = ⟪A x, y⟫_𝕜