theorem extracted_formal_statement_2.{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)
  (this : ⟪A x, A x⟫_ℝ = 0) : x ∈ LinearMap.ker A := sorry

State before:
case h.mpr
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
this : ⟪A x, A x⟫_ℝ = 0
⊢ x ∈ LinearMap.ker A

Tactic: apply inner_self_eq_zero.1 this

State after:

Relevant lemmas:
inner_self_eq_zero : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_6 : NormedAddCommGroup E]
  [inst_7 : InnerProductSpace 𝕜 E] {x : E}, ⟪x, x⟫_𝕜 = 0 ↔ x = 0