theorem extracted_formal_statement_3.{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] (fullrank : Injective ⇑A) : LinearMap.ker A = ⊥ := 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
fullrank : Injective ⇑A
⊢ LinearMap.ker A = ⊥

Tactic: apply KerA_bot A fullrank

State after:

Relevant lemmas:
KerA_bot : ∀ {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X] [inst_6 : InnerProductSpace ℝ X]
  [inst_7 : NormedAddCommGroup Y] [inst_8 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y), Injective ⇑A → LinearMap.ker A = ⊥