∀ {𝕜 : Type u_3} {E : Type u_2} {F : Type u_1} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace 𝕜 F] (T : E →ₗ.[𝕜] F)
  {a b : F},
  a ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)} →
    b ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)} → a + b ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}