Let ℓ² = {(𝑥₁, 𝑥₂, 𝑥₃, … ) ∶ 𝑥_𝑛 ∈ ℝ for all 𝑛 ∈ ℕ and \(\rm \Sigma_{n=1}^\infty x_n^2<\infty \}\)

For a sequence (𝑥₁, 𝑥₂, 𝑥₃, … ) ∈ ℓ² , define ‖(𝑥₁, 𝑥₂, 𝑥₃, … )‖₂ = \(\rm (\Sigma_{n=1}^\infty x_n^2)^{\frac{1}{2}}\)

Let 𝑆 ∶ (ℓ² , ‖⋅‖₂) → (ℓ² , ‖⋅‖₂) and 𝑇 ∶ (ℓ² , ‖⋅‖₂) → (ℓ² , ‖⋅‖₂) be defined by

𝑆(𝑥₁, 𝑥₂, 𝑥₃, … ) = (𝑦₁, 𝑦₂, 𝑦₃, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n=1\\\ x_{n-1},& n\ge2\end{matrix}\right.\)

𝑇(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦_𝑛 = \(\rm \left\{\begin{matrix}0,&n\ is\ odd\\\ x_{n},& n\ is\ even\end{matrix}\right.\)

Then