Let $(e_n)$ be an orthonormal basis in $ℓ^2$. Show that $S_{e_n}=e_{n+1}$ defines a bounded linear operator $S: ℓ^2 \rightarrow ℓ^2$ and that $||S||=1$.
Well, I need to show that $||S_{e_n}||_{ℓ^2} \leq C||e_n||_{ℓ^2}$ for a positive constant $C$. I know the definition of the $ℓ^2$-norm but I don't see how it could help me. All the other proofs I have looked for don't use the norm definition. But how could I prove this?