Supremum Operator Norm of Cesàro Summation Operator

Theorem
Let $A : \ell^\infty \to \ell^\infty$ be the Cesàro summation operator.

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

Then $\norm A = 1$.

Proof
By Cesàro Summation Operator is Continuous Linear Transformation:


 * $\norm {A \mathbf x}_\infty \le \norm {\mathbf x}_\infty$

where $\norm {\, \cdot \,}_\infty$ denotes the supremum norm.

By Universal Upper Bound greater than Supremum Operator Norm:


 * $\norm A \le 1$

Let $\mathbf 1 := \tuple {1, 1, \ldots} \in \ell^\infty$

Then:

Altogether, $\norm A = 1$.