Cesàro Summation Operator is Continuous Linear Transformation

Theorem
Let $\ell^\infty$ be the space of bounded sequences.

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

Then $A$ is a continuous linear transformation.

Well-Definedness
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$.

Then:

Thus:

Hence:


 * $A \mathbf x \in \ell^\infty$

Linearity
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N}, \mathbf y = \sequence {y_n}_{n \mathop \in \N} \in \ell^\infty$.

Let $\lambda \in \C$.

By Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space:


 * $\mathbf x + \lambda \mathbf y \in \ell^\infty$

Then:

By definition, $A$ is a linear transformation.

Continuity
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$.

Then:

By continuity of linear transformations in normed vector space, $A$ is continuous.

Altogether:


 * $A \in \map {CL} {\ell^\infty}$