# 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.

## Proof

### Well-Definedness

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

Then:

 $\ds \forall n \in \N: \,$ $\ds \size {x_n}$ $\le$ $\ds \sup_{n \mathop \in \N} \size {x_n}$ $\ds$ $=$ $\ds \norm {\mathbf x}_\infty$ Definition of Supremum Norm

Thus:

 $\ds \size {\sum_{i \mathop = 1}^n\frac{x_i} n }$ $\le$ $\ds \frac 1 n \sum_{i \mathop = 1}^n \size {x_i}$ General Triangle Inequality for Complex Numbers $\ds$ $\le$ $\ds \frac 1 n \sum_{i \mathop = 1}^n \sup_{n \mathop \in \N} \size {x_i}$ $\ds$ $=$ $\ds \frac 1 n n \norm {\mathbf x}_\infty$ Definition of Supremum Norm $\ds$ $=$ $\ds \norm {\mathbf x}_\infty$

Hence:

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

$\Box$

### Linearity

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

Let $\lambda \in \C$.

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

Then:

 $\ds \map A {\mathbf x + \lambda \mathbf y}$ $=$ $\ds \tuple {x_1 + \lambda y_1, \frac {x_1 + x_2 + \lambda \paren {y_1 + y_2} } 2, \frac {x_1 + x_2 + x_3 + \lambda \paren {y_1 + y_2 + y_3} } 3, \ldots}$ Definition of Cesàro Summation Operator $\ds$ $=$ $\ds \tuple {x_1, \frac {x_1 + x_2} 2, \frac {x_1 + x_2 + x_3} 3, \ldots} + \tuple {\lambda y_1, \frac {\lambda \paren {y_1 + y_2} } 2, \frac {\lambda \paren {y_1 + y_2 + y_3} } 3, \ldots}$ Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space $\ds$ $=$ $\ds \tuple {x_1, \frac {x_1 + x_2} 2, \frac {x_1 + x_2 + x_3} 3, \ldots} + \lambda \tuple {y_1, \frac {y_1 + y_2} 2, \frac {y_1 + y_2 + y_3} 3, \ldots}$ $\ds$ $=$ $\ds \map A {\mathbf x} + \lambda \map A {\mathbf y}$ Definition of Cesàro Summation Operator

By definition, $A$ is a linear transformation.

$\Box$

### Continuity

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

Then:

 $\ds \norm {A \mathbf x}_\infty$ $=$ $\ds \sup_{n \mathop \in \N} \size {\sum_{i \mathop = 1}^n \frac {x_i} n }$ Definition of Cesàro Summation Operator $\ds$ $\le$ $\ds \norm {\mathbf x}_\infty$

$\Box$

Altogether:

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

$\blacksquare$