# Derivative Operator is Linear Mapping

## Theorem

Let $I := \closedint a b$ be a closed real interval.

Let $\CC \closedint a b$ be the space of real-valued functions continuous on $I$.

Let $\CC^1 \closedint a b$ be the space of real-valued functions continuously differentiable on $I$.

Let $D$ be the derivative operator such that:

$D : \map {\CC^1} I \to \map \CC I$

and $Dx := x'$.

Then $D$ is a linear mapping.

## Proof

### Distributivity

 $\ds \map D {x + y}$ $=$ $\ds \paren {x + y}'$ Definition $\ds$ $=$ $\ds x' + y'$ Sum Rule for Derivatives $\ds$ $=$ $\ds Dx + Dy$ Definition

$\Box$

### Positive homogenity

 $\ds \map D {\alpha x}$ $=$ $\ds \paren {\alpha x}'$ Definition $\ds$ $=$ $\ds \alpha x'$ Derivative of Constant Multiple $\ds$ $=$ $\ds \alpha Dx$ Definition

$\Box$

By definition, $D$ is a linear mapping

$\blacksquare$