# Derivative Operator is Linear Mapping

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## 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

$\blacksquare$

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations