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