Derivative Operator is Linear Mapping

Theorem

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

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

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

Let $D$ be the derivative operator such that:

$D : \map {C^1} I \to \map C I$

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