Derivative Operator on Continuously Differentiable Function Space with C^1 Norm is Continuous

Theorem
Let $I = \closedint 0 1$ be a closed real interval.

Let $\map \CC I$ be the real-valued, continuous on $I$ function space.

Let $\map {\CC^1} I$ be the continuously differentiable function space.

Let $x \in \map {\CC^1} I$ be a continuoulsly differentiable real-valued function.

Let $D : \map {\CC^1} I \to \map \CC I$ be the derivative operator such that:


 * $\forall t \in \closedint 0 1 : \map {Dx} t := \map {x'} t$

Suppose $\map \CC I$ and $\map {\CC^1} I$ are equipped with $C^0$ and $C^1$ norms respectively

Then $D$ is continuous.

Proof
We have that the derivative operator is linear mapping.

By definition and continuity of linear transformations, $D$ is continuous.