Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Normed Vector Space

Theorem
Space of Continuously Differentiable on Closed Interval Real-Valued Functions with $C^1$ norm forms a normed vector space.

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

We have that:


 * Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space


 * $\map {C^1} I$ norm on the space of continuously differentiable on closed interval real-valued functions is a norm

By definition, $\struct {\map {C^1} I, \norm {\, \cdot \,}_{1, \infty} }$ is a normed vector space.