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.

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces