Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space

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

The space of continuous real-valued functions on $I$ with supremum norm forms a normed vector space.

Proof
We have that:


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


 * Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function

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