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

Theorem
Space of continuous on closed interval real-valued functions with supremum norm forms a normed vector space.

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

We have that:


 * Space of continuous on closed interval real-valued functions with pointwise addition and pointwise scalar multiplication form vector space


 * Supremum norm on the space of continuous on closed interval real-valued functions is a norm

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