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

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

Let $\struct {\map C I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, continuous on $I$.

Let $\map x t \in \map C I$ be a continuous real function.

Let $\size {\, \cdot \,}$ be the absolute value.

Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$.

Then $\norm {\, \cdot \,}_\infty$ is a norm over $\struct {\map C I, +, \, \cdot \,}_\R$.

Positive definiteness
Suppose $\norm x_\infty = 0$.

Then:

Therefore:


 * $\forall t \in I : \map x t = 0$

Positive homogeneity
Let $\alpha \in \R$.