Supremum Norm is Norm

Theorem
Let $S$ be a set.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $K \in \set {\R, \C}$.

Let $\BB$ be the set of bounded mappings $S \to X$.

Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\BB$.

Then $\norm {\, \cdot \,}_\infty$ is a norm on $\BB$.

Proof
First:

Now let $\lambda \in K, f \in \BB$

We have:

Finally let $f, g \in \BB$.

We have:

Thus $\norm {\, \cdot \,}_\infty$ has the defining properties of a norm.