Supremum Norm is Norm

Theorem
Let $S$ be a set, and $(X,\| \cdot \|)$ a normed vector space over $K \in \{\R,\C\}$.

Let $\mathcal B$ be the set of bounded functions $S \to X$.

Let $\| \cdot\|_{\infty}$ be the sup norm on $\mathcal B$.

Then $\| \cdot \|_\infty$ is a norm on $\mathcal B$.

Proof
First we have

Now let $\lambda \in K$, $f \in \mathcal B$. We have:

Finally let $f,g \in \mathcal B$. We have:

Therefore $\| \cdot \|_\infty$ has the defining properties of a norm.