Definition Suppose $X$ is a nonempty set. Then a function $f:X\to\mathbb{C}$ is a bounded function if there exist a $C<\infty$ such that $|f(x)|<C$ for all $x\in X$. The set of all bounded functions on $X$ is usually denoted by $B(X)$ ([1], pp. 61).

Under standard point-wise addition and point-wise multiplication by a scalar, $B(X)$ is a complex vector space.

If $f\in B(X)$, then the sup-norm, or uniform norm, of $f$ is defined as

It is straightforward to check that $||\cdot||_{\infty}$ makes $B(X)$ into a normed vector space, i.e., to check that $||\cdot||_{\infty}$ satisfies the assumptions for a norm.