bounded function

Definition Suppose $X$ is a nonempty set. Then a function$f:X\to ℂ$ is a if there exist a such that for all $x\in X$. Theset of all bounded functions on $X$ is usuallydenoted by $B(X)$ ([1], pp. 61).

Under standard point-wise addition and point-wise multiplication bya 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

$${||f||}_{\mathrm{\infty }}=\underset{x\in X}{sup}|f(x)|.$$

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

0.0.1 Example

Suppose $X$ is a compact topological space. Further, let $C(X)$ be theset of continuous complex-valued functions on $X$ (with the same vectorspace structure as $B(X)$). Then $C(X)$ is a vector subspace of $B(X)$.