# bounded function

Definition Suppose $X$ is a nonempty set. Then a function $f:X\to\mathbb{C}$ is a if there exist a $C<\infty$ such that $|f(x)| 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

 $||f||_{\infty}=\sup_{x\in X}|f(x)|.$

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.

## 0.0.1 Example

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

## References

• 1 C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
 Title bounded function Canonical name BoundedFunction Date of creation 2013-03-22 13:44:06 Last modified on 2013-03-22 13:44:06 Owner Koro (127) Last modified by Koro (127) Numerical id 7 Author Koro (127) Entry type Definition Classification msc 46-00 Defines supremum norm Defines sup norm Defines sup-norm Defines uniform norm Defines bounded function Defines unbounded function