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# bounded function

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

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

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.

## Mathematics Subject Classification

46-00*no label found*

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