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 $x\in X$. 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

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

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

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