bounded function
Definition Suppose $X$ is a nonempty set. Then a function $f:X\to \u2102$ 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 $B(X)$.
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 |