|
|
|
|
bounded function
|
(Definition)
|
|
|
Definition Suppose $X$ is a nonempty set. Then a function $f:X\to \sC$ 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.
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 $B(X)$ .
- 1
- C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
|
"bounded function" is owned by Koro. [ full author list (2) | owner history (1) ]
|
|
(view preamble | get metadata)
| Also defines: |
supremum norm, sup norm, sup-norm, uniform norm, bounded function, unbounded function |
|
|
Cross-references: vector subspace, structure, continuous, topological space, compact, norm, normed vector space, vector space, complex, scalar, multiplication, addition, function
There are 36 references to this entry.
This is version 4 of bounded function, born on 2003-07-06, modified 2004-12-11.
Object id is 4426, canonical name is BoundedFunction.
Accessed 28324 times total.
Classification:
| AMS MSC: | 46-00 (Functional analysis :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
