|
|
|
|
bounded function
|
(Definition)
|
|
|
Definition Suppose  is a nonempty set. Then a function
 is a bounded function if there exist a  such that  for all  . The set of all bounded functions on  is usually denoted by  ([1], pp. 61).
Under standard point-wise addition and point-wise multiplication by a scalar,  is a complex vector space.
If  , then the sup-norm, or uniform norm, of  is defined as
It is straightforward to check that
 makes  into a normed vector space, i.e., to check that
 satisfies the assumptions for a norm.
Suppose  is a compact topological space. Further, let  be the set of continuous complex-valued functions on  (with the same vector space structure as  ). Then  is a vector subspace of  .
- 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)
| 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 28 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 21140 times total.
Classification:
| AMS MSC: | 46-00 (Functional analysis :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
