continuous linear mapping
If there is a nonnegative constant such that for each , we say that is . This should not be confused with the usual terminology referring to a bounded function as one that has bounded range. In fact, bounded linear mappings usually have unbounded ranges.
If is bounded, then , so is a Lipschitz function. Now suppose is continuous. Then there exists such that when . For any , we then have
hence ; so is bounded.
- 1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
|Title||continuous linear mapping|
|Date of creation||2013-03-22 13:15:41|
|Last modified on||2013-03-22 13:15:41|
|Last modified by||Koro (127)|
|Synonym||bounded linear mapping|
|Defines||bounded linear transform|
|Defines||bounded linear operator|