Bounded Operator (original) (raw)
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A bounded operator 
between two Banach spaces satisfies the inequality
| 
| (1) |
| ---------------------------------------------------------------------------------------------------------- | --- |
where 
is a constant independent of the choice of 
. The inequality is called a bound. For example, consider
, which has L2-norm 
. Then 
is a bounded operator,
 |
(2) |
|---|
from L2-space to L1-space. The bound
| 
| (3) |
| ---------------------------------------------------------------------------------------------------------------------- | --- |
holds by Hölder's inequalities.
Since a Banach space is a metric space with its norm, a continuous linear operator must be bounded. Conversely, any bounded linear operator must be continuous, because bounded operators preserve the Cauchy property of a Cauchy sequence.
See also
Banach Space, Continuous, Hilbert Space, Linear Operator, L-_p_-Space
This entry contributed by Todd Rowland
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Rowland, Todd. "Bounded Operator." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BoundedOperator.html