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.
