Let 
be a normed (Banach) algebra. An algebraic left 
-module 
is said to be a normed (Banach) left 
-module if 
is a normed (Banach) space and the outer multiplication is
jointly continuous, i.e., if there is a nonnegative number 
such that 
. If 
has an identity 
, then 
is called unital if 
for all 
. A normed (Banach) right module can be similarly defined.
For example, every closed left ideal 
of a normed algebra 
can be regarded as a Banach left 
-module with the product of 
giving the module multiplication.
