A normed (Banach) left 
-module over a normed (Banach) associative
algebra 
is an algebraic left A-module 
that is a normed space (Banach space, respectively) over the same field 
or 
, with a bilinear action.
The action is jointly continuous, i.e., there is
a nonnegative constant 
such that 
for
all 
and 
.
If 
has a multiplicative identity 
, then 
is called a unital A-module
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 normed left 
-module, with the product of 
giving the module multiplication.
If 
is a Banach algebra, then 
is a Banach left 
-module.
