A quasi-Hilbert algebra is an associative involutive algebra 
over the field 
of complex numbers, equipped
with an inner product 
and completion 
. Its antilinear involution 
satisfies 
. It also carries a bijective linear mapping 
with inverse 
such that the following
axioms hold:
1. 
for all 
.
2. 
for all 
.
3. For each 
,
the map 
is continuous.
4. The collection of all products 
of elements 
is dense in 
.
5. If 
are elements of 
such that 
for every 
, there exists a sequence 
in 
such that 
and 
.
When all components of a quasi-Hilbert algebra need to be explicitly acknowledged, one may write 
.
