Let 
be a T2 associative inner product space over the field 
of complex
numbers with completion 
, and assume that 
comes with an antilinear involution 
and a bijective linear mapping 
with inverse 
. 
is said to be a quasi-Hilbert algebra if the following axioms
are satisfied:
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 
.
