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 .