There are at least two distinct (though related) notions of the term Hilbert algebra in functional analysis.
In some literature, a linear manifold of a (not necessarily separable) Hilbert space is a Hilbert algebra if the following conditions are satisfied:
1. is dense in .
2. is a ring so that, for any , there is defined an element such that , , , and for any complex number .
3. For any , there exists an adjoint element such that , and .
4. For any , there exists a positive number such that for all .
5. For every , there exists a unique bounded linear operator on such that for all . Moreover, if for an element and for all , then .
At least one author defines a Hilbert algebra to be a quasi-Hilbert algebra
for which for all (Dixmier 1981).