Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Vector space
Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.
Complete inner product spaces are known as "Hilbert spaces", in honor of David Hilbert.
The Hilbert space "L"(Ω), with inner product given by
where formula_53 denotes the complex conjugate of "g"("x"), is a key case.