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.
Hilbert space
the supremum being taken over all finite subsets of . It follows that, for this sum to be finite, every element of has only countably many nonzero terms. This space becomes a Hilbert space with the inner product
for all . Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.