Dirichlet space

In mathematics, the Dirichlet space on the domain (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space , for which the Dirichlet integral, defined by

is finite (here dA denotes the area Lebesgue measure on the complex plane ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on . It is not a norm in general, since whenever f is a constant function.