Dirichlet space

Summary

The Dirichlet space $\mathcal{D}$ is the Hilbert space of holomorphic functions $f$ on the unit disc whose derivative is square-integrable with respect to area. The Dirichlet integral $\int_{\mathbb{D}} \abs{f'}^2 \, dA$ is the area of the image $f(\mathbb{D})$, counted with multiplicity. In terms of Taylor coefficients, $f = \sum a_n z^n$ lies in $\mathcal{D}$ exactly when $\sum n \abs{a_n}^2 < \infty$, so $\mathcal{D}$ is contained in the Hardy space $H^2$. It is a model space in operator theory and is linked to logarithmic capacity on the circle.