Bloch space

Summary

The Bloch space is the space of holomorphic functions $f$ on the unit disc for which $(1 - \abs{z}^2)\abs{f'(z)}$ stays bounded. With a natural norm it is a Banach space, and the seminorm $\sup_z (1 - \abs{z}^2)\abs{f'(z)}$ does not change under composition with disc automorphisms. It contains all bounded holomorphic functions and also some unbounded ones such as $\log(1 - z)$. It is closely tied to Bloch's theorem, to the space of bounded mean oscillation, and to the duality theory of Bergman spaces.