Bloch space

In the mathematical field of complex analysis, the Bloch space, named after André Bloch and denoted $\mathcal{B}$ or ℬ, is the space of holomorphic functions f defined on the open unit disc D in the complex plane, such that the function

$(1-|z|^2)|f^\prime(z)|$

is bounded.^[1] $\mathcal{B}$ is a Banach space, with the norm defined by

$\|f\|_\mathcal{B} = |f(0)| + \sup_{z \in \mathbf{D}} (1-|z|^2) |f'(z)|.$

This is referred to as the Bloch norm and the elements of the Bloch space are called Bloch functions.

Notes

1. Wiegerinck, J. (2001), "Bloch function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4