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

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