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

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

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

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

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