In the mathematical field of complex analysis, the Bloch space, named after French mathematician André Bloch and denoted B {\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
( 1 − | z | 2 ) | f ′ ( z ) | {\displaystyle (1-|z|^{2})|f^{\prime }(z)|}
is bounded. B {\displaystyle {\mathcal {B}}} is a type of Banach space, with the norm defined by
‖ f ‖ B = | f ( 0 ) | + sup z ∈ D ( 1 − | z | 2 ) | f ′ ( z ) | . {\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.
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