Bloch functions on the unit ball of a Banach space

Abstract: The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions $f$ on the unit ball $B_E$ of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of $(1-|x|^2) |f'(x)|$ on $B_E$ and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms $\phi$ of $B_E$. We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on $B_{E}$ are Bloch functions in both ways.