  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.
