  Let $\Om$ be a unbounded domain in a Banach space. In this work, we wish to
impose {\it local conditions} on boundary point of $\Om$ (including the point
at infinity) that guarantee complete hyperbolic of $\Om.$ We also search for
local boundary conditions so that Vitali properties hold true for $\Om.$ These
properties might be considered as analogues of the taut property in the finite
dimensional case.
