  The AdS/CFT correspondence can be realized in spaces that are globally
different but share the same asymptotic behavior. Two known cases are: a
compact AdS space and the space generated by a large number of coincident
branes. We discuss the physical consistency, in the sense of the Cauchy
problem, of these two formulations. We show that the role of the boundary in
the compact AdS space is equivalent to that of the flat asymptotic region in
the brane space. We also show, by introducing a second coordinate chart for the
pure AdS space, that a point at its spatial infinity corresponds to a horizon
in the brane system.
