  This paper establishes bounds on norms of all orders for solutions on the
global attractor of the 2D Navier-Stokes equations, complexified in time.
Specifically, for periodic boundary conditions on $[0,L]^2$, and a force
$g\in\calD(A^{\frac{\alpha-1}{2}})$, we show there is a fixed strip about the
real time axis on which a uniform bound $|A^{\alpha}u|<
m_\alpha\nu\kappa_0^\alpha$ holds for each $\alpha \in \bN$. Here $\nu$ is
viscosity, $\k0=2\pi/L$, and $m_\alpha$ is explicitly given in terms of $g$ and
$\alpha$. We show that if any element in $\calA$ is in $\D(A^\alpha)$, then all
of $\calA$ is in $\D(A^\alpha)$, and likewise with $\D(A^\alpha)$ replaced by
$C^\infty(\Omega)$. We demonstrate the universality of this "all for one, one
for all" law on the union of a hierarchal set of function classes. Finally, we
treat the question of whether the zero solution can be in the global attractor
for a nonzero force by showing that if this is so, the force must be in a
particular function class.
