  Given a state space, Assignment of Observables involves Taking Function
spaces Thereover. At the classical level, the state space in question is phase
space or configuration space. This assignment picks up nontrivialities when
whichever combination of constraints and the quantum apply.
  For Finite Theories, weak observables equations are inhomogeneous-linear
first-order PDE systems. Their general solution thus splits into complementary
function plus particular integral: strong and nontrivially-weak observables
respectively. We provide a PDE analysis for each of these. In the case of
single observables equations - corresponding to single constraints - the Flow
Method readily applies. Finding all the observables requires free
characteristic problems. For systems, this method can be applied sequentially,
due to integrability conferred by Frobenius' Theorem. In each case, the first
part of this approach is Lie's Integral Theory of Geometrical Invariants, or
the physical counterpart thereof. The second part finds the function space
thereover, giving the entire space of (local) observables.
  We also outline the Field Theory counterpart. Here one has functional
differential equations. Banach (or tame Fr\'{e}chet) Calculus is however
sufficiently standard for the Flow Method and free characteristic problems to
still apply. These calculi support the Lie-theoretic combination of machinery
that our Local Resolution of the Problem of Time requires, by which Field
Theory and GR are included.
