  A mathematically rigorous Hamiltonian formulation for classical and quantum
field theories is given. New results include clarifications of the structure of
linear fields, and a plausible formulation for nonlinear fields. Many
mathematical formulations of field theory suffer greatly from either a failure
to explicitly define the field configuration space, or else from the choice to
define field operators as distributions. A solution to such problems is given
by instead using locally square-integrable functions, and by paying close
attention to this space's topology. One benefit of this is a clarification of
the field multiplication problem: The pointwise product of fields is still not
defined for all states, but it is densely defined, and this is shown to be
sufficient for specifying dynamics. Significant progress is also made, through
this choice of configuration space, in appropriately representing field states
with `infinitely many particles', or those which do not go to zero at infinity.
