  The in-in path integral of a scalar field propagating in a fixed background
is formulated in a suitable function space. The free kinetic operator, whose
inverse gives the propagators of the in-in perturbation theory, becomes
essentially self adjoint after imposing appropriate boundary conditions. An
explicit spectral representation is given for the scalar in the flat space and
the standard propagators are rederived using this representation. In this way
the subtle boundary path integral over the field configurations at the return
time is handled straightforwardly. It turns out that not only the values of the
forward (+) and the backward (-) evolving fields but also their time
derivatives must be matched at the return time, which is mainly overlooked in
the literature. This formulation also determines the field configurations that
are included in the path integral uniquely. We show that some of the recently
suggested instanton-like solutions corresponding to the stationary phases of
the cosmological in-in path integrals can be rigorously identified as limits of
sequences in the function space.
