  This study shows how Aronszajn's theory of reproducing kernels can be of use
for the construction the Hilbert spaces of quantum theory. We show that the
Feynman propagator is an example of a reproducing kernel under a boundedness
condition. To every Lagrangian thus corresponds a Hilbert space that does not
need to be postulated \emph{a priori}. For the free non-relativistic particle,
we justify mathematically the concept of space-time granularity. Reproducing
kernels allow for a functional, rather than distributional, description of the
Hilbert spaces of quantum theory, including the Fock space.
