  This paper is the first in a series of three papers that aim at understanding
the scaling behaviour of hydrodynamic turbulence. We present in this paper a
perturbative theory for the structure functions and the response functions of
the hydrodynamic velocity field in real space and time. Starting from the
Navier-Stokes equations (at high Reynolds number Re) we show that the standard
perturbative expansions that suffer from infra-red divergences can be exactly
resummed using the Belinicher-L'vov transformation. After this exact (partial)
resummation it is proven that the resulting perturbation theory is free of
divergences, both in large and in small spatial separations. The hydrodynamic
response and the correlations have contributions that arise from mediated
interactions which take place at some space- time coordinates. It is shown that
the main contribution arises when these coordinates lie within a shell of a
"ball of locality" that is defined and discussed. We argue that the real
space-time formalism developed here offers a clear and intuitive understanding
of every diagram in the theory, and of every element in the diagrams. One major
consequence of this theory is that none of the familiar perturbative mechanisms
may ruin the classical Kolmogorov (K41) scaling solution for the structure
functions. Accordingly, corrections to the K41 solutions should be sought in
nonperturbative effects. These effects are the subjects of papers II and III in
this series, that will propose a mechanism for anomalous scaling in turbulence,
which in particular allows multiscaling of the structure functions.
