  In this series of papers we aim to provide a mathematically comprehensive
framework to the Hamiltonian pictures of quantum field theory in curved
spacetimes. Our final goal is to study the kinematics and the dynamics of the
theory from the point of differential geometry in infinite dimensions.In this
first part we introduce the tools of Gaussian analysis in infinite dimensional
spaces of distributions. These spaces will serve the basis to understand the
Schr\"odinger and Holomorphic pictures, over arbitrary Cauchy hypersurfaces,
using tools of Hida-Malliavin calculus. Here the Wiener-Ito decomposition
theorem provides the QFT particle interpretation. Special emphasis is done in
the applications to quantization of these tools in the second part of this
paper. We devote a section to introduce Hida test functions as a notion of
second quantized test functions. We also analyze of the ingredients of
classical field theory modeled as distributions paving the way for quantization
procedures that will be analyzed in the second part of this series.
