  The purpose is to formulate a Fourier transformation for the space of
functionals, as an infinitesimal meaning. We extend ${\bf R}$ to $
^{\star}(^{\ast}{\bf R})$ under the base of nonstandard methods for the
construction. The domain of a functional is the set of all internal functions
from a $ ^{\ast}$-finite lattice to a $ ^{\ast}$-finite lattice with a double
meaning. Considering a $ ^{\ast}$-finite lattice with a double meaning, we find
how to treat the domain for a functional in our theory of Fourier
transformation, and calculate two typical examples.
