This is the first of a series of papers that presents a kinetic theory of fluctuations in equilibrium classical fluids that makes extensive use of diagrammatic techniques in its development and that will facilitate the use of diagrammatic techniques in the derivation of approximate kinetic theories. We develop the theory for atomic liquids, but the results are easily generalizable to molecular liquids. The fundamental fluctuating quantity in the theory is f(R,P), the density of particles (atoms) at a point in single-particle phase space. The time correlation function for fluctuations of this quantity from its average, , is the most basic correlation function of concern in generalized kinetic theories of fluctuations in liquids. In this paper, we investigate the properties of a basis set of vectors for the Hilbert space of classical dynamical variables that was suggested by Gross, Boley, and Lindenfeld. In later papers, this basis set, which we call the "fluctuation basis", will be used to construct a diagrammatic theory for this correlation function, its generalizations, and its memory function.