Density functional theory in finite basis sets tends to degenerate to one-body reduced (1RDM) functional theory. As all calculations are done in finite basis sets, a rigorous foundation of 1RDM functional theory is desirable. To avoid uniqueness problems in the potential to 1RDM mapping, I will discuss the foundations of 1RDM functional theory in finite basis sets at finite temperatures, both for fermions and bosons. The fermionic case turns out to be relatively straightforward, but the bosonic case requires more care. The main result is that we can rigorously proof v-representability and functional differentiability in this setting.