I like to think of DFT in terms of Legendre-Fenchel transforms. In short, from the concavity and continuity of the ground-state energy v → E[v] as a function of the external potential v ∈ L 3/2 (R 3 ) + L ∞ (R 3 ) follows the existence of a universal density functional ρ → F[ρ] as a function of the electron density ρ ∈ L 3 (R 3 ) ∩ L 1 (R 3 ) such that (4)