Density functional theory in terms of a Legendre transformation for beginners

The derivation of the Hohenberg-Kohn (HK) theorem by means of a simplified version of Lieb's Legendre transformation is presented, with the stress on physically problematic aspects, and caring about the definition of the domains of the respective functionals only to the extent, that this is absolutely necessary. The take-home lesson consists in a critical analysis of some statements often found in the DFT literature. As a simple illustration for a Legendre transformation we discuss that in parameter space for the family of n-electron isoelectronic atomic ions, with the nuclear charge as parameter.