  Density functional theory (DFT) has become a basic tool for the study of
electronic structure of matter, in which the Hohenberg-Kohn theorem plays a
fundamental role in the development of DFT. Unfortunately, the existing proofs
are incomplete even incorrect; besides, the statement of the Hohenberg-Kohn
theorem for many-electron Coulomb systems is not perfect. In this paper, we
shall restate the Hohenberg-Kohn theorem for Coulomb type systems and present a
rigorous proof by using the Fundamental Theorem of Algebra.
