The Bogoliubov inequality is used to prove the nonexistence of the excitonic insulating phase in one and two dimensions for systems of electrons described by three different many-band models with interactions. The proof is carried through (1) for arbitrary two-particle interactions with an interaction potential which goes to zero faster than [Formula: see text] (n being the number of dimensions) at large [Formula: see text], (2) for two-particle interactions of arbitrary range, provided only the so-called Coulomb and exchange parts of the interaction are retained and the remainder is neglected, and (3) for the simplified isotropic two-band model suggested for the excitonic insulator by Des Cloizeaux.