We examine the validity of the first Hohenberg-Kohn theorem, namely, the one-to-one relationship between an external potential and the 1-particle density, when it is applied to finite subspaces and consider the stability of these subspaces with respect to external potentials. This is done by analysing the DFT description of some simple atoms (eg, H, He, Li, and Be) provided by the solution of the Kohn-Sham equation in a finite Gaussian basis set. We show that in the finite subspace generated from the finite basis set, it is possible to construct external potentials that differ from one another by more than a constant but which associate with the same 1-particle density. We carry out the specific construction of these potentials for the above atoms using the wave functions resulting from the application of the B3LYP functional. We comment on the fact that these instability potentials seem to be prominent only in the outer region of the atom where the density tends to zero. We also discuss the implications that the instability potentials have in relation to the Kohn-Sham formalism and equations.