  Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and $\Omega,
\omega $ be bounded domains in ${\mathbb R}^n$, $n \geq 1$, such that $\omega
\subset \Omega$ and the complement $\Omega \setminus \omega$ has no (non-empty)
compact components in $\Omega$. We prove that this is the necessary and
sufficient condition for the space $H^{2s,s} _{\mathcal H} (\Omega \times
(T_1,T_2))$ of solutions to the heat operator ${\mathcal H} $ in a cylinder
domain $\Omega \times (T_1,T_2)$ from the anisotropic Sobolev space $H^{2s,s}
(\Omega \times (T_1,T_2))$ to be dense in the space $L^{2} _{\mathcal H}(\omega
\times (T_1,T_2))$, consisting of solutions in the domain $\omega \times
(T_1,T_2)$ from the Lebesgue class $L^{2} (\omega \times (T_1,T_2))$. As an
important corollary we obtain the theorem on the existence of a basis with the
double orthogonality property for the pair of the Hilbert spaces $H^{2s,s}
_{\mathcal H} (\Omega \times (T_1,T_2))$ and $L^{2} _{\mathcal H}(\omega \times
(T_1,T_2))$ .
