Let U, V be two symmetric convex bodies in ${ℝ}^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n\in U$ such that, for each choice of signs ${\epsilon }_1,...,{\epsilon }_n=\pm 1$, one has ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\notin rV$ where $r={\left(2\pi e^2\right)}^{-1/2}{n}^{1/2}\left(\right|U|/{\left|V\right|)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left(u_n\right)$ such that the series ${\sum }_{n=1}^{\infty }{\epsilon }_n{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon }_n=\pm 1$ and any permutation π of indices.

Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $\left(e_n\right)$ resp. $\left(f_n\right)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n{e}_{\pi \left(k_n\right)}$ where $\left(t_n\right)$ is a scalar sequence, π is a permutation of ℕ and $\left(k_n\right)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $\left(U_n\right)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $li{m}_n\left(d_{n+1}(U_q,U_p)\right)/\left(d_n(U_r,U_s)\right)=0$ where...