Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $(e_n)$ resp. $(f_n)$, 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 $t_n e_{π(k_n)}$ where $(t_n)$ is a scalar sequence, π is a permutation of ℕ and $(k_n)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $(U_n)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $lim_n (d_{n+1}(U_q, U_p))/(d_n(U_r, U_s)) = 0$ where $d_n(U,V)$ denotes the nth Kolmogorov diameter.