BESSAGA CONJECTURE IN UNSTABLE KOTHE SPACES AND PRODUCTS

1993-01-01
NURLU, Z
SARSOUR, J
Let F be a complemented subspace of a nuclear Frechet 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(pi(kn)) where (t(n)) is a scalar sequence, pi 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 there exists s for-all p there exists q for-all r lim(n) d(n+1)(U(q),U(p)0 / d(n)(U(r),U(s)) = 0 where d(n)(U, V) denotes the nth Kolmogorov diameter.