Block sequences in nuclear Fr�chet spaces with basis

Abstract

It is shown that a block sequence in a nuclear Frechet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Frechet space is isomorphic toΩ = RN, N = {1,2,...} if it has a basis such that any block sequence with blocks of length ≤2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.