On rearrangement theorems in Banach spaces

It is shown that every infinite-dimensional real Banach space X contains a sequence (x(n))(n is an element of N) with the following properties: (a) Some subsequence of (Sigma(n)(k=1) x(k))(n is an element of N) converges in X and sup(n is an element of N)parallel to Sigma(n)(k=1) x(k)parallel to <= 1; (b) Sigma(infinity)(k=1)parallel to x(k)parallel to(p) < infinity for every p is an element of]2, +infinity[; (c) for any permutation pi : N -> N and any sequence (theta(n))(n is an element of N) with theta(n) epsilon {-1, 1}, n = 1, 2, ... , the series Sigma(infinity)(k=1)theta(k)x(pi(k)) diverges in X. This result implies, in particular, that the rearrangement theorem and the Dvoretzky-Hanani theorem fail drastically for infinite-dimensional Banach spaces.