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It is shown that every in nite-dimensional real Banach space contains a sequence ( ) ∈ℕ with the following properties: (a) Some subsequence of (∑ =1 ) ∈ℕ converges in and sup ∈ℕ ‖∑ =1 ‖ ≤ 1; (b) ∑ ∞ =1 ‖ ‖ < ∞ for every ∈ ]2, +∞[; (c) for any permutation : ℕ → ℕ and any sequence ( ) ∈ℕ with ∈ {−1, 1}, = 1, 2, . . ., the series ∑ ∞ =1 ( ) diverges in . This result implies, in particular, that the rearrangement theorem and the Dvoretzky-Hanani theorem fail drastically for in nite-dimensional Banach spaces.

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In the paper, we introduce a characteristic of the Sylvester matrix and find its explicit values. We describe some applications of the introduced characteristic.The Sylvester matrices find applications in various domains of mathematics and, in particular, in discrete mathematics (see, for example, [1]). Note also that the Sylvester matrices were used in the construction of an unconditionally converging series in the Danach space /i which does not converges absolutely [2]. In [3], with the use of the Sylvester (Hadamard) matrices series of special type in the Banach spaces l p , 1 < p < oo, are introduced and a criterion of the unconditional convergence of these series are given.The Sylvester matrices S^n\ n = 1,2,... , are defined by the recurrence relations (see [1])