On the $$c_0$$-equivalence and permutations of series

Abstract

Assume that a convergent series of real numbers $$\sum \limits _{n=1}^\infty a_n$$ ∑ n = 1 ∞ a n has the property that there exists a set $$A\subseteq {\mathbb {N}}$$ A ⊆ N such that the series $$\sum \limits _{n \in A} a_n$$ ∑ n ∈ A a n is conditionally convergent. We prove that for a given arbitrary sequence $$(b_n)$$ ( b n ) of real numbers there exists a permutation $$\sigma :{\mathbb {N}}\rightarrow {\mathbb {N}}$$ σ : N → N such that $$\sigma (n) = n$$ σ ( n ) = n for every $$n \notin A$$ n ∉ A and $$(b_n)$$ ( b n ) is $$c_0$$ c 0 -equivalent to a subsequence of the sequence of partial sums of the series $$\sum \limits _{n=1}^\infty a_{\sigma (n)}$$ ∑ n = 1 ∞ a σ ( n ) . Moreover, we discuss a connection between our main result with the classical Riemann series theorem.