A simple inductive proof of Levy-Steinitz theorem

Abstract

We present a relatively simple inductive proof of the classical Levy-Steinitz Theorem saying that for a sequence $(x_n)_{n=1}^\infty$ in a finite-dimensional Banach space $X$ the set of all sums of rearranged series $\sum_{n=1}^\infty x_{σ(n)}$ is an affine subspace of $X$. This affine subspace is not empty if and only if for any linear functional $f:X\to \mathbb R$ the series $\sum_{n=1}^\infty f(x_{σ(n)})$ is convergent for some permutation $σ$ of $\mathbb N$. This gives an answer to a problem of Vaja Tarieladze, posed in Lviv Scottish Book in September, 2017. Also we construct a sequence $(x_n)_{n=1}^\infty$ in the torus $\mathbb T\times\mathbb T$ such that the series $\sum_{n=1}^\infty x_{σ(n)}$ is divergent for all permutations $σ$ of $\mathbb N$ but for any continuous homomorphism $f:\mathbb T^2\to\mathbb T$ to the circle group $\mathbb T:=\mathbb R/\mathbb Z$ the series $\sum_{n=1}^\infty f(x_{σ_f(n)})$ is convergent for some permutation $σ_f$ of $\mathbb N$. This example shows that the second part of Levy-Steinitz Theorem (characterizing sequences with non-empty set of potential sums) does not extend to locally compact Abelian groups.