2017/11/11 by Banakh, Taras
#40A05 #46B15 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · doi:10.48550/arxiv.1711.04136
We present a relatively simple inductive proof of the classical Levy-Steinitz Theorem saying that for a sequence (xn)n=1^∞ in a finite-dimensional Banach space X the set of all sums of rearranged series ∑n=1^∞ xσ(n) is an affine subspace of X. This affine subspace is not empty if and only if for any linear functional f:X→ \mathbb R the series ∑n=1^∞ f(xσ(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 (xn)n=1^∞ in the torus \mathbb T×\mathbb T such that the series ∑n=1^∞ xσ(n) is divergent for all permutations σ of \mathbb N but for any continuous homomorphism f:\mathbb T2→\mathbb T to the circle group \mathbb T:=\mathbb R/\mathbb Z the series ∑n=1^∞ f(xσf(n)) is convergent for some permutation σf 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.