A simple inductive proof of Levy-Steinitz theorem

الأصل · EN

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→ R the series ∑ₙ₌₁∞ f(xσ₍ₙ₎) is convergent for some permutation σ of 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 T such that the series ∑ₙ₌₁∞ xσ₍ₙ₎ is divergent for all permutations σ of N but for any continuous homomorphism f:T² to the circle group T:=R/Z the series ∑ₙ₌₁∞ f(xσf₍ₙ₎) is convergent for some permutation σf of 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.