On Levy-Steinitz theorem

Author: George Chelidze
Co-authors: Vaja Tarieladze
Keywords: Levy-Steinitz theorem, rearranged series
Annotation:

We call a series ∑▒x_n is a Banach space "good" if there exists a permutation σ:N→N such that the rearranged series ∑▒x_(σ(n)) converges. In the present paper we give an answer on the question of Vaja Tarieladze posed in Lviv Scottish Book: Find a simple proof of the following theorem which was proved by E.Steinitz in 1913 according to V.Kadets. Theorem. A series ∑▒x_n in a finite-dimensional Banach space X is "good" if and only if for every linear function f:X→R the series ∑▒〖〖f(x〗_n)〗 is "good".