room: A1-33, 10:30–11:30

Taras Banakh (Ivan Franko National University of Lviv and Jan Kochanowski University in Kielce)
A simple inductive proof of Levy–Steinitz theorem

Summary. 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 \(\sum_{n=1}^\infty x_{\sigma(n)}\) is an affine subspace of \(X\). This affine subspace is not empty if and only if for any linear functional \(f\colon X\to\mathbb{R}\) the series \(\sum_{n=1}^\infty f(x_{\sigma(n)})\) is convergent for some permutation \(\sigma\) of \(\mathbb{N}\). This gives an answer to a problem of Vaja Tarieladze, posed in Lviv Scottish Book in September, 2017. More details can be found in arxiv.

Coffee and the at 10:00 in the Professors’ club.