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Let us call a series $\sum_n x_n$ is a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges.

Find a simple proof of the following theorem (which was proved by E.Steinitz in 1913 according to V.Kadets).

Theorem. A series $\sum_n x_n$ in a finite-dimensional Banach space $X$ is "good" if and only if for every linear function $f:X\to\mathbb R$ the series $\sum_n f(x_n)$ is "good".

(This problem was posed 24.09.2017 by Vaja Tarieladze from Tbilisi on page 72 of Volume 1 of the Lviv Scottish Book).

Let us call a series $\sum_n x_n$ in a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges.

Find a simple proof of the following theorem (which was proved by E.Steinitz in 1913 according to V.Kadets).

Theorem. A series $\sum_n x_n$ in a finite-dimensional Banach space $X$ is "good" if and only if for every linear function $f:X\to\mathbb R$ the series $\sum_n f(x_n)$ is "good".

(This problem was posed 24.09.2017 by Vaja Tarieladze from Tbilisi on page 72 of Volume 1 of the Lviv Scottish Book).

Let us call a series $\sum_n x_n$ is a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges.

Find a simple proof of the following theorem (which was proved by E.Steinitz in 1913 according to V.Kadets).

Theorem. A series $\sum_n x_n$ in a finite-dimensional Banach space $X$ is "good" if and only if for every linear function $f:X\to\mathbb R$ the series $\sum_n f(x_n)$ is "good".

(This problem was posed in Lviv Scottish Book by Vaja Tarieladze from Tbilisi on 24.09.2017, see page 72 of http://www.math.lviv.ua/szkocka/view.php).

Let us call a series $\sum_n x_n$ is a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges.

Find a simple proof of the following theorem (which was proved by E.Steinitz in 1913 according to V.Kadets).

Theorem. A series $\sum_n x_n$ in a finite-dimensional Banach space $X$ is "good" if and only if for every linear function $f:X\to\mathbb R$ the series $\sum_n f(x_n)$ is "good".

(This problem was posed 24.09.2017 by Vaja Tarieladze from Tbilisi on page 72 of Volume 1 of the Lviv Scottish Book).