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](http://www.math.lviv.ua/szkocka/viewpage.php?vol=1&page=72) of [Volume 1](http://www.math.lviv.ua/szkocka/viewbook.php?vol=1) of the [Lviv Scottish Book](http://www.math.lviv.ua/szkocka)).



Is "weakly good" series in a finite-dimensional Banach space "good"?
A relatively short inductive proof of Steinitz Theorem can be founded in [this paper](https://arxiv.org/abs/1711.04136). Here we present a sketch of the proof, which is based on 3 lemmas whose proof is left to the reader. First we recall the formulation of the result.


**Theorem.** A series $\sum\_{n=1}^\infty x\_n$ in a finite-dimensional Banach space $X$ is "good" if and only if for any linear functional $f:X\to\mathbb R$ the series $\sum\_{n=1}^\infty f(x\_n)$ is "good".


*Proof.* This theorem will be proved by induction on the dimension of $X$. The theorem is trivially true for Banach spaces of dimension $\le 1$.
Assume that for some number $d$ the theorem has been proved for all Banach spaces of dimension $<d$.


Fix a Banach space $X$ of dimension $d$. We can assume that $X=\mathbb R^d$ and the norm $\|\cdot\|$ of $X$ is generated by the standard inner product $\langle\cdot,\cdot\rangle$.


Let $\sum\_{n=1}^\infty x\_n$ be a series in $X$ such that for every linear functional $f:X\to\mathbb R$ the series $\sum\_{n=1}^\infty f(x\_n)$ is "good". We lose no generality assuming that $x\_n\ne 0$ for all $n$.


**Lemma 1.** $\lim\_{n\to\infty}x\_n=0$.


**Definition.** A point $s$ on the unit sphere $S=\{x\in X:\|x\|=1\}$ is called a *divergence direction* of the series $\sum\_{n=1}^\infty x\_n$ if for any neighborhood $U\subset S$ of $x$ the set $\mathbb N\_U=\{n\in\mathbb N:\frac{x\_n}{\|x\_n\|}\in U\}$ is infinite and $\sum\_{n\in\mathbb N\_U}\|x\_n\|=\infty$.


Let $D$ be the (closed) set of all divergence directions of $(x\_n)\_{n\in\mathbb N}$.


If $D=\emptyset$, then by the compactness of the sphere $S$, $\sum\_{n=0}^\infty \|x\_n\|<\infty$ and the series $\sum\_{n=1}^\infty x\_n$ is convergent, so is "good".


So, we assume that $D$ is not empty.
Using Hahn-Banach Theorem it can be shown that the convex hull $conv(D)$ of $D$ contains zero.


Let $D\_0$ be a subset of smallest cardinality in $D$ such that $0\in conv(D\_0)$.


It can be shown that the set $D\_0$ is affinely independent and zero is contained in the interior of the simplex $conv(D\_0)$.


Let $X\_0$ be the linear hull of the set $D\_0$ and $X\_1$ be the orhogonal complement of $X\_0$ in the Euclidean space $X$.
 So, $X=X\_0\oplus X\_1$.


For $i\in\{0,1\}$ let $pr\_i:X\to X\_i$ be the orthogonal projections of $X$ onto $X\_i$.


**Lemma 2.** There exists a subset $\Omega\subset \mathbb N$ such that


1) $\sum\_{n\in\Omega}\|pr\_1(x\_n)\|<+\infty$;


2) for any $s\in D$ and a neighborhood $U\subset S$ of $s$ the set $\Omega\_U=\{n\in\Omega:\frac{x\_n}{\|x\_n\|}\in U\}$ is infinite and $\sum\limits\_{n\in\Omega\_U}\|pr\_0(x\_n)\|=\infty$.


**Lemma 3.** There exists a constant $C$ (dependent on $D\_0$) such that for any $x\in X\_0$, $\varepsilon>0$, and a finite set $\Omega\_0\subset \Omega$ there exists a finite set $F\subset \Omega\setminus \Omega\_0$ such that


1) $\|x-\sum\_{n\in F}pr\_0(x\_n)\|<\varepsilon$;


2) $\sum\_{n\in F}\|x\_n\|\le C\max\{\|x\|,\varepsilon\}$.


Since $X\_1$ has dimension $<d$, we can apply the inductive assumption and conclude that the series $\sum\_{n\in\mathbb N\setminus\Omega}pr\_1(x\_n)$ is "good", which means that for some permutation $\sigma\_1$ of $\mathbb N\setminus\Omega$ the series $\sum\_{n\in\mathbb N\setminus\Omega}pr\_1(x\_{\sigma\_1(n)})$ is convergent in the Banach space $X\_1$.


Then using Lemma 3, we can extend the permutation $\sigma\_1$ of $\mathbb N\setminus\Omega$ to a permutation $\sigma$ of $\mathbb N$ such that $\sum\_{n=1}^\infty pr\_0(x\_{\sigma(n)})=0$.


Since $\sum\_{n\in\Omega}\|pr\_1(x\_n)\|<+\infty$, the series
$\sum\_{n\in\Omega}pr\_1(x\_{\sigma(n)})$ converges and then the series $\sum\_{n\in\mathbb N}pr\_1(x\_{\sigma(n)})=\sum\_{n\in\mathbb N\setminus\Omega}pr\_1(x\_{\sigma(n)})+\sum\_{n\in\Omega}pr\_1(x\_{\sigma(n)})$ converges, too.


Then the series $\sum\_{n\in\mathbb N}x\_{\sigma(n)}=\sum\_{n\in\mathbb N}pr\_0(x\_{\sigma(n)})+pr\_1(x\_{\sigma(n)})$ also converges and hence the series $\sum\_{n\in\mathbb N}x\_n$ is "good".


