show/hide this revision's text 5 edited body; edited body

To a conditionally convergent series $\sum_{n\geq 1}v_n$ in $\mathbb{R^d}$ \mathbb{R}^d$ one can attach so called convergence functionals $f$, which are linear functionals $f:\mathbb{R^d}\to\mathbb{R}$ f:\mathbb{R}^d\to\mathbb{R}$ with the property $\sum_{n=1}^{\infty}|f(v_n)|<\infty$. Let $\Gamma ((v_n))$ be the set of all these functionals. Then the set of values of the possible rearrangements of the series $\sum_{n=0}^{\infty}v_n$ is exactly the affine space $\sum_{n=0}^{\infty}v_n + \Gamma ((v_n))_0$, where $\Gamma ((v_n))_0$ denotes the annihilator of $\Gamma ((v_n))$, i.e. $\bigcap_{f\in\Gamma ((v_n))}\mathrm{ker}(f)$. This is precisely the Steinitz` Theorem mentioned by KConrad. Let me just add that this result does not hold in general for infinite-dimensional spaces. However, a genarilization generalization of Steinitz theorem seems very approachable for locally convex spaces -> see e.g. "The Steinitz theorem on rearrangement of series for nuclear spaces" by W. Banaszczyk (1990), in Journal für die reine und angewandte Mathematik 403, 187-200.

EDIT: added the condition on $v_n$ per KConrad´s comment.
show/hide this revision's text 4 edited body

To a conditionally convergent series $\sum_{n\geq 1}a_n$ 1}v_n$ in $\mathbb{R^d}$ one can attach so called convergence functionals $f$, which are linear functionals $f:\mathbb{R^d}\to\mathbb{R}$ with the property $\sum_{n=1}^{\infty}|f(a_n)|\sum_{n=1}^{\infty}|f(v_n)|<\infty$. Let $\Gamma ((a_n))$ (v_n))$ be the set of all these functionals. Then the set of values of the possible rearrangements of the series $\sum_{n=0}^{\infty}a_n$ \sum_{n=0}^{\infty}v_n$ is exactly the affine space $\sum_{n=0}^{\infty}a_n \sum_{n=0}^{\infty}v_n + \Gamma ((a_n))_0$, (v_n))_0$, where $\Gamma ((a_n))_0$ (v_n))_0$ denotes the annihilator of $\Gamma ((a_n))$, (v_n))$, i.e. $\bigcap_{f\in\Gamma ((a_n))}\mathrm{ker}(f)$. (v_n))}\mathrm{ker}(f)$. This is precisely the Steinitz` Theorem mentioned by KConrad. Let me just add that this result does not hold in general for infinite-dimensional spaces. However, a genarilization of Steinitz theorem seems very approachable for locally convex spaces -> see e.g. "The Steinitz theorem on rearrangement of series for nuclear spaces" by W. Banaszczyk (1990), in Journal für die reine und angewandte Mathematik 403, 187-200.

EDIT: added the condition on $a_n$ v_n$ per KConrad´s comment.
show/hide this revision's text 3 deleted 2 characters in body

To a conditionally convergent series $\sum_{n\geq 1}a_n$ in $\mathbb{R^d}$ one can attach so called convergence functionals $f$, which are linear functionals $f:\mathbb{R^d}\to\mathbb{R}$ with the property $\sum_{n=1}^{\infty}|f(a_n)|<\infty$. Let $\Gamma ((a_n))$ be the set of all these functionals. Then the set of values of the possible rearrangements of the series $\sum_{n=0}^{\infty}a_n$ is exactly the affine space $\sum_{n=0}^{\infty}a_n + \Gamma ((a_n))_0$, where $\Gamma ((a_n))_0$ denotes the annihilator of $\Gamma ((a_n))$, i.e. $\bigcap_{f\in\Gamma ((a_n))}\mathrm{ker}(f)$. This is precisely the Steinitz` Theorem mentioned by KConrad. Let me just add that this result does not hold in general for infinite-dimensional spaces. However, a genarilization of Steinitz theorem seems very approachable for locally convex spaces -> see e.g. "The Steinitz theorem on rearrangement of series for nuclear spaces" by W. Banaszczyk (1990), in Journal für die reine und angewandte Mathematik 403, 187-200.

EDIT: added the condition on $a_n$ per the post of KConradKConrad´s comment.
show/hide this revision's text 2 added 37 characters in body; added 63 characters in body
show/hide this revision's text 1