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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $(x_n)_{n \\ge 0}$ be a sequence in the [[Euclidean space|0XPR1E85W0]] $\\R^d$.\nLet $S$ be the set of all $s \\in \\R^d$ such that some [[rearrangement|0HRJ2Y74T3]] $\\sum_n x_{\\sigma(n)}$ converges to $s$.\nLet $\\Gamma$ be the set of [[linear functionals|0M856Q1AMZ]] $f \\colon \\R^d \\to \\R$ such that $\\sum_n \\abs{f(x_n)} \u003c \\infty$.\nIf $S$ is not empty, then\n$$S = s_0 + \\set{v \\in \\R^d : f(v) = 0 \\text{ for all } f \\in \\Gamma}$$\nfor any $s_0 \\in S$.\nIn particular, $S$ is either empty or an [[affine subspace|0FBXFBZBH8]] of $\\R^d$.\n\n## Remarks\n\nFor $d = 1$ this is the [[Riemann rearrangement theorem|0E7VBYGFF7]].\nIf the series converges [[absolutely|0W3K3QPEWT]], then $\\Gamma$ contains all functionals and $S$ is a single point.\nIf it converges conditionally, then $\\Gamma = \\set{0}$ and $S = \\R$.\nExample in $\\R^2$: take $x_n = ((-1)^n/(n+1), 1/2^n)$.\nThe second coordinate converges absolutely, the first only conditionally, so $S$ is a horizontal line $\\set{(t, 2) : t \\in \\R}$.\nThe proof is harder than in dimension one.\nIts core is a lemma of Steinitz on finitely many vectors in $\\R^d$ with sum zero.\nThey can be ordered so that every partial sum has norm at most $C_d$ times the largest norm of the vectors, where $C_d$ depends only on $d$.\nIn infinite-dimensional Banach spaces the conclusion can fail: the set of sums need not be an affine subspace.\n\n## Explanations\n\n### L1 human\n\nReordering a convergent series in the plane or in space can change its sum.\nThe theorem says that the possible sums always form a point, a line, a plane, or a higher flat piece of space.\nThe directions in which the sum cannot move are the directions in which the series converges absolutely.\n\n### L2 human\n\nThe theorem extends Riemann's rearrangement theorem from $\\R$ to $\\R^d$.\nThe subspace of possible shifts is cut out by the functionals that make the series absolutely convergent.\nIt explains why unconditional convergence and absolute convergence coincide in $\\R^d$: the set $S$ is a single point exactly when every functional is in $\\Gamma$.\n",
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