TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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The set of rearrangement sums of a convergent series in $\mathbb{R}^n$ is a nonempty subset of the affine subspace $s + \Gamma^{\perp}$

Statement

Let $n \in \mathbb{N}$ with $n \ge 1$ , let $(x_k)$ be a sequence in $\mathbb{R}^{n}$ whose series converges (Series of vectors in $\mathbb{R}^n$ , absolute convergence, rearrangement, and the set of rearrangement sums) and write $s := \sum_{k=0}^{\infty}x_k$ . Let $\Gamma$ and $\Gamma^{\perp}$ be as in The subspace $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$ . Then:

Nonemptiness. $s \in \mathcal{S}(x)$ , so $\mathcal{S}(x) \ne \varnothing$ .
Containment. $\mathcal{S}(x) \;\subseteq\; s + \Gamma^{\perp},$ the affine subspace through $s$ with direction $\Gamma^{\perp}$ (The subspace $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$ ). Equivalently, $t - s \in \Gamma^{\perp}$ for every rearrangement sum $t$ .
The absolutely convergent case. If $\sum x_k$ converges absolutely then $\Gamma = \mathbb{R}^{n}$ , $\Gamma^{\perp} = \{0\}$ , the affine subspace is the single point $\{s\}$ , and $\mathcal{S}(x) = \{s\}$ .
The one-dimensional conditionally convergent case. Let $n = 1$ and identify $\mathbb{R}^{1}$ with $\mathbb{R}$ as in Series of vectors in $\mathbb{R}^n$ , absolute convergence, rearrangement, and the set of rearrangement sums. If $\sum x_k$ converges conditionally (Absolutely convergent and conditionally convergent series, and the general starting index) then $\Gamma = \{0\}$ , $\Gamma^{\perp} = \mathbb{R}^{1}$ , and the containment of clause 2 is an equality, $\mathcal{S}(x) = s + \Gamma^{\perp} = \mathbb{R}^{1}$ , by the published The Riemann series theorem: a conditionally convergent real series has, for every $c \in \mathbb{R}$ , a rearrangement with sum $c$ , and rearrangements diverging to $+\infty$ , to $-\infty$ , and oscillating with any prescribed $\liminf \le \limsup$ in $\overline{\mathbb{R}}$ .

What this theorem does not say, stated here and repeated in the Remarks. It proves a containment and nothing more. Whether $\mathcal{S}(x)$ is all of $s + \Gamma^{\perp}$ when $n \ge 2$ is not settled anywhere on this page, and no item on this page asserts anything about it in either direction. Clause 4 is the case $n = 1$ , where the answer is supplied by a published theorem about the real line; it is not evidence for any statement in higher dimensions.

Facts & Assumptions

Given: A natural $n \ge 1$ ; a sequence $(x_k)$ in $\mathbb{R}^{n}$ with $\sum x_k$ convergent of sum $s$ ; a bijection $\sigma$ of $\mathbb{N}$ ; a vector $a \in \Gamma$ ; the partial sums $s_N = \sum_{k<N}x_k$ and $s^{\sigma}_N = \sum_{k<N}x_{\sigma(k)}$ .

[L1]

Series of vectors, absolute convergence, rearrangement, $\mathcal{S}(x)$ , and the identification of $\mathbb{R}^{1}$ with $\mathbb{R}$ (Series of vectors in $\mathbb{R}^n$ , absolute convergence, rearrangement, and the set of rearrangement sums, Rearrangement of a series along a bijection of $\mathbb{N}$ , and unconditional convergence, Injection, surjection, bijection, Isometry, isometric embedding, and the subspace metric on a subset).

[L2]

$\Gamma$ and $\Gamma^{\perp}$ are linear subspaces; $a \in \Gamma$ means $\sum_k|\langle a,x_k\rangle|$ converges; $\Gamma = \mathbb{R}^{n}$ exactly when $\sum x_k$ converges absolutely; and $s + W$ denotes the coset of a linear subspace $W$ (The subspace $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$ , Linear subspace of a vector space).

[L3]

The inner product is bilinear and symmetric, $\langle y,y\rangle = \lVert y\rVert_2^{2}$ , positive definiteness gives $\langle y,y\rangle = 0$ only for $y = 0$ , and Cauchy-Schwarz gives $|\langle a,y\rangle| \le \lVert a\rVert_2\lVert y\rVert_2$ (The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ , Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$ , the parallelogram law and polarisation, The $p$ -norms $\lVert x\rVert_p$ for rational $p \ge 1$ , and $\lVert x\rVert_\infty$ , A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).

[L4]

Laws of finite sums and the induction principle (Laws of finite sums and finite products, Finite sums and finite products, by recursion, The principle of mathematical induction); finite sums in $\mathbb{R}^{n}$ are pointwise (The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$ ; hence $\dim_F F^{n} = n$ , and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$ clause 1).

[L5]

Dirichlet's rearrangement theorem: an absolutely convergent real series has, for every bijection $\sigma$ of $\mathbb{N}$ , a rearrangement converging to the same sum (Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum, Absolutely convergent and conditionally convergent series, and the general starting index).

[L6]

The Riemann series theorem: a conditionally convergent real series has, for every $c \in \mathbb{R}$ , a rearrangement converging to $c$ (The Riemann series theorem: a conditionally convergent real series has, for every $c \in \mathbb{R}$ , a rearrangement with sum $c$ , and rearrangements diverging to $+\infty$ , to $-\infty$ , and oscillating with any prescribed $\liminf \le \limsup$ in $\overline{\mathbb{R}}$ clause 1); and over $\mathbb{R}$ a convergent series is absolutely convergent or conditionally convergent and not both (For a series of real numbers, unconditional convergence and absolute convergence are the same property, Absolutely convergent and conditionally convergent series, and the general starting index).

[L7]

Convergence in $(\mathbb{R}^{n},d_2)$ and in $\mathbb{R}$ , uniqueness of limits, and the componentwise criterion (Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$ , Limits and Cauchy sequences of reals, A sequence in a metric space has at most one limit, For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm).

[L8]

An absolutely convergent series in $\mathbb{R}^{n}$ converges, every rearrangement converges to the same sum, and $\mathcal{S}(x)$ is then a single point (An absolutely convergent series in $\mathbb{R}^n$ converges, and every rearrangement converges to the same sum).

[L9]

Absolute value and order arithmetic: $|uv| = |u||v|$ , $|u| \ge 0$ , and $u>0$ gives $u^{-1}>0$ (Basic properties of the absolute value, Inverses of positives are positive, and reciprocation reverses order).

Proof

technique · direct

1.1

The identity map of $\mathbb{N}$ is a bijection and the rearrangement along it is the original series, so $s \in \mathcal{S}(x)$ and clause 1 holds.

1.2

For every $a \in \mathbb{R}^{n}$ and every finite list $u : p \to \mathbb{R}^{n}$ , $\bigl\langle a, \sum_{j<p}u_j\bigr\rangle = \sum_{j<p}\langle a, u_j\rangle$ : at $p=0$ both sides are $0$ , and the successor step is additivity of the inner product in its second argument.

L3L4

1.3

If $u_N \to u$ in $(\mathbb{R}^{n},d_2)$ then $\langle a, u_N\rangle \to \langle a,u\rangle$ in $\mathbb{R}$ , since $\bigl|\langle a,u_N\rangle - \langle a,u\rangle\bigr| = \bigl|\langle a, u_N-u\rangle\bigr| \le \lVert a\rVert_2\,\lVert u_N-u\rVert_2$ , so a tolerance $\varepsilon/(\lVert a\rVert_2+1)$ on the right serves for $\varepsilon$ on the left.

L3L7L9

1.4

Now let $n = 1$ and suppose $\sum x_k$ converges conditionally, so the real series $\sum_k (x_k)_0$ converges and $\sum_k|(x_k)_0|$ diverges. For $a \in \mathbb{R}^{1}$ , $\langle a, x_k\rangle = a_0 (x_k)_0$ and $|\langle a,x_k\rangle| = |a_0|\,|(x_k)_0|$ ; if $a_0 \ne 0$ then convergence of $\sum_k |a_0||(x_k)_0|$ would give convergence of $\sum_k |(x_k)_0|$ after multiplying by the positive $1/|a_0|$ , which is false, so $a \in \Gamma$ forces $a_0 = 0$ ; and $a = 0$ does lie in $\Gamma$ . Hence $\Gamma = \{0\}$ .

L1L2L9

2.1

Let $t \in \mathcal{S}(x)$ , say $s^{\sigma}_N \to t$ for a bijection $\sigma$ , and let $a \in \Gamma$ . By steps 1.2 and 1.3, $\langle a,t\rangle = \lim_N \langle a, s^{\sigma}_N\rangle = \lim_N \sum_{k<N}\langle a, x_{\sigma(k)}\rangle$ , so the real series $\sum_k\langle a,x_{\sigma(k)}\rangle$ converges with sum $\langle a,t\rangle$ .

step 1.2step 1.3L1L7

2.2

In the same way $\langle a, s\rangle = \lim_N\langle a, s_N\rangle = \lim_N\sum_{k<N}\langle a,x_k\rangle$ , so $\sum_k\langle a,x_k\rangle$ converges with sum $\langle a,s\rangle$ .

step 1.2step 1.3L7

2.3

With $\Gamma = \{0\}$ the condition defining $\Gamma^{\perp}$ is $\langle 0,y\rangle = 0$ , which holds for every $y$ , so $\Gamma^{\perp} = \mathbb{R}^{1}$ and $s + \Gamma^{\perp} = \mathbb{R}^{1}$ .

step 1.4L2L3

3.1

The real sequence $k \mapsto \langle a, x_{\sigma(k)}\rangle$ is the rearrangement along $\sigma$ of the sequence $k \mapsto \langle a,x_k\rangle$ , and the latter series converges absolutely because $a \in \Gamma$ ; so by Dirichlet's theorem the two series have the same sum.

step 2.1step 2.2L2L5

3.2

By the Riemann series theorem applied to the conditionally convergent real series $\sum_k (x_k)_0$ , every real $c$ is the sum of some rearrangement of it; transporting along the identification of $\mathbb{R}$ with $\mathbb{R}^{1}$ , every element of $\mathbb{R}^{1}$ lies in $\mathcal{S}(x)$ . So $\mathcal{S}(x) = \mathbb{R}^{1} = s + \Gamma^{\perp}$ , which with steps 1.4 and 2.3 is clause 4.

step 1.4step 2.3L1L6L7

4.1

Combining steps 2.1, 2.2 and 3.1 gives $\langle a,t\rangle = \langle a,s\rangle$ , hence $\langle a, t-s\rangle = 0$ by bilinearity.

step 2.1step 2.2step 3.1L3

5.1

Since $a \in \Gamma$ was arbitrary, $t-s \in \Gamma^{\perp}$ , that is $t \in s + \Gamma^{\perp}$ ; as $t \in \mathcal{S}(x)$ was arbitrary, clause 2 holds.

step 4.1L2

6.1

Suppose $\sum x_k$ converges absolutely. Then $\Gamma = \mathbb{R}^{n}$ , so any $y \in \Gamma^{\perp}$ satisfies $\langle y,y\rangle = 0$ and hence $y = 0$ ; thus $\Gamma^{\perp} = \{0\}$ and $s + \Gamma^{\perp} = \{s\}$ . Moreover $\mathcal{S}(x) = \{s\}$ by [L8], so clause 3 holds and the containment of clause 2 is an equality in this case.

step 5.1L2L3L8

7.1

Clauses 1, 2, 3 and 4 are steps 1.1, 5.1, 6.1 and 3.2.

step 1.1step 5.1step 6.1step 3.2∎

Remarks

This theorem proves containment only, and the reverse inclusion is not proved, assumed, or asserted anywhere on this page. For $n \ge 2$ the question whether every point of $s + \Gamma^{\perp}$ is a rearrangement sum is open as far as this library is concerned. It is not open in the mathematical literature, and this page deliberately states nothing about what the literature says, exactly as the published The same question in $\mathbb{R}^d$ : what the set of rearrangement sums looks like, and why that answer is not reachable at this point in the reading order declines to. What is missing here is machinery, not effort: every route known to the author of this page passes through the orthogonal decomposition of a finite-dimensional inner product space and through a separation argument for convex sets, and neither exists in this library — the first belongs to a page earlier in the plan order that is not yet built, and the second to no planned page at all. See Conventions of this page, the standing $n \ge 1$ hypothesis, and what is taken up elsewhere in the reading order.
The title claims exactly clause 2 and clause 1, and no more. A title asserting that $\mathcal{S}(x)$ is the affine subspace would assert the reverse inclusion, which is not proved here.
Clause 4 is the published one-dimensional dichotomy seen from this page. Over $\mathbb{R}$ a convergent series is either absolutely convergent, and then $\Gamma$ is everything and $\mathcal{S}$ is a point (clause 3), or conditionally convergent, and then $\Gamma$ is $\{0\}$ and $\mathcal{S}$ is the whole line (clause 4). Both extremes are consistent with clause 2, and both are equalities; that is a fact about dimension $1$ , where a linear subspace of $\mathbb{R}^{1}$ is $\{0\}$ or everything and there is no room in between.
What the containment already rules out. Even without the reverse inclusion, clause 2 forbids a rearrangement sum from leaving the affine subspace. That is enough to refute the naive $\mathbb{R}^{n}$ analogue of the Riemann series theorem, and the companion page does so with an elementary witness, using clause 2 and nothing further.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 213 results over 44 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Levy-Steinitz theorem (Wikipedia) (standard reference, not scraped)
Riemann series theorem (Wikipedia) (standard reference, not scraped)
T. Banakh, A Simple Inductive Proof of the Levy-Steinitz Theorem (standard reference, not scraped)

The set of rearrangement sums of a convergent series in Rn\mathbb{R}^nRn is a nonempty subset of the affine subspace s+Γ⊥s + \Gamma^{\perp}s+Γ⊥