Series of vectors in $\mathbb{R}^n$, absolute convergence, rearrangement, and the set of rearrangement sums

Let $n \in \mathbb{N}$ with $n \ge 1$, so that $\mathbb{R}^{n}$ carries the Euclidean metric $d_2$ ($\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page). A sequence of vectors is a function $x : \mathbb{N} \to \mathbb{R}^{n}$, written $(x_k)$ with $x_k := x(k)$; as everywhere in this library $\mathbb{N}$ contains $0$ and a sequence is indexed from $0$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$).

Partial sums and convergence

The partial sums of $(x_k)$ are

$s_N \;:=\; \sum_{k<N} x_k \;\in\; \mathbb{R}^{n} \qquad (N \in \mathbb{N}),$

the finite sum of the vector space $\mathbb{R}^{n}$ (Linear combination of a finite list, and the span $\operatorname{span}(S)$ as the smallest linear subspace containing $S$), so $s_0 = 0$ and $s_{N+1} = s_N + x_N$. No third notion of finite sum is introduced: by 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 the vector sum is computed pointwise, $(s_N)(j) = \sum_{k<N} x_k(j)$ for $j<n$, the right-hand side being the real finite sum of Finite sums and finite products, by recursion.

The series $\sum x_k$ converges to $s \in \mathbb{R}^{n}$ when $s_N \to s$ in $(\mathbb{R}^{n}, d_2)$ (Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$), and then $s$ is the sum, written $\sum_{k=0}^{\infty}x_k$. The symbol denotes a single vector, because a sequence in a metric space has at most one limit (A sequence in a metric space has at most one limit). The series diverges when $(s_N)$ does not converge.

Absolute convergence

$\sum x_k$ converges absolutely when the real series $\sum \lVert x_k\rVert_2$ converges (Series, partial sums, convergence and the sum, divergence, and the tail series); since $\lVert x_k\rVert_2 \ge 0$ (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms), this is a statement about a series of nonnegative terms, exactly as in Absolutely convergent and conditionally convergent series, and the general starting index.

The choice of norm is immaterial. If $N$ is any norm on $\mathbb{R}^{n}$ then $c\lVert x_k\rVert_2 \le N(x_k) \le C\lVert x_k\rVert_2$ for fixed $c, C > 0$ (For $n \ge 1$ all norms on $\mathbb{R}^n$ are equivalent, Equivalent norms, and the dictionary with equivalent metrics), so $\sum N(x_k)$ converges exactly when $\sum\lVert x_k\rVert_2$ does, both being series of nonnegative terms. The notion defined above therefore depends on $\mathbb{R}^{n}$ and not on the norm chosen to test it.

Rearrangement and the set of rearrangement sums

Let $\sigma : \mathbb{N} \to \mathbb{N}$ be a bijection (Injection, surjection, bijection). The rearrangement of $\sum x_k$ along $\sigma$ is the series $\sum x_{\sigma(k)}$ of the sequence $k \mapsto x_{\sigma(k)}$, verbatim as in Rearrangement of a series along a bijection of $\mathbb{N}$, and unconditional convergence one dimension down. The set of rearrangement sums of $(x_k)$ is

$\mathcal{S}(x) \;:=\; \Bigl\{\, s \in \mathbb{R}^{n} \;:\; \text{some rearrangement of } \sum x_k \text{ converges to } s \,\Bigr\}.$

Taking $\sigma$ to be the identity shows that a convergent $\sum x_k$ has its own sum in $\mathcal{S}(x)$, so $\mathcal{S}(x) \ne \varnothing$ for a convergent series.

Agreement with the one-dimensional theory

$\mathbb{R}^{1}$ is the set of functions $1 \to \mathbb{R}$ and is not literally $\mathbb{R}$. The map $\theta : \mathbb{R} \to \mathbb{R}^{1}$ sending $t$ to the function with value $t$ at $0$ is a bijection; it preserves addition and scalar multiplication, since both are computed pointwise (Vector space over a field, 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 $0$), and $d_2(\theta(s),\theta(t)) = |s-t|$, so it is an isometric bijection (Isometry, isometric embedding, and the subspace metric on a subset). Under that identification, and for $n = 1$:

• the partial sums above are the partial sums of Series, partial sums, convergence and the sum, divergence, and the tail series;
• convergence and the sum are those of Series, partial sums, convergence and the sum, divergence, and the tail series;
• absolute convergence is that of Absolutely convergent and conditionally convergent series, and the general starting index, since $\lVert\theta(t)\rVert_2 = |t|$;
• rearrangement is that of Rearrangement of a series along a bijection of $\mathbb{N}$, and unconditional convergence;
• $\mathcal{S}(x)$ is the image under $\theta$ of the set of rearrangement sums that the published remark 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 writes $\mathcal{S}(a)$.

Every comparison on this page between $\mathbb{R}^{n}$ and the published one-dimensional theory goes through this identification, and it is stated each time.

• Where $n \ge 1$ comes from. Convergence is tested with $d_2$, and $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it defines the metrics on $\mathbb{R}^{n}$ only for $n \ge 1$. The algebra above — partial sums, rearrangement, the set $\mathcal{S}(x)$ as a set of vectors — makes sense at $n = 0$ as well, but nothing on this page is asserted there.

• Convergence is componentwise. By 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 clause 1 and the pointwise formula for partial sums, $\sum x_k$ converges to $s$ if and only if the $n$ real series $\sum (x_k)_j$ converge, with sums $s_j$. That is the form every proof below uses, and it is what reduces the vector theory to the published scalar theory rather than duplicating it.

• Unconditional convergence is not defined here. The one-dimensional notion is Rearrangement of a series along a bijection of $\mathbb{N}$, and unconditional convergence, and over $\mathbb{R}$ it coincides with absolute convergence (For a series of real numbers, unconditional convergence and absolute convergence are the same property). Whether that coincidence survives to $\mathbb{R}^{n}$ for $n \ge 2$ is not settled anywhere on this page, and nothing here asserts it in either direction. What is proved is that absolute convergence implies convergence of every rearrangement to the same sum (An absolutely convergent series in $\mathbb{R}^n$ converges, and every rearrangement converges to the same sum).