Conventions of this page, the standing $n\ge 1$ hypothesis, and what is taken up elsewhere in the reading order

The published ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it defines ${\mathbb{R}}^n$ together with the metrics $d_1$, $d_2$, $d_{\mathrm{\infty }}$ only for $n\ge 1$, and says why: at $n=0$ the value $d_{\mathrm{\infty }}(x,y)$ would be a maximum over the empty index set, which does not exist. Everything downstream of that item inherits the hypothesis, and this page inherits it too. In particular $\mathbb{R}$ and ${\mathbb{R}}^n$ for $n\ge 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $\mathbb{R}$ and Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line are stated for $n\ge 1$ and are never cited here for all $n$.

The boundary runs between the algebra and the metric, not where a reader would guess. The following items of this page carry no hypothesis on the dimension:

• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms — including the observation that the zero space carries exactly one norm;
• The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, whose sum is the empty sum at $n=0$;
• Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation, all four of whose clauses hold for every $n$, apart from the closing sentence of clause 2 identifying the induced metric with $d_2$;
• Equivalent norms, and the dictionary with equivalent metrics, which is about an arbitrary real vector space;
• clause 1 of Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page, that each $\parallel \cdot {\parallel }_p$ with rational $p\ge 1$ is a norm;
• clause 1 of The finite and reverse triangle inequalities for a norm; and for $n\ge 1$ every norm $N$ on ${\mathbb{R}}^n$ satisfies $N(x)\le C\parallel x{\parallel }_1$ and is Lipschitz, hence continuous, for $d_2$, the finite and reverse triangle inequalities for a norm on any real vector space.

The remaining items all carry $n\ge 1$ (or $m\ge 1$ for the codomain of a vector-valued function), and each states it in its own Statement: The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$ for $\parallel \cdot {\parallel }_{\mathrm{\infty }}$; clauses 2 and 3 of Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page; clauses 2, 3, 4 of The finite and reverse triangle inequalities for a norm; and for $n\ge 1$ every norm $N$ on ${\mathbb{R}}^n$ satisfies $N(x)\le C\parallel x{\parallel }_1$ and is Lipschitz, hence continuous, for $d_2$; For $n\ge 1$ all norms on ${\mathbb{R}}^n$ are equivalent; 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; For $n\ge 1$ every bounded sequence in ${\mathbb{R}}^n$ has a convergent subsequence; Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions; A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions; The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral; For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable; The mean value inequality: if $f:[a,b]\to {\mathbb{R}}^m$ is continuous and differentiable on $(a,b)$ with $\parallel f^{\mathrm{\prime }}{\parallel }_2\le M$, then $\parallel f(b)-f(a){\parallel }_2\le M(b-a)$; If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz; Series of vectors in ${\mathbb{R}}^n$, absolute convergence, rearrangement, and the set of rearrangement sums; An absolutely convergent series in ${\mathbb{R}}^n$ converges, and every rearrangement converges to the same sum; The subspace $\mathrm{\Gamma }$ of directions along which a series converges absolutely, and its orthogonal complement ${\mathrm{\Gamma }}^{\perp }$; Steinitz's polygonal confinement theorem: finitely many vectors of norm at most $1$ summing to $0$ can be ordered so that every partial sum has norm at most $n$; and The set of rearrangement sums of a convergent series in ${\mathbb{R}}^n$ is a nonempty subset of the affine subspace $s+{\mathrm{\Gamma }}^{\perp }$.

Where a statement about $n=0$ is nevertheless true, it is proved here from scratch rather than imported: see the second remark of 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 for completeness of ${\mathbb{R}}^0$.

Rational powers $a^r$ of a positive base supplies $a^r$ for a positive base and any rational exponent, together with $0^r$ for rational $r>0$; real exponents do not exist at this point in the reading order; Why real exponents are deferred on the rational-powers page records why. Consequently The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$ defines $\parallel \cdot {\parallel }_p$ for rational $p\ge 1$ only, and the published Minkowski inequality it rests on is itself stated for rational $p$. No statement on this page is written with $p$ ranging over a real interval, and the phrase "for $p\in [1,\mathrm{\infty })$" appears nowhere.

${\mathbb{R}}^n$ is the set of functions $n\to \mathbb{R}$ (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$, ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it), so ${\mathbb{R}}^1$ is not literally $\mathbb{R}$: its elements are functions on the one-element set $1$. Every comparison on this page between the theory in ${\mathbb{R}}^n$ and the published one-dimensional theory therefore goes through the isometric bijection $\theta :\mathbb{R}\to {\mathbb{R}}^1$ sending $t$ to the function with value $t$ at $0$, and each item that makes such a comparison states the identification explicitly: For $n\ge 1$ every bounded sequence in ${\mathbb{R}}^n$ has a convergent subsequence, Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions, Series of vectors in ${\mathbb{R}}^n$, absolute convergence, rearrangement, and the set of rearrangement sums and The set of rearrangement sums of a convergent series in ${\mathbb{R}}^n$ is a nonempty subset of the affine subspace $s+{\mathrm{\Gamma }}^{\perp }$. Coordinates are indexed from $0$ throughout, as The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$ fixes.

Each item below is a statement about where material sits in this library's reading order, and none of them is a claim about mathematics that this library denies.

• Linear maps, the operator norm, and "a linear map between finite-dimensional normed spaces is Lipschitz". Abstract linear maps are defined on the earlier linear-algebra page in Linear map between vector spaces over the same field. This page neither defines an operator norm nor proves that every linear map between finite-dimensional normed spaces is Lipschitz. The later total-derivative treatment uses a concrete Euclidean formulation; identifying that formulation with the abstract definition requires an explicit agreement argument rather than a second silent meaning of "linear map".
• Inner product spaces and orthogonality. The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ defines the concrete Euclidean form on ${\mathbb{R}}^n$ and claims nothing about abstract inner product spaces. Orthonormal bases, Gram-Schmidt, orthogonal projection, orthogonal complements and finite-dimensional orthogonal decomposition are now developed on a separate published linear algebra page. In particular nothing on this page asserts that ${\mathbb{R}}^n$ is the direct sum of $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\perp }$ (The subspace $\mathrm{\Gamma }$ of directions along which a series converges absolutely, and its orthogonal complement ${\mathrm{\Gamma }}^{\perp }$).
• Uniform convergence, the total derivative, and integration over subsets of ${\mathbb{R}}^n$ all come later in this track.
• The classical mean value witness. The crispest counterexample to the equality form of the mean value theorem for vector-valued functions is $t\mapsto (\cos t,\sin t)$ on $[0,2\pi ]$. The trigonometric functions are introduced later in the reading order than this page, so the companion page uses the polynomial curve $t\mapsto (t^2,t^3)$ on $[0,1]$ instead. The substitution is recorded in the companion item that carries the witness, not here, so that a reader meeting the polynomial curve is told at once why the classical one is absent. The witness refutes the vector-valued equality generalisation of The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$; the inequality that survives is The mean value inequality: if $f:[a,b]\to {\mathbb{R}}^m$ is continuous and differentiable on $(a,b)$ with $\parallel f^{\mathrm{\prime }}{\parallel }_2\le M$, then $\parallel f(b)-f(a){\parallel }_2\le M(b-a)$.

The set of rearrangement sums of a convergent series in ${\mathbb{R}}^n$ is a nonempty subset of the affine subspace $s+{\mathrm{\Gamma }}^{\perp }$ proves that the set $\mathcal{S}(x)$ of rearrangement sums of a convergent series in ${\mathbb{R}}^n$ is nonempty and contained in the affine subspace $s+{\mathrm{\Gamma }}^{\perp }$, and Steinitz's polygonal confinement theorem: finitely many vectors of norm at most $1$ summing to $0$ can be ordered so that every partial sum has norm at most $n$ proves Steinitz's polygonal confinement lemma in full. The reverse inclusion is not proved on this page, and this page asserts nothing about it in either direction, for any $n\ge 2$. No recorded-not-proved item has been created for it either.

The reverse inclusion still needs an argument beyond the containment proof on this page. Orthogonal decomposition and convex separation are now available elsewhere in the library, but their existence alone does not establish the rearrangement construction. Any discharge here must supply that construction and its convergence estimates explicitly.

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 raised this question on the series page and declined to state what the literature answers; this page answers the part it can and continues to decline the rest. What a reader is protected from meanwhile is the wrong guess: the companion page refutes outright the naive ${\mathbb{R}}^n$ analogue of the Riemann series theorem, using the containment half and nothing more.

Steinitz's polygonal confinement theorem: finitely many vectors of norm at most $1$ summing to $0$ can be ordered so that every partial sum has norm at most $n$ is Steinitz's polygonal confinement lemma from his 1913 paper on conditionally convergent series. It is not the Steinitz exchange lemma of linear algebra, which is published in this library under the id thm-steinitz-exchange and additionally carries the alias lem-steinitz. The two are different theorems by the same author; the ids do not collide, and no item on this page uses the bare alias.