R^n as a Normed Space; Vector-Valued Functions

Throughout this page $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property)) constructed in this library, in particular a field, so that "vector space" below always means vector space over $\mathbb{R}$ (Vector space over a field).

Let $V$ be a vector space over $\mathbb{R}$, with zero vector $0_V$. A norm on $V$ is a function $N:V\to \mathbb{R}$ such that for all $u,v\in V$ and all $\lambda \in \mathbb{R}$:

• (N1) Separation. $N(v)=0$ if and only if $v=0_V$.
• (N2) Absolute homogeneity. $N(\lambda v)=\mathrm{\mid }\lambda \mathrm{\mid }N(v)$, the absolute value being that of Absolute value in an ordered field.
• (N3) Triangle inequality. $N(u+v)\le N(u)+N(v)$.

A normed space is a pair $(V,N)$ consisting of a vector space $V$ over $\mathbb{R}$ and a norm $N$ on it. When only one norm is in play we write $\parallel v\parallel$ for $N(v)$; when several are, the norm is always named.

The values of a norm are real numbers. The codomain is $\mathbb{R}$, so $N(v)$ is an honest element of the complete ordered field and no infinite value is permitted. This is the same convention Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here records for metrics.

Nonnegativity is a theorem, not an axiom

Many texts add a fourth condition $N(v)\ge 0$. It is redundant. Applying (N2) with $\lambda =-1$ gives $N(-v)=\mathrm{\mid }-1\mathrm{\mid }N(v)=N(v)$ (Basic properties of the absolute value, In any vector space $0_{F}v=0_V$, $\lambda 0_V=0_V$, $(-\lambda )v=-(\lambda v)$, $(-1_F)v=-v$, and $\lambda v=0_V$ forces $\lambda =0_F$ or $v=0_V$ for $(-1)v=-v$), and then (N3) with $u=v$ and $-v$ gives

$0=N(0_V)=N(v+(-v))\le N(v)+N(-v)=N(v)+N(v),$

where $N(0_V)=0$ is (N1). So $N(v)+N(v)\ge 0$, and if $N(v)<0$ then $N(v)+N(v)<0$ by addition of inequalities, which trichotomy forbids (Complete ordered field (least-upper-bound property)). Hence $N(v)\ge 0$ for every $v\in V$.

Consequently the verification of a candidate norm has three things to check and not four, exactly as the verification of a candidate metric has three and not four (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, Nonnegativity of a metric is a consequence of the other axioms, not an axiom). No item in this library assumes nonnegativity of a norm before the argument above.

The induced metric

Let $N$ be a norm on $V$ and define

$d_N(u,v):=N(u-v)(u,v\in V),$

where $u-v=u+(-v)$ (Vector space over a field). Then $d_N$ is a metric on $V$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), and the three axioms are the three conditions above, in order:

• (M1) $d_N(u,v)=0$ means $N(u-v)=0$, which by (N1) says $u-v=0_V$, that is $u=v$; and conversely $d_N(v,v)=N(0_V)=0$.
• (M2) $d_N(v,u)=N(v-u)=N((-1)(u-v))=\mathrm{\mid }-1\mathrm{\mid }N(u-v)=d_N(u,v)$, by (N2), Basic properties of the absolute value and $(-1)w=-w$ (In any vector space $0_{F}v=0_V$, $\lambda 0_V=0_V$, $(-\lambda )v=-(\lambda v)$, $(-1_F)v=-v$, and $\lambda v=0_V$ forces $\lambda =0_F$ or $v=0_V$).
• (M3) $d_N(u,w)=N((u-v)+(v-w))\le N(u-v)+N(v-w)=d_N(u,v)+d_N(v,w)$, by (N3).

A normed space is therefore a metric space, and every notion defined for metric spaces — open set (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), convergence, Cauchyness, continuity, compactness — is available in it with no further definition. This library never introduces a second notion of any of them for normed spaces.

Two properties an arbitrary metric need not have

The metric $d_N$ satisfies, for all $u,v,w\in V$ and $\lambda \in \mathbb{R}$:

• translation invariance, $d_N(u+w,v+w)=N((u+w)-(v+w))=N(u-v)=d_N(u,v)$;
• absolute homogeneity, $d_N(\lambda u,\lambda v)=N(\lambda (u-v))=\mathrm{\mid }\lambda \mathrm{\mid }{d}_N(u,v)$, by (N2).

Not every metric on a vector space arises from a norm, and homogeneity is what fails. The published bounded remetrisation $\min (d,1)$ and $d\mathrm{/}(1+d)$ are metrics uniformly equivalent to $d$, so every metric space carries a bounded metric with the same topology replaces a metric $d$ by $d^{\mathrm{\prime }}=\min \{d,1\}$, a metric with the same topology whose values never exceed $1$; on a vector space $V$ containing a vector $v$ with $d(v,0_V)>0$ this $d^{\mathrm{\prime }}$ cannot be $d_M$ for any norm $M$, since absolute homogeneity would force $d^{\mathrm{\prime }}(\lambda v,0_V)=\mathrm{\mid }\lambda \mathrm{\mid }{d}^{\mathrm{\prime }}(v,0_V)$, which is unbounded in $\lambda$, while $d^{\mathrm{\prime }}$ is bounded by $1$. So the passage from norms to metrics is not reversible, and a statement about a metric on a vector space is strictly weaker than the corresponding statement about a norm.

• Why (N1) is stated as an equivalence. The direction $N(0_V)=0$ is forced by (N2) with $\lambda =0$, since $0v=0_V$ (In any vector space $0_{F}v=0_V$, $\lambda 0_V=0_V$, $(-\lambda )v=-(\lambda v)$, $(-1_F)v=-v$, and $\lambda v=0_V$ forces $\lambda =0_F$ or $v=0_V$) gives $N(0_V)=\mathrm{\mid }0\mathrm{\mid }N(v)=0$. Only the direction "$N(v)=0$ implies $v=0_V$" is a genuine assumption, and dropping it gives what is usually called a seminorm, a notion this library does not use. The situation is exactly the one Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here describes for (M1) and the pseudometric.

• The zero space carries exactly one norm. If $V=\{0_V\}$ then the only function $V\to \mathbb{R}$ satisfying (N1) is the one with value $0$, and it satisfies (N2) and (N3) trivially. In particular ${\mathbb{R}}^0$, the function space on the empty index set (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$), is a normed space, although the metrics of the published metric theory on ${\mathbb{R}}^n$ are defined only for $n\ge 1$.

• What is not defined here. This item does not define linear maps; their published definition is Linear map between vector spaces over the same field. It also does not define operator norms, dual spaces, or abstract inner product spaces. Conventions of this page, the standing $n\ge 1$ hypothesis, and what is taken up elsewhere in the reading order records the remaining scope boundaries and what each later development would license.

Let $n\in \mathbb{N}$. A natural number is a von Neumann natural, that is a set, and $n=\{0,1,\dots ,n-1\}$ (The natural numbers $\mathbb{N}$ (von Neumann), On $\mathbb{N}$ the order is membership: $m<n⟺m\in n$), so

${\mathbb{R}}^n=\{x:x\text{ is a function }n\to \mathbb{R}\}$

is the function space of The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$ at $F=\mathbb{R}$ and $X=n$, a vector space over $\mathbb{R}$ under the pointwise operations (Vector space over a field). We write $x_k:=x(k)$ for $k<n$, and two elements of ${\mathbb{R}}^n$ are equal exactly when they agree at every $k<n$. This is the same set that ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it calls ${\mathbb{R}}^n$.

The Euclidean inner product of $x,y\in {\mathbb{R}}^n$ is the real number

$⟨x,y⟩:={\sum }_{k<n}{x}_k{y}_k,$

the finite sum of Finite sums and finite products, by recursion applied to the list $k\mapsto x_k{y}_k$ (extended by $0$ beyond $n$, as every finite list in this library is). The Euclidean norm of $x$ is

$\parallel x{\parallel }_2:=\sqrt{⟨x,x⟩},$

which is defined because $⟨x,x⟩={\sum }_{k<n}{x}_k^2\ge 0$ (a sum of nonnegative terms, Laws of finite sums and finite products clause 4 and Squares of nonzero elements are positive, the case $x_k=0$ giving $x_k^2=0$ by Integer powers $a^m$) and every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

Both are defined for every $n$, including $n=0$

At $n=0$ the set ${\mathbb{R}}^0$ has exactly one element, the empty function, and it is the zero vector space (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 $0$ clause 5); the sum above is the empty sum, so $⟨x,y⟩=0$ and $\parallel x{\parallel }_2=0$. This is the first place on this page where the two index regimes diverge, and the divergence is deliberate. The published metrics $d_1$, $d_2$, $d_{\mathrm{\infty }}$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it are defined only for $n\ge 1$, because $d_{\mathrm{\infty }}$ would otherwise be a maximum over the empty index set; the algebra above needs no such restriction. The boundary in this page runs between the algebra and the metric, not where a reader would guess, and Conventions of this page, the standing $n\ge 1$ hypothesis, and what is taken up elsewhere in the reading order lists exactly which items inherit $n\ge 1$.

The algebra of the inner product

For all $x,y,z\in {\mathbb{R}}^n$ and $\lambda \in \mathbb{R}$:

1. Symmetry. $⟨x,y⟩=⟨y,x⟩$, since $x_k{y}_k=y_k{x}_k$ termwise.
2. Additivity in the first argument. $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩$: the list $k\mapsto (x_k+y_k)z_k$ is the termwise sum of $k\mapsto x_k{z}_k$ and $k\mapsto y_k{z}_k$, so Laws of finite sums and finite products clause 1 applies.
3. Homogeneity in the first argument. $⟨\lambda x,y⟩=\lambda ⟨x,y⟩$, by Laws of finite sums and finite products clause 2.
4. Bilinearity. Clauses 2 and 3 together with symmetry give the same two laws in the second argument.
5. Positive definiteness. $⟨x,x⟩\ge 0$, and $⟨x,x⟩=0$ if and only if $x=0$. Indeed a vanishing sum of nonnegative terms has every term $0$ (Laws of finite sums and finite products clause 4), so $x_k^2=0$ for every $k<n$, and a nonzero real has a positive square (Squares of nonzero elements are positive), whence $x_k=0$ for every $k<n$ and $x=0$.
6. Agreement with the published Euclidean metric. For $n\ge 1$ and $x,y\in {\mathbb{R}}^n$, $\parallel x-y{\parallel }_2=\sqrt{{\sum }_{k<n}(x_k-y_k{)}^2}=d_2(x,y)$, the two sides being the same expression (${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it). In particular $\parallel x{\parallel }_2=d_2(x,0)$.

That $\parallel \cdot {\parallel }_2$ is a norm in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms is proved in 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, where the triangle inequality is obtained from the Cauchy-Schwarz inequality; it is not assumed here.

• Scope: the concrete form only. What is defined above is the Euclidean inner product on ${\mathbb{R}}^n$ and nothing more. The general theory of inner product spaces — abstract inner products, orthonormal bases, Gram-Schmidt, orthogonal projection and orthogonal complements of arbitrary subspaces — is planned for a page of this library that comes earlier in the plan order and is not yet built. No item on this page claims anything about abstract inner product spaces, and no item on this page introduces the general notion.

• The standard basis and coordinates. For $i<n$ the standard unit vector $e_i\in {\mathbb{R}}^n$ has $e_i(i)=1$ and $e_i(j)=0$ for $j\mathrm{\ne }i$ (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 $0$). Then $⟨x,e_i⟩=x_i$: the list $k\mapsto x_k(e_i{)}_k$ vanishes except at $k=i$, where its value is $x_i$, and a list vanishing off one index sums to its value there (Laws of finite sums and finite products clause 3, splitting the range at $i$). So the coordinates of $x$ are recovered by testing against the standard basis, which is the form used repeatedly below.

• Powers here are integer powers. $x_k^2$ means the integer power of Integer powers $a^m$, and $\parallel x{\parallel }_2^2=⟨x,x⟩$ by Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$ and Laws of integer exponents.

Let $n\in \mathbb{N}$ and let $x,y\in {\mathbb{R}}^n$, with the Euclidean inner product and the Euclidean norm as in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Then:

1. Cauchy-Schwarz. $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2,$ with equality if and only if there is a pair $(\lambda ,\mu )\mathrm{\ne }(0,0)$ of reals with $\lambda x_k=\mu y_k$ for every $k<n$.
2. $\parallel \cdot {\parallel }_2$ is a norm on ${\mathbb{R}}^n$ (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms), for every $n\in \mathbb{N}$; the metric it induces is $d_2$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it whenever $n\ge 1$.
3. Parallelogram law. $\parallel x+y{\parallel }_2^2+\parallel x-y{\parallel }_2^2=2\parallel x{\parallel }_2^2+2\parallel y{\parallel }_2^2\mathrm{.}$
4. Polarisation. $⟨x,y⟩=\frac{1}{4}(\parallel x+y{\parallel }_2^2-\parallel x-y{\parallel }_2^2),$ so the inner product is recovered from the norm it induces.

Clause 1 is a citation, not a new proof. The inequality and its equality case are the published The Cauchy-Schwarz inequality for finite sums, stated there for two lists of reals; all that happens below is that it is read in the vector notation of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Re-proving it here would put two proofs of one statement in the library.

• Which route to the triangle inequality was taken. The proof above obtains (N3) by expanding $\parallel x+y{\parallel }_2^2$ and applying Cauchy-Schwarz. The alternative is to quote Minkowski's inequality for finite sums (rational exponent) at the rational exponent $p=2$, which states the same inequality directly; that route is equally legitimate and is the one 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 uses for a general exponent. Only one of the two is used here, so no statement is proved twice.

• Clause 1 holds at $n=0$, where it reads $0\le 0$, and the equality case is then satisfied by every pair $(\lambda ,\mu )\mathrm{\ne }(0,0)$, the condition quantifying over no indices. Clause 2 also holds at $n=0$, the zero space carrying exactly one norm (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). What is not available at $n=0$ is the metric $d_2$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it, which is why the last sentence of clause 2 carries $n\ge 1$.

• Clauses 3 and 4 are what the companion page uses. The parallelogram law is an identity satisfied by every norm of the form $\sqrt{⟨\cdot ,\cdot ⟩}$, so a norm violating it is not of that form; that is how the companion page rules out $\parallel \cdot {\parallel }_1$ on ${\mathbb{R}}^2$. Polarisation says the inner product carries no information the norm does not.

Let $n\in \mathbb{N}$ and let ${\mathbb{R}}^n$ be the function space of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, with $x_k:=x(k)$ for $k<n$.

The $p$-norm, for a rational exponent $p\ge 1$

Let $p\in \mathbb{Q}$ with $p\ge 1$. For $x\in {\mathbb{R}}^n$ put

$\parallel x{\parallel }_p:=({\sum }_{k<n}\mathrm{\mid }{x}_k{\mathrm{\mid }}^p{)}^{1\mathrm{/}p},$

where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the absolute value (Absolute value in an ordered field), the sum is the finite sum of Finite sums and finite products, by recursion, and both powers are the rational powers of Rational powers $a^r$ of a positive base.

Every power written here is defined. Each base $\mathrm{\mid }{x}_k\mathrm{\mid }$ is a nonnegative real and $p>0$, so $\mathrm{\mid }{x}_k{\mathrm{\mid }}^p$ is given by Rational powers $a^r$ of a positive base for $\mathrm{\mid }{x}_k\mathrm{\mid }>0$ and by its supplementary clause $0^p=0$ for $\mathrm{\mid }{x}_k\mathrm{\mid }=0$; the sum of these nonnegative terms is nonnegative (Laws of finite sums and finite products clause 4), and $1\mathrm{/}p$ is a positive rational, so the outer power is defined for the same two reasons. The value does not depend on which representative of $p$ or of $1\mathrm{/}p$ is used (Rational powers do not depend on the representative).

The exponent is a rational, and that is not a matter of taste. Rational powers $a^r$ of a positive base supplies $a^r$ for a nonnegative base and a rational exponent only; real exponents do not exist at this point in the reading order, and Why real exponents are deferred on the rational-powers page records exactly why. This is also why the published Minkowski inequality Minkowski's inequality for finite sums (rational exponent), which is what makes the triangle inequality work below, is itself stated for rational $p\ge 1$. No statement on this page is written for $p$ ranging over a real interval.

The maximum norm

For $n\ge 1$ and $x\in {\mathbb{R}}^n$ put

$\parallel x{\parallel }_{\mathrm{\infty }}:=\max \{\mathrm{\mid }{x}_k\mathrm{\mid }:k<n\},$

the maximum of a nonempty finite set of reals, which exists and is one of its elements (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

The hypothesis $n\ge 1$ is required and propagates. At $n=0$ the set $\{\mathrm{\mid }{x}_k\mathrm{\mid }:k<n\}$ is empty and has no maximum (Maximum and minimum of a set). This is the same restriction 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 carries, for the same reason, and every statement on this page that mentions $\parallel \cdot {\parallel }_{\mathrm{\infty }}$ inherits it. The $p$-norms for rational $p\ge 1$ carry no such restriction: at $n=0$ each is the empty sum raised to a positive rational power, hence $0$.

The three cases the rest of the page uses

• $\parallel x{\parallel }_1={\sum }_{k<n}\mathrm{\mid }{x}_k\mathrm{\mid }$, since $t^1=t$ for $t\ge 0$ (Laws of rational exponents, and $0^1=0$ by the supplementary clause).
• $\parallel x{\parallel }_2=({\sum }_{k<n}\mathrm{\mid }{x}_k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}=\sqrt{{\sum }_{k<n}{x}_k^2}$, which is exactly the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$: the exponent $2$ agrees with the integer power, so $\mathrm{\mid }{x}_k{\mathrm{\mid }}^2=x_k^2$ (Basic properties of the absolute value), and $t^{1\mathrm{/}2}$ is the unique nonnegative square root of $t$, which is $\sqrt{t}$ (Rational powers $a^r$ of a positive base, Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$). The two notations denote the same function and no second Euclidean norm is introduced.
• $\parallel x{\parallel }_{\mathrm{\infty }}$ as above, for $n\ge 1$.

That each of these is a norm in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, and that the metrics they induce are exactly the published $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it, is 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; it is proved there and is not assumed here.

• Why $p\ge 1$. The triangle inequality for $\parallel \cdot {\parallel }_p$ is Minkowski's inequality, and Minkowski's inequality for finite sums (rational exponent) is stated for rational $p\ge 1$. For $0<p<1$ the displayed expression is still defined but is not a norm on ${\mathbb{R}}^n$ for $n\ge 2$; nothing on this page asserts anything about that range, and the expression is never written with such an exponent.

• Monotonicity in the base is what makes the comparisons below work. For a fixed positive rational $r$ the map $a\mapsto a^r$ is strictly increasing on the positive reals (Monotonicity of $r\mapsto a^r$ and of $a\mapsto a^r$ clause 2), so an inequality between nonnegative sums passes through the outer power. That is the only property of rational powers used in the comparison chain 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 subscript $\mathrm{\infty }$ is a name, not a number. No arithmetic is performed with it, and $\parallel \cdot {\parallel }_{\mathrm{\infty }}$ is not $\parallel \cdot {\parallel }_p$ for any exponent; it is a separately defined function that happens to sit at the end of the family. This is the same refusal to extend $\mathbb{R}$ silently that Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length records for the interval notation.

Let $n\in \mathbb{N}$ and let $p\in \mathbb{Q}$ with $p\ge 1$, with the norms of The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$. Then:

1. $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$ (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).
2. For $n\ge 1$, $\parallel \cdot {\parallel }_{\mathrm{\infty }}$ is a norm on ${\mathbb{R}}^n$.
3. The dictionary. For $n\ge 1$ and all $x,y\in {\mathbb{R}}^n$, $\parallel x-y{\parallel }_1=d_1(x,y),\parallel x-y{\parallel }_2=d_2(x,y),\parallel x-y{\parallel }_{\mathrm{\infty }}=d_{\mathrm{\infty }}(x,y),$ where $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are the metrics of 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. So the metric induced by each of these three norms (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms) is the correspondingly named published metric, not merely one equivalent to it.