A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

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. The same real-valued codomain is required of a metric by Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric.

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 analogous weakening in Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric retains symmetry and the triangle inequality but replaces separation by $p(x,x)=0$; that defines a 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.