Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function

$d(u,v) = \lVert u – v \rVert$, $u,v \in V$.

My question is whether every metric on a linear space can be induced by norm? I know answer is no but I need proper justification.

Edit: Is there any method to check whether a given metric space is induced by norm ?

Thanks for help

Answer

Let $V$ be a vector space over the field $\mathbb{F}$. A norm
$$\| \cdot \|: V \longrightarrow \mathbb{F}$$
on $V$ satisfies the homogeneity condition
$$\|ax\| = |a| \cdot \|x\|$$
for all $a \in \mathbb{F}$ and $x \in V$. So the metric
$$d: V \times V \longrightarrow \mathbb{F},$$
$$d(x,y) = \|x – y\|$$
defined by the norm is such that
$$d(ax,ay) = \|ax – ay\| = |a| \cdot \|x – y\| = |a| d(x,y)$$
for all $a \in \mathbb{F}$ and $x,y \in V$. This property is not satisfied by general metrics. For example, let $\delta$ be the discrete metric
$$\delta(x,y) = \begin{cases} 1, & x \neq y, \\ 0, & x = y. \end{cases}$$
Then $\delta$ clearly does not satisfy the homogeneity property of the a metric induced by a norm.


To answer your edit, call a metric
$$d: V \times V \longrightarrow \mathbb{F}$$
homogeneous if
$$d(ax, ay) = |a| d(x,y)$$
for all $a \in \mathbb{F}$ and $x,y \in V$, and translation invariant if
$$d(x + z, y + z) = d(x,y)$$
for all $x, y, z \in V$. Then a homogeneous, translation invariant metric $d$ can be used to define a norm $\| \cdot \|$ by
$$\|x\| = d(x,0)$$
for all $x \in V$.

Attribution
Source : Link , Question Author : Srijan , Answer Author : Henry T. Horton

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