This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me.

This time I am making things more concrete: I am esp. interested in the difference between a metric and a norm. I understand that the metric gives the distance between two points as a real number. The norm gives the length of a a vector as a real number (see def. e.g. here). I further understand that all normed spaces are metric spaces (for a norm induces a metric) but not the other way around (please correct me if I am wrong).

Here I am only talking about vector spaces. As an example lets talk about Euclidean distance and Euclidean norm. Wikipedia says:

A vector can be described as a

directed line segment from the origin

of the Euclidean space (vector tail),

to a point in that space (vector tip).

If we consider that its length is

actually the distance from its tail to

its tip, it becomes clear that the

Euclidean norm of a vector is just a

special case of Euclidean distance:

the Euclidean distance between its

tail and its tip.What confuses me is that they seem to be having it backwards: The Euclidean metric induces the Euclidean norm: You measure the distance between tip and tail and get the length out of that. What makes my confusion complete is that $L^2$

distanceis also called the Euclideannorm(see here).I would very much appreciate it if somebody could clear the haze.

**Answer**

The metric $d(u,v)$ induced by a vector space norm has additional properties that are not true of general metrics. These are:

**Translation Invariance:** $d(u+w,v+w)=d(u,v)$

**Scaling Property:** For any real number $t$, $d(tu,tv)=|t|d(u,v)$.

Conversely, if a metric has the above properties, then $d(u,0)$ is a norm.

More informally, the metric induced by a norm “plays nicely” with the vector space structure. The usual metric on $\mathbb{R}^n$ has the two properties mentioned above. But there are metrics on $\mathbb{R}^n$ that are *topologically equivalent* to the usual metric, but not translation invariant, and so are not induced by a norm.

**Attribution***Source : Link , Question Author : vonjd , Answer Author : André Nicolas*