- A metric $d$ has to verify the 4 axioms of separability, identity of indiscernibles, symmetry and sub-additivity, and takes value in $[0, +\infty[$.
- I saw that it is possible to also extend the definition to the extended positive real numbers allowing for $d$ to take the value $+\infty$, where the axioms still make sense from the operations defined on these extended numbers.
- Some metrics can be found to arise from a norm, by defining $d(x,y) = \|x - y\|$. For some metrics it is not the case, as for example when $d$ is bounded, it cannot be homogeneous.
So now I am wondering: Is it possible for an extended metric to arise from a norm ?
Sure, as long as we are making up definitions, let's go all the way. Let's say an extended norm is a thing that takes values in $[0, \infty]$ and still satisfies the axioms of a norm, where $0\cdot \infty$ is interpreted as $0$.
An extended norm induces an extended metric by $d(x, y) = \|x-y\|$.
An example of an extended norm would be $\|f\|=\int_X |f|$ on the space of all measurable functions on some measure space $X$. Another example is $\|x\| = \sup_n |x_n|$ on the space of all sequences $\{x_n\}$. One can use all kinds of familiar norms on function spaces and sequence spaces, allowing elements for which the norm is infinite.
I don't know what it gets us, besides having one more definition.