General Topology – Metric Spaces and Normed Vector Spaces

Question

Studying I learned that there are some theorems and definitions that need a metric structure on the space in which we are working, for example the definition of local maximum needs a metric space or the theorems that states the equivalence of local and global maxima of concave functionals needs a normed vector space.

I know that every normed vector space has a metric structure and that distances can be generated by norms, so which are the differences between this two concepts?

Is there a hierarchy between them, i.e. normed vector spaces is the general concept, whereas metric space the particular one?

How should I choose where to work when dealing with a problem?

Answer 1

Metric spaces are much more general than normed spaces. Every normed space is a metric space, but not the other way round. This can happen for two reasons:

1. Many metric spaces are not vector spaces. Since a norm is always taken over a vector space, these can't be normed spaces.
2. Even if We're dealing with a vector space over $\mathbb{R}$ or $\mathbb{C}$, the metric structure might nor "play nice" with the linear structure. For example, you might take the discrete metric on $\mathbb{R}$. This is a metric but is certainly not induced by a norm.

In terms of what to choose when dealing with a specific problem... As stated above, if you're not working in a vector space you have no hope of finding a norm. If you are, then norms are usually more useful because they allow you to take advantage of the linear structure when dealing with distances. But often it's actually more useful to forget this structure, in which case metrics are fine... Really depends on the application.