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?