# Metric spaces and normed vector spaces

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?

• I think metric is just the generalization of concept of norm.. – Meow Jan 11 '16 at 11:27
• You can think of a norm as the distant from 0. The problem is not all spaces has a natural "0" so we need a more general notion, as Mr. Prajakta has mentioned. Also, a distant function need not be linear even if it induce the same topology on the space. – BigbearZzz Jan 11 '16 at 11:38

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.