The Relationship between norms and metrics

Question

Are normed spaces a subset of metric spaces?

As norms give rise to metrics is it right to say that the set of all normed spaces forms a subset of all metric spaces?

Answer 1

Depends on the notion of subset (or rather: subclass) in this context.

In formal context, a normed space is typically defined as a pair of a vector space and a norm on it, i.e., a specific map $V\to\Bbb R$ with the well-known properties. A metric space is a pair of a set and a metric, i.e., a specific map $X\times X\to\Bbb R$. Now, neither "is" a vectors space a set (as a vector space already "is" a tuple of a set and operations), nor is a map from an object to the reals a map from the product of an object with itself to the reals.

However, we have what is called a functor from the category of normed vector spaces to the category of metric spaces, which sends a normed space to the metric space having the underlying set of the vector space as its set and the metric between two vectors given by the norm of the difference. Unfortunately, we cannot reconstruct the vector space structure from the metric - not even the zero vector (because the metric is translation invariant). There are also other less trivial problems that prevent this association from being one-to-one, which would be the least to require if one wanted to at least view the normed vectors spaces as a subclass of the metric spaces.

Answer 2

Since a norm induces a metric, every normed space is a metric space.

However, not every metric space is a normed space.

See this picture from Wikipedia: