What is the difference between a normed linear space and a metric space?
When I was going through definition I saw that a metric space is defined on a set. But a normed space is defined on a vector space.
Is there any metric space where the metric is defined on a set but not on a vector space?
Best Answer
Take a normed space and remove the origin. Now you have a metric space which is not a normed space. Alternatively, consider any non-contractible Riemannian manifold. For example, take a sphere with its standard round metric. This will be a metric space which is certainly not a normed space.