A metric is a function that give the distance, i.e. a real number, between any two elements of a space.
A metric may be induced from a norm as shown at: Section "Metric induced by a norm".
Because a norm can be induced by an inner product, and the inner product given by the matrix representation of a positive definite symmetric bilinear form, in simple cases metrics can also be represented by a matrix.
Canonical example: Euclidean space.
TODO examples:
Figure 1.
Hierarchy of topological, metric, normed and inner product spaces
. Source.
In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.
One notable example where completeness matters: Lebesgue integral of is complete but Riemann isn't.
Subcase of a normed vector space, therefore also necessarily a vector space.
Appears to be analogous to the dot product, but also defined for infinite dimensions.
Vs metric:
  • a norm is the size of one element. A metric is the distance between two elements.
  • a norm is only defined on a vector space. A metric could be defined on something that is not a vector space. Most basic examples however are also vector spaces.
An inner product induces a norm with:
In a vector space, a metric may be induced from a norm by using subtraction:
Metric space but where the distance between two distinct points can be zero.
Notable example: Minkowski space.