Product metric
Encyclopedia
In mathematics
, the product metric is a definition of metric
on the Cartesian product
of two metric spaces. As described below, the p product metric of the Cartesian product of n metric spaces is the p norm
of the n-vector of the norms of the n subspaces:
and 
be metric spaces and let 
. Define the 
-product metric 
on 
by
for 
for
, 
.
s, using the L2 norm gives rise to the Euclidean metric in the product space; however, any other choice of p will lead to a topologically equivalent metric space. In the category of metric spaces
, the sup norm is used.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the product metric is a definition of metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
on the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of two metric spaces. As described below, the p product metric of the Cartesian product of n metric spaces is the p norm
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
of the n-vector of the norms of the n subspaces:
Definition
Let and be metric spaces and let . Define the -product metric on by for for
, .Choice of norm
For Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s, using the L2 norm gives rise to the Euclidean metric in the product space; however, any other choice of p will lead to a topologically equivalent metric space. In the category of metric spaces
Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
, the sup norm is used.