Table of Contents
What is the difference between normed space and metric space?
While a metric provides us with a notion of the distance between points in a space, a norm gives us a notion of the length of an individual vector. A norm can only be defined on a vector space, while a metric can be defined on any set.
What defines a metric?
Definition: A metric is a quantifiable measure that is used to track and assess the status of a specific process. This difference becomes especially obvious when metric becomes “business metric,” and thereby becomes a “quantifiable measure” that is used to track and assess the status of a specific business process.
Does every norm induce a metric?
Norm induced metric Norms on vector spaces are equivalent to certain metrics, namely homogeneous, translation-invariant ones. In other words, every norm determines a metric, and some metrics determine a norm.
How do you find metric norms?
A norm can be turned into a metric, via d (x,y) = w (x − y). This is called the induced metric. ≤ w (x − y)+ w (y − z) = w (x − y)+|−1|w (y − z) = d (x,y)+ d (y, z).
Is a metric a vector?
No, not necessarily. A vector space with no additional structure has no metric, and is thus not a metric space. You can give a vector space more structure so that it is also a metric space. A vector space over a field has the following properties.
Can every metric on a vector space be obtained from a norm?
This theorem illustrates an important fact: Every metric on a vector space might not necessarily be obtained from a norm.
What is the difference between a norm and a metric space?
Okay, to sum up. All norms are metrics, and normed spaces (vector spaces with a norm) have a lot more structure than general metric spaces. Anything that holds in a metric space will also hold for a normed space.
Is the norm of a graph a metric?
Not so in a metric space. It is easy to see that a norm is a metric on V, because length is the same as “distance from 0.” To check, simply replace x by v − w in the definition of a norm and say ‖ v − w ‖ = d ( v, w). (Note that v − w ∈ V because V is a linear space, so we can take it’s norm).
What is the difference between norm and translation invariant metric?
So a norm is exactly the same thing as a translation invariant metric on a vector space, and indeed a translation invariant metric on an affine space is exactly the same thing as a norm on its tangent vector space of translations. One of the concepts that “metrics” model, is distance. Another is “cost”.
How to determine if a vector space is a metric space?
The pair (X;d) is called a metric space. Remark: If jjjjis a norm on a vector space V, then the function d: V V !R. + de ned by d(x;x0) := jjx x0jjis a metric on V In other words, a normed vector space is automatically a metric space, by de ning the metric in terms of the norm in the natural way.