Metric and Norm

[Nine Divines]

Can I say a metric is the same as a norm? They appear to share identical properties. For example, the Euclidean metric is the same as a Euclidean norm. The absolute value of a real number is another.

 

[Dr.Peterson]

Can I say a metric is the same as a norm? They appear to share identical properties. For example, the Euclidean metric is the same as a Euclidean norm. The absolute value of a real number is another.

Please state the definitions you are using for "norm" and "metric", and list the properties you consider identical.

There are definitely relationships and similarities, but they are quite different. A norm is, in effect, a unary operator, while a metric is a function of two variables.

 

[HallsofIvy]

No, they are not at all the same thing. A "norm" applies to one object (if the object is a vector then it is the length of the vector, if the object is a point in a space with an "origin" then it is the distance from the point to the origin) while a metric applies to two objects and gives the distance between them. If a we have a norm, ||x||, then we can define a metric by d(x, y)= ||x- y|| but there exist metrics that do not correspond to a norm: "d(x,y)= 0 if x equals y, 1 if x not equal to y" is a metric that cannot be derived from a norm.

 

[pka]

Can I say a metric is the same as a norm? They appear to share identical properties. For example, the Euclidean metric is the same as a Euclidean norm. The absolute value of a real number is another.

Here are two of the most adhortative references: Number I & Number II
Please study both. You have already been told the information in both.

 

[Nine Divines]

Thank you for your correspondence. This post has made me realise I need to use precise definitions. I'll adjust and repost accordingly.