Are the axioms of a Norm different from those of a Metric?(adsbygoogle = window.adsbygoogle || []).push({});

For instance Wikipedia says:

a NORM is a function p: V → R s.t. V is a Vector Space, with the following properties:

For all a ∈ F and all u, v ∈ V, p(av) = |a| p(v), (positive homogeneity or positive scalability).

p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).

If p(v) = 0 then v is the zero vector (separates points).

While a metric is defined as a distance function between elements of a set with the 3 familiar axioms, of positive definition, triangular inequality and symmetry.

The terms are however used loosely in many math books and make you think they are one and the same.

The reason I am asking is I am faced with a question where given a norm definition on the set ##\mathbb{Q}## I am asked to determine if the operation defines a norm? So I was wondering which axioms do I check, those of the Metric or those of the VS norm?

Thank you

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# Norm vs. a Metric

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