A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V.

We often omit p or ‖·‖ and just write V for a space if it is clear from the context what (semi) norm we are using.

In a more general sense, a vector norm can be taken to be any real-valued vector that satisfies these three properties. The properties 1. and 2. together imply that if and only if x = 0.

A useful variation of the triangle inequality is for any vectors x and y.

This also shows that a vector norm is a continuous function.