Normed Vector Space

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

A seminormed vector space is a pair (V,p) where V is a vector space and p a seminorm 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 function that satisfies these three properties. The properties 1. and 2. together imply that

if and only if .

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.