3.3.3 Normed Vector Spaces
The best sort of a norm is one which comes from an inner product. However, any vector
space, V which has a function,
) is called a normed vector space if
. That is
The last inequality above is called the triangle inequality. Another version of this
To see that 3.19 holds, note
and now switching z and w, yields
which implies 3.19.
Any normed vector space is a metric space, the distance given by
This satisfies all the axioms of a distance. Therefore, any normed linear space is a metric
space with this metric and all the theory of metric spaces applies.