Due to
fact,
the norm associated to an inner product is a norm in the sense of the following definition. In particular, a vector space with an inner product is a normed vector space.
A
-vector space
is called a
normed vector space
if a
norm
is defined on it.
On a euclidean vector space, the norm given via the the inner product is also called the euclidean norm. For
,
endowed with the standard inner product, we have
For a vector
, ,
in a normed vector space , the vector is called the corresponding normalized. Such a normalized vector has norm . Passing to the normalized vector is also called normalization.