There exist other structures which a vector space may have. A norm on the vector space is a linear functional, say , with the following three properties:
Such a function is usually designated by , a norm of the vector . The existence of such a norm gives rise to the following definition:
A linear space equipped with a norm is called a normed linear space.
Example 1: Every inner product of an inner product space determines the norm given by
which, as we have seen, satisfies the triangle inequality,
Thus an inner product space is always a normed linear space with the inner product norm. However, a normed linear space is not necessarily an inner product space.
Example 2: Consider the vector space of matrices . Then
is a norm on this vector space.
Example 3: Consider the vector space of all infinite sequences
of real numbers satisfying the convergence condition
where is a real number. Let the norm be defined by
One can show that (Minkowski's inequality)
i.e., that the triangle inequality,
holds. Hence is a norm for this vector space. The space of -summable real sequences equipped with the above norm is called and the norm is called the -norm.
This -norm gives rise to geometrical objects with unusual properties. consider the following
Example 4: The surface of a unit sphere centered around the origin of a linear space with the -norm is the locus of points for which
Consider the intersection of this sphere with the finite dimensional subspace , which is spanned by .
a) When , this intersection is the locus of points for which
This is the familiar ( )-dimensional unit sphere in -dimensional Euclidean space whose distance function is the Pythagorean distance
b) When , this intersection is the locus of points for which
This is the ( )-dimensional unit sphere in -dimensional vector space endowed with a different distance function, namely one which is the sum of the differences
between the two locations in , instead of the sum of squares. This distance function is called the Hamming distance, and one must use it, for example, when travelling in a city with a rectangular grid of streets. With such a distance function a circle in is a square standing on one of its vertices. See Figure 1.2. A 2-sphere in is a cube standing on one of its vertices, etc.
, this intersection is the locus of points for which