Normed Linear Spaces

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:

- Positive definiteness:
for all nonzero vectors
in
, and
.
- Linearity:
for all vectors
and for all complex numbers
.
- Triangle inequality: for all vectors and in .

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

**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 (

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

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

c) When
, this intersection is the locus of points for which

Such a unit sphere is based on the distance function

This is called the