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# 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:

1. Positive definiteness:   for all nonzero vectors in , and .

2. Linearity:   for all vectors and for all complex numbers .

3. 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 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.

Lecture 3

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.

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 Chebyshev distance. It is simply the maximum coordinate difference regardless of any other differences. With such a distance function a circle in is a square. See Figure 1.3. A 2-sphere in is a cube, etc.

Next: Metric Spaces Up: Infinite Dimensional Vector Spaces Previous: Inner Product Spaces   Contents   Index
Ulrich Gerlach 2010-12-09