On-Line Computer Graphics Notes

## Normed Linear Spaces

Overview

A normed linear space is a linear (vector) space in which we can associate a ``length'' to each object in the space.

Definition of a Normed Linear Space

A space is called a normed linear space if it is a linear space and there is a length function , called the norm, that satisfies the following three relations: If f, and g are members of and c is a constant, then

• || f || >= 0
• || f || = 0 if and only if p is the zero element of the linear space.
• -- the triangle inequality.
• || c f || = |c| || f ||

A Normed Space is a Metric Space

A normed space is a metric space since we can define a distance function by

This function satisfies all the axioms of a metric space

An Example -- Vectors in 2-dimensional space

The is one of the standard examples of a vector space that every student studies in a basic linear algebra class. The length of a vector , calculated by

satisfies all the axioms above. The triangle inequality in this case just states that the sum of the lengths of two sides of triangle is greater that the length of the third side.

An Example -- Polynomials

If we consider polynomial functions of one variable on the interval , with the following norm,

then we can verify the axioms above by

• , which is clearly true, as the absolute value function produces only nonnegative numbers

• || p || = 0 if and only if is the zero polynomial on the unit interval is actually easy to show. If is the zero polynomial then it is trivially true, and if ||p|| = 0 then it is clear that for all otherwise the maximum will be greater than zero.

• -- the triangle inequality is shown by

• || c p || = |c| || p || is shown by