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


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


This document maintained by Ken Joy

Comments to the Author

All contents copyright (c) 1996, 1997
Computer Science Department,
University of California, Davis
All rights reserved.



Ken Joy Mon Dec 9 08:40:02 PST 1996