On-Line Computer Graphics Notes
Overview
An inner product space is a linear (vector) space with a function that serves a purpose much like the dot product in two and three-dimensional space. With this inner product, we can define a norm on the space that makes it a normed linear space. In these notes, we review the properties that define these spaces.
Definition of a Inner Product Space
A space 
is called an inner product space if it is a
Linear Space
and for any two elements 
and 
of 
there is associated
a number 
-- which is called the inner product, dot product,
or scalar product -- that has the following properties: If p, 
,
, and 
are arbitrary members of 
then
for any scalar c
for all
.
We Define a Norm the Inner Product Space
If we define the norm of an element p in 
to be
This satisfies the first two properties of a norm -- namely
and
For an two elements 
and 
of an inner product space 
,
the inequality
holds. Equality holds only if 
.
To see this we calculate
and for
the contents of the bracket are zero and
Where we are assuming that 
is not the zero element of the vector
space (If so, then the identity clearly holds). This equation then
translates to
and is is clear that equality only happens when 
is
the zero element in the linear space.
The Triangle Inequality
For any two elements 
and 
of an inner product space 
,
the inequality
holds. Equality only exists when one element is the zero element of
the linear space, or 
.
To see this notice that if 
is the zero element of the vector
space, then the result is trivial. If 
is not zero, then
where the last substitution is via the Schwartz Inequality. The triangle inequality follows by dividing
both sides of the equation by 
.
Since the triangle inequality holds for the norm 
, this implies that every inner
product space is a normed linear space
The Parallelogram Equation
Any two elements 
and 
of an inner product space satisfy the
parallelogram equation
This is straightforward to show because
If 
and 
are interpreted as being in the inner product space
of all vectors in two dimensional space, then the parallelogram
equation expresses the fact that in a parallelogram, the sum of the
squares of the lengths of the diagonals is equal to the sum of the
squares of the lengths of the four sides.
Normed Linear Spaces are almost Inner-Product Spaces
We can almost define an inner product on a
normed linear space. We define this
inner product as follows: Suppose that 
is a normed linear space
and that 
and 
are members of 
, then we can define
This is almost an inner product for a normed linear space. It
actually is an inner product if the
parallelogram equation holds
in the space 
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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