On-Line Computer Graphics Notes
Overview
The value of the development of the concepts of a linear space ../Vector-Spaces/Vector-Spaces.html, the normed linear space../Normed-Linear-Spaces/Normed-Linear-Spaces.html and the inner product space ../Inner-Product-Spaces/Inner-Product-Spaces.html becomes clear when studying spaces of continuous functions. In these notes, we examine the space of real valued continuous functions over a closed interval. The concepts here become very useful when studying topics in the areas of geometric modeling.
We will call this space by its normal mathematical notation of 
.
C[a,b] is a Linear Space
To be a linear space, we must define an addition and scalar
multiplication of on the space. This is actually easily done in
.
If 
and 
are functions in 
, then we define the
function 
to be simply
and, for a scalar c, we define scalar multiplication 
to be
It is straightforward to check these operations against the
axioms of a linear space
../Vector-Space-Axioms/Vector-Space-Axioms.html. We note that 
is an infinite-dimensional vector space -- that is, it has no finite
set of elements that serve as a basis.
C[a,b] is an Inner Product Space
The inner product on 
is given by the following: If 
and 
are two elements of 
, then
It is straightforward to check that this inner product satisfies the definition of an inner product ../Inner-Product-Properties/Inner-Product-Properties.html#definition.
The Norm on C[a,b]
An inner product space is natrually a normed space. In this case, the norm is
This is commonly called the 
norm of 
, and it arises
naturally from the inner product. Notice in this case
the Schwartz Inequality
../Inner-Product-Space-Properties/Inner-Product-Space-Properties.html#schwartzinequality states that
and the triangle inequality ../Inner-Product-Space-Properties/Inner-Product-Space-Properties.html#triangleinequality states that
Other Norms on C[a,b]
We can define other norms on 
which make it a useful normed
linear space. Three of these norms are
norm, which is given by
norm, which is given by
where
or ``sup'' norm, which is given by
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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