Difference between revisions of "Inner product"
(Somebody who knows more about inner products should write this article) |
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For the vector space <math> \mathbb{R}^n </math>, the [[dot product]] is perhaps the most familiar example of an inner product. | For the vector space <math> \mathbb{R}^n </math>, the [[dot product]] is perhaps the most familiar example of an inner product. | ||
| − | In addition, for the vector space <math> \displaystyle A </math> of functions mapping some interval <math> I \mapsto \mathbb{R} </math>, the operator <math> \displaystyle \int_{I} f(x)g(x) dx </math> is an inner product for <math> f, g \in A </math>. | + | In addition, for the vector space <math> \displaystyle A </math> of continuous functions mapping some interval <math> I \mapsto \mathbb{R} </math>, the operator <math> \displaystyle \int_{I} f(x)g(x) dx </math> is an inner product for <math> f, g \in A </math>. |
== Resources == | == Resources == | ||
Latest revision as of 15:46, 15 April 2007
For a vector space 
over 
(or 
), an inner product is a binary operation 
(or ) which satisfies the following axioms:
- For all
, 
. - For all
, 
, 
. - For all
, 
.
, 
.
, 
, 
.
, 
.From these three axioms we can also conclude that 
and 
.
- For all
, 
, with equality if and only if 
.
, 
, with equality if and only if 
.This is reasonable because from the first axiom, we must have 
.
Note that from these axioms we may also obtain the following result:
for all 
if and only if 
.
for all 
if and only if 
.This is occasionally listed as an axiom in place of the condition that equality holds on the condition 
exactly when 
.
Examples
For the vector space 
, the dot product is perhaps the most familiar example of an inner product.
In addition, for the vector space 
of continuous functions mapping some interval 
, the operator 
is an inner product for 
.
Resources
This article is a stub. Help us out by expanding it.
