# 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 .

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 .

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.*