Inner product
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Revision as of 20:46, 15 April 2007 by Boy Soprano II (Talk | contribs)
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.
