Inner product
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
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