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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