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# Inner product.

Definition 12.1.1   Let be an vector space and let be a mapping

The mapping is an inner product on if it verifies the following properties:
1.
Linearity in the first variable:
(a)
(b)
.
2.
Linearity in the second variable:
(a)
(b)
.
3.
Symmetry:
4.
is positive: .
5.
is definite: .

Generally, a given inner product is denoted instead of .

A mapping which is linear in the first variable and in the second variable is called bilinear. We summarize the above definition by saying that an inner product is a bilinear symmetric definite positive from to .

Remark 12.1.2   If a mapping from to is linear in one variable and symmetric, then it is bilinear.

Example 12.1.3   Let . For and , define

Then this mapping is an inner product on .

Example 12.1.4

 IS an inner product IS NOT an inner product

Next: Verification of the axioms Up: Euclidean spaces. Previous: Euclidean spaces.
Noah Dana-Picard
2001-02-26