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
.

IS an inner product | IS NOT an inner product |