An inner product is a map
- Conjugate symmetry:
- Linearity in the first argument:
- iff u = 0.
For the vectors
the inner product is computed as
Since the conjugate of is equal to for real numbers, if all elements of both vectors have no imaginary components this merely reduces to the dot product. In this sense, the inner product can be thought of an extension of the dot product to the complex plane. They are also similar to the outer product in that whereas an inner product is given by
where H is the conjugate transpose, the outer product is given by
As such, the inner product of two vectors is the trace of the outer product.
If this is equal to 0, the functions are said to be orthogonal on the interval (unlike with vectors, this has no geometric significance). This definition is useful in Fourier analysis.