Complex Inner Product

linear-algebra

Definition

Complex Inner Product

Let be a complex vector space. A complex inner product is an inner product

satisfying, for all and :

1. Conjugate symmetry:

2. Sesquilinearity (linear in the first argument):

3. Positive definiteness:

For , the standard complex inner product is

Einstein Notation:

Contrast with the Real Case

In a real inner product space, conjugate symmetry reduces to ordinary symmetry because for . Sesquilinearity becomes ordinary bilinearity.