Definition
Inner Product
An inner product is a function that takes in two vectors of a vector space
and outputs an element of a scalar field ( being either or ): with the following properties:
- Sesquilinearity: The operation must respect linearity in the first argument, i.e.:
- Positive definiteness: The inner product of a vector with itself must be non-negative, and it is zero iff the vector itself is the zero vector, i.e.:
Bra-Ket Notation
In bra-ket notation, the inner product
is
- sesquilinear with the ket
as the linear side, and - the bra
as the semilinear side.
Properties
Let
Skew-Symmetry
where
denotes the complex conjugate.