Inner Product

linear-algebra

Definition

Inner Product

An inner product is a function that takes in two vectors of a vector space $V$ and outputs an element of a scalar field $\mathbb{F}$ ($\mathbb{F}$ being either $\mathbb{R}$ or $\mathbb{C}$):

$$⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$$

with the following properties:

1. Conjugate symmetry:

$$⟨u,v⟩=\overline{⟨v,u⟩}\forall u,v\in V$$

2. Sesquilinearity: The operation must respect linearity in the first argument, i.e.:

$$⟨cu+v,w⟩=c⟨u,w⟩+⟨v,w⟩\forall u,v,w\in V,c\in \mathbb{F}$$

3. 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.:

$$\begin{array}{rl}⟨v,v⟩& \ge 0\forall v\in V\\ ⟨v,v⟩& =0⟺v=0\forall v\in V\end{array}$$

Bra-Ket Notation

In bra-ket notation, the inner product

$$⟨x|y⟩:=⟨y,x⟩$$

is

• sesquilinear with the ket $|y⟩$ as the linear side, and
• the bra $⟨x|$ as the semilinear side.

Properties

Let $v_1,v_2$ be two vectors.

Skew-Symmetry

$$⟨v_1,v_2⟩=\overline{⟨v_1,v_2⟩}$$

where $\overline{\cdot }$ denotes the complex conjugate.