InnerProductSpace

Inner Product Space

An Inner Product Space is a vector space $V$ over a field $F$ (typically the field of Real or Complex numbers) which additionally has an operation called an inner product.

This inner product operation is a function that takes two vectors from $V$ and returns a scalar from $F$. It is denoted as

$$\langle . , . \rangle : V \times V \rightarrow F$$

The inner product has the following properties for all $u, v, w \in V$ and $a \in F$:

1. Conjugate symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$. In real vector spaces where there is no conjugation, this just becomes

$$\langle u, v \rangle = \langle v, u \rangle$$

2. Linearity in the first argument:

$$\langle au + v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle$$

3. Positive-definiteness: $\langle v, v \rangle \geq 0$ with equality if and only if $v$ is the zero vector.

An inner product induces a norm, and hence a metric, on the vector space, making it a metric space. The norm $|v|$ of a vector $v$ is defined by the inner product as

$$|v| = \sqrt{\langle v, v \rangle}$$