Definition

Let $V$ be a vector space over a field $\mathbb{K}$ where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$

A semi-inner product on $V$ is a function $\;\langle \cdot , \cdot \rangle : V \times V \longrightarrow \mathbb{K}\;$ that satisfies the following conditions:

1. $\langle \lambda_1 v_1 + \lambda_2 v_2 , w \rangle = \lambda_1 \langle v_1 , w \rangle + \lambda_2 \langle v_2 , w \rangle\;$ for every $v_1, v_2, w \in V$ and $\lambda_1, \lambda_2 \in \mathbb{K}$
2. $\langle v ,w \rangle = \overline{\langle w ,v \rangle}\;$ for every $v, w \in V$ where the line above means complex conjugation.
3. $\langle v ,v \rangle \geq 0$ (positive semi definite).

Hence, a semi-inner product on a vector space is just like an inner product, but for which $\langle v ,v \rangle$ can be zero (even if $v \neq 0$ .

A semi-inner product space is just a vector space endowed with a semi-inner product.

Topology

Every semi-inner product space $V$ can be given a topology associated with the semi-inner product. In fact, a semi-norm $\| \cdot \|$ can be defined in $V$ by

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality is valid for semi-inner product spaces:

Properties

Let $V$ be a semi-inner product space and $W:=\{v \in V : \langle v , v \rangle = 0\}$ It is not difficult to see, using the Cauchy-Schwarz inequality, that $W$ is a vector subspace.

The semi-inner product in $V$ induces a well defined semi-inner product in the quotient $V/W$ which is, in fact, an inner product. Thus, the quotient $V/W$ is an inner product space.