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. 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. 2.

   $\langle v,w\rangle=\overline{\langle w,v\rangle}\;$ for every $v,w\in V$, where the line above means complex conjugation.

3. 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.

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

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

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.