Vector space/K/Inner product/Definition

Inner product

Let ${}V$ be a ${}{\mathbb {K} }$-vector space. An inner product on ${}V$ is a mapping

$V\times V\longrightarrow {\mathbb {K} },(v,w)\longmapsto \left\langle v,w\right\rangle ,$

satisfying the following properties:

1. We have

   ${}\left\langle \lambda _{1}x_{1}+\lambda _{2}x_{2},y\right\rangle =\lambda _{1}\left\langle x_{1},y\right\rangle +\lambda _{2}\left\langle x_{2},y\right\rangle \,$

   for all ${}\lambda _{1},\lambda _{2}\in {\mathbb {K} }$, ${}x_{1},x_{2},y\in V$, and

   ${}\left\langle x,\lambda _{1}y_{1}+\lambda _{2}y_{2}\right\rangle ={\overline {\lambda _{1}}}\left\langle x,y_{1}\right\rangle +{\overline {\lambda _{2}}}\left\langle x,y_{2}\right\rangle \,$

   for all ${}\lambda _{1},\lambda _{2}\in {\mathbb {K} }$, ${}x,y_{1},y_{2}\in V$.

2. We have

   ${}\left\langle v,w\right\rangle ={\overline {\left\langle w,v\right\rangle }}\,$

   for all ${}v,w\in V$.

3. We have ${}\left\langle v,v\right\rangle \geq 0$ for all ${}v\in V$, and ${}\left\langle v,v\right\rangle =0$ if and only if ${}v=0$ holds.