| context | $V$ |
| definiendum | $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ |
| context | $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ |
| context | $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ |
| $u,v,w\in V$ |
| $a,b\in \mathbb C$ |
| postulate | $\overline{\langle v|w \rangle}=\langle w|v \rangle$ |
| postulate | $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ |
| postulate | $v = 0 \Rightarrow \langle v|v \rangle = 0 $ |
| postulate | $\langle u|a\cdot v+b\cdot w \rangle = a\cdot \langle u|v \rangle+b\cdot \langle u|w \rangle $ |
| postulate | $\langle a\cdot v+b\cdot w | u \rangle = \overline a\cdot \langle v|u \rangle+\overline b \cdot \langle w|u \rangle $ |
Wikipedia: Inner product space