Hermitian form

A sesquilinear form over a pair of complex vector spaces $(V,W)$ is a function $B:V\times W\to ℂ$ satisfying the following properties:

1. $B({𝐯}_1+{𝐯}_2,𝐰)=B({𝐯}_1,𝐰)+B({𝐯}_2,𝐰)$

2. $B(𝐯,{𝐰}_1+{𝐰}_2)=B(𝐯,{𝐰}_1)+B(𝐯,{𝐰}_2)$

3. $B(c𝐯,d𝐰)=cB(𝐯,𝐰)\overline{d}$

for all $𝐯,{𝐯}_1,{𝐯}_2\in V$, $𝐰,{𝐰}_1,{𝐰}_2\in W$, and $c,d\in ℂ$. The vector spaces $V$ and $W$ are often identical, although the definition does not require them to be the same vector space.

A sesquilinear form $B:V\times V\to ℂ$ over a single vector space $V$ is called a Hermitian form if it is complex conjugate symmetric: namely, if $B({𝐯}_1,{𝐯}_2)=\overline{B({𝐯}_2,{𝐯}_1)}$.

An inner product over a complex vector space is a positive definite Hermitian form.