40 - Sesquilinear Form

Sesquilinear form

Let $V$ be a complex vector space
A function $⟨-,-⟩:V\times V\to \mathbb{C}$ is a sesquilinear form if

1. $⟨{\alpha }_1{v}_1+{\alpha }_2{v}_2,v_3⟩={\alpha }_1⟨v_1,v_3⟩+{\alpha }_2⟨v_2,v_3⟩$ for all $v_1,v_2,v_3\in V$ and ${\alpha }_1,{\alpha }_2\in \mathbb{C}$
2. $⟨v_1,v_2⟩=\overline{v_2,v_1}\text{ for all }{v}_1,v_2\in V$