Week of 1/21/08

Definition of complex (Hermitian) inner product

$\langle u+v,w \rangle \equiv \langle u,w \rangle + \langle v,w \rangle$

$\langle u,v+w \rangle \equiv \langle u,v \rangle + \langle u,w \rangle$

$\langle \alpha u,v \rangle \equiv \alpha \langle u,v \rangle$

$\langle u,\alpha v \rangle \equiv \alpha^* \langle u,v \rangle$

$\langle u,v \rangle \equiv \langle v,u \rangle ^*$

$\langle u,u \rangle = 0,$ with equality only if $u \equiv 0$

The basic example is the form

$h(z,w) \equiv \sum z_i w^* _i$