Elementary Properties of a Complex Inner Product

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $0_{V}$ be its zero vector, let $u,v,w\in V$, and let $\lambda$ be a complex number with conjugate $\overline{\lambda}$. Then the following hold.

1. (Additivity in the first argument) $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$.

2. (Conjugate homogeneity in the first argument) $\langle\lambda u,v\rangle=\overline{\lambda}\,\langle u,v\rangle$.

3. (Zero vector) $\langle 0_{V},v\rangle=0$ and $\langle v,0_{V}\rangle=0$.

4. (Definiteness) $\langle v,v\rangle=0$ if and only if $v=0_{V}$.