Complex Inner Product Space

Let $\mathbb{C}$ be the field of complex numbers, let $V$ be a complex vector space, let $0_{V}$ be its zero vector, and for $\lambda\in\mathbb{C}$ let $\overline{\lambda}$ denote its complex conjugate.

An inner product on $V$ is a map assigning to each pair $u,v$ of elements of $V$ a complex number $\langle u,v\rangle$, subject to the following conditions for all $u,v,w\in V$ and all $\lambda\in\mathbb{C}$.

1. (Conjugate symmetry) $\langle u,v\rangle=\overline{\langle v,u\rangle}$.

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

3. (Homogeneity in the second argument) $\langle u,\lambda v\rangle=\lambda\langle u,v\rangle$.

4. (Positive definiteness) $\langle v,v\rangle$ is a real number satisfying $0\le\langle v,v\rangle$, and $\langle v,v\rangle=0$ only if $v=0_{V}$.

A complex inner product space is a complex vector space together with an inner product on it.