Definition 12.6.3 (Hermitian Form).label Let $E$ be a vector space over $K \in \RC$ and $\lambda: E \times E \to K$, then $\lambda$ is a Hermitian form if

1. (H1)

   For each $x, y, z \in E$, $\lambda(x + y, z) = \lambda(x, z) + \lambda(y, z)$.

2. (H2)

   For any $x, y \in E$ and $\mu \in K$, $\lambda(\mu x, y) = \mu \lambda(x, y)$.

3. (H3)

   For every $x, y \in E$, $\lambda(x, y) = \ol{\lambda(y, x)}$.