Definition 11.2.4 (Hermitian).label Let $E$ be a vector space over $\real$, $*: \complex(E) \to \complex(E)$ be the canonical complex conjugation map, and $\phi \in \hom(\complex(E); \complex)$, then the following are equivalent:

1. (1)

   $\phi|_{E} \in \hom(E; \real)$.

2. (2)

   For each $x \in E$, $\dpn{x, \phi}{\complex(E)}= \ol{\dpn{x^*, \phi}{\complex(E)}}$.

If the above holds, then $\phi$ is Hermitian.