Definition 12.6.5 (Inner Product).label Let $E$ be a vector space over $K$ and $\inp_{E}: E \times E \to K$, then $\inp_{E}$ is an inner product if:

1. (H1)

   For each $x, y, z \in E$, $\angles{x + y, z}_{E} = \dpn{x, z}{E}+ \dpn{y, z}{E}$.

2. (H2)

   For any $x, y \in E$ and $\mu \in K$, $\dpn{\mu x, y}{E}= \mu \dpn{x, y}{E}$.

3. (H3)

   For every $x, y \in E$, $\dpn{x, y}{E}= \ol{\dpn{y, x}{E}}$.

4. (I)

   For each $x \in E$, $\dpn{x, x}{E}\ge 0$, with equality if and only if $x = 0$.