Definition 5.20.1C5880C6F61FA

Definition 5.20.1.C5880C6F61FA Let $V$ be a $1$-dimensional complex linear space. A Hermitian form $h$ on $V$ is a map $h\colon V\times V\rightarrow \mathbb {C}$ such that

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   $h$ is $\mathbb {C}$-linear in the second variable and conjugate linear in the first, and

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   \[ |v|_ h^2\coloneqq h(v,v)\in \mathbb {R}_{>0} \]

   for each $v\in V\setminus \{ 0\} $.

We usually identify $h$ with the quadratic form $V\rightarrow \mathbb {R}$ sending $v$ to $|v|_ h^2$. We write $|v|_ h=\sqrt{|v|_ h^2}$ for any $v\in V$.

The singular Hermitian form on $V$ is the map $V\rightarrow \{ 0,\infty \} $ sending $0$ to $0$ and other elements to $\infty $.