Definition 5.20.290891EB78EA0

Definition 5.20.2.90891EB78EA0 Let $V_1$ and $V_2$ be $1$-dimensional complex linear spaces. Given two maps $h_ i\colon V_ i\rightarrow [0,\infty ]$ ($i=1,2$), each of which is either a Hermitian form or a singular Hermitian form, we define the tensor product $h_1\otimes h_2\colon V_1\otimes V_2\rightarrow [0,\infty ]$ as follows:

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   If either $h_1$ or $h_2$ is singular, we define $h_1\otimes h_2$ as the singular Hermitian form;

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   otherwise, define $h_1\otimes h_2$ as the usual tensor product: For any $v_1\in V_1$, $v_2\in V_2$, set

   \[ h_1\otimes h_2(v_1\otimes v_2)=h_1(v_1)h_2(v_2). \]