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If we have given an inner product space $(V,g)$, where $V$ is vector space and $g$ is inner product. What will be corresponding bi-linear form $g'$ on $C\otimes V$.

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up vote 3 down vote accepted

I assume $V$ is a real vector space. I'll write the inner product as $\langle, \rangle$. Now, there are (at least) two options for extending it to $\mathbb{C} \otimes V$:

Option 1: Make it $\mathbb{C}$-bilinear in both variables. That is, set $$\langle c_1 \otimes v_1, c_2 \otimes v_2 \rangle_{\mathbb{C}} = c_1 c_2 \langle v_1, v_2 \rangle$$

and extend linearly. Unfortunately, the result will not be positive-definite.

Option 2: Make it conjugate-linear in the first variable and linear in the second. That is, set $$\langle c_1 \otimes v_1, c_2 \otimes v_2 \rangle_{\mathbb{C}} = \overline{c_1} c_2 \langle v_1, v_2 \rangle$$

and extend linearly. The result will be positive-definite.

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