How to indicate the space of sesquilinear forms?

Question

Is there a canonical way to indicate the space of sesquilinear forms? For bilinear forms I can use the (0,2) Tensor space, but since sesquilinear forms are not linear in one variable I don't know how to indicate the space where they live. Does anybody have an idea? Thank you

Answer 1

You can use the construction of the complex conjugate vector space and identify a sesquilinear form with an element of $V^{*} \otimes \overline{V}^{*}$. An elementary tensor $\varphi \otimes \psi \in V^{*} \otimes \overline{V}^{*}$ gives rise to a sesquilinear form $g_{\varphi \otimes \psi} \colon V \times V \rightarrow \mathbb{C} $ defined by

$$ g(v, w) := \varphi(v) \psi(w) $$

which will be antilinear in the second variable.