# How to indicate the space of sesquilinear forms?

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

## 1 Answer

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.

• Ok that's a nice trick, does it have also have a name or is it just like that? – Dac0 Jan 26 '16 at 12:23
• I never heard of a name attributed to it - this is just part of complex multilinear algebra. – levap Jan 26 '16 at 12:33