# Contents

## Idea

On a complex vector space a sesquilinear map is a function of two arguments which is a linear function in one argument and complex anti-linear in the other.

More generally:

Let $A$ be a star algebra. Then every left $A$-module $\rho_l \colon A \otimes V \longrightarrow V$ canonically becomes a right module $\rho_r \colon V \otimes A \longrightarrow A$ by setting

$\rho_r(v,a) \coloneqq a^\ast v$

and vice versa.

With this operation understood to turn a left module $V$ into a right module, then a sesquilinear form on $V$ is simply an element in the tensor product of modules

$V^\ast \otimes_A V^\ast$

of the $A$-linear dual $V^\ast$ of $V$ with itself.

## References

Last revised on September 6, 2016 at 09:51:05. See the history of this page for a list of all contributions to it.