1
$\begingroup$

I know that the dual of a tensor product is the tensor product of the duals. For example, the dual space of $V\otimes V^*$ is $V^* \otimes V$. I also know that the induced dual space mapping of a tensor product is the product of the individual dual space maps: $\langle \boldsymbol{\omega}\otimes \mathbf{v}, \mathbf{w}\otimes\boldsymbol{\gamma}\rangle = \langle \boldsymbol{\omega},\mathbf{w}\rangle \langle\boldsymbol{\gamma},\mathbf{v}\rangle$.

What is the induced dual space mapping for an element of $\Lambda^p(V)$? By my calculation it's dual space is $\Lambda_p(V)$ and the dual space mapping is $p!$ multiplied by a determinant of the matrix of all the simple dual space maps. Does anyone know the formulae in "common use" if there is one? I am flummoxed by all of the conventions in use.

Improve this question
$\endgroup$
3
  • $\begingroup$ You're thinking of finite dimensional vector spaces only (the very first assertion is false otherwise). $\endgroup$
    – Teri
    Commented Mar 12, 2017 at 15:18
  • 1
    $\begingroup$ Yes, finite dimensional only! Thank you. My work involves finite dimensional tensor product spaces! $\endgroup$
    – BoundaryCondition
    Commented Mar 12, 2017 at 15:22
  • $\begingroup$ Here is a pretty good discussion of this: mathoverflow.net/questions/68004/… $\endgroup$
    – Deane Yang
    Commented Mar 12, 2017 at 15:58

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .