definition of the tensor product of dual spaces

Question

I have $V$ a real finite dimensional vectorspace.

I would like to know what exactly $V^*\otimes \cdots \otimes V^*$ ($r$ times) is.

I know that given a vectorspace $W$, $W\otimes W$ is just the vectorspacer spanned by the elements $e_i\otimes e_j$ where $\{e_i\}$ is a basis of $V$. However, $V^*$ is not just vectorspace, it also has an action on $V$ so I would expect $V^*\otimes \cdots \otimes V^*$ to also have an action on $V^r$ (or on $V\otimes \cdots \otimes V$ maybe).

How is this defined?

I would say $$(\alpha_1\otimes\cdots\otimes\alpha_r)(v_1,\cdots,v_r) = \sum\alpha_i(v_i)$$

But I don't know if that's it

Answer 1

The tensor product of (multi)linear forms is defined by the product of their images, i.e., $$ (\alpha_1\otimes\cdots\otimes\alpha_r)(v_1,\cdots,v_r) = \prod_{i=1}^r\alpha_i(v_i) $$

You can look up more definitions of tensor products here.