PF tensor product space equation

[Rasalhague]

In the first equation on this page,

https://www.physicsforums.com/library.php?do=view_item&itemid=335

is there a loss of generality when there exists a metric tensor, since in that case

$$V \otimes V \otimes V^* \neq V \otimes V^* \otimes V,$$

because

$$T^{ij}\;_{k} \neq T^{i}\;_{k}\;^{j}.$$

 

[Fredrik]

I haven't seen that definition before. It seems weird and awkward compared to the one I'm used to. Also, it says that a vector is a linear function from V to the underlying field, but that would be a covector.