PF tensor product space equation

Rasalhague
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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}.
 
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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.
 

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