PF tensor product space equation

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SUMMARY

The discussion centers on the PF tensor product space equation, specifically addressing the implications of having a metric tensor on the equation V ⊗ V ⊗ V* ≠ V ⊗ V* ⊗ V. The participants highlight that the presence of a metric tensor introduces complexities in the relationships between tensors, as indicated by the inequality T^{ij} k ≠ T^{i} k j. Additionally, there is a clarification regarding the definition of vectors and covectors, emphasizing that a vector is defined as a linear function from V to the underlying field, which contrasts with the traditional understanding of covectors.

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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

[tex]V \otimes V \otimes V^* \neq V \otimes V^* \otimes V,[/tex]

because

[tex]T^{ij}\;_{k} \neq T^{i}\;_{k}\;^{j}.[/tex]
 
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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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