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I am currently reading Chapter 12: Multilinear Algebra ... ...

I need help in order to fully understand the definition and nature of the set of covariant tensors of rank \(\displaystyle r\) ... as described in Browder, Section 12.7 and 12.8 ...

Te relevant text reads as follows:

My questions related to the above text are as follows:

**Question 1**Given that \(\displaystyle T^r\) is the set of all (multilinear) maps from \(\displaystyle V^r\) to \(\displaystyle \mathbb{R}\) can we conclude that \(\displaystyle T^r\) is equal to the dual space of \(\displaystyle V^r\), visually \(\displaystyle (V^r)^*\) ... ?

**Question 2**In Section 12.8 Browder writes the following ...

\(\displaystyle T^r = V^* \otimes \ ... \ \otimes V^*\) (r times)

What meaning can we give to this notation ... what clues does it give us about the nature of \(\displaystyle T^r\) ... indeed why is Browder mentioning/using this notation ... ?

Hope someone can help ...

Peter