# Tensor product of dual vectors and vectors

I am reading "General Relativity" by Wald. At first he defines a tensor of type $(k,l)$ to be a multilinear map $T: V^* \times \cdots \times V^* \times V \times \cdots \times V \rightarrow \mathbb{R}$, but then he literally says this on the next page: "Thus, one way of constructing tensors is to take outer products of vectors and dual vectors. A tensor which can be expressed as such an outer product is called simple. If $\{v_{\mu}\}$ is a basis of $V$ and $\{v^{\nu^*}\}$ is its dual basis, it is easy to show that the $n^{k+l}$ simple tensors $\{v_{\mu_1} \otimes \cdots \otimes v_{\mu_k} \otimes v^{{\nu_1}^*} \otimes \cdots \otimes v^{{\nu_k}^*} \}$ yield a basis for $\mathcal{T}(k,l)$."

My question is, when he first defined a tensor, he defined it as a multilinear map on $(V^*)^k \times V^l$. But then if you look at the quote above, in his basis for the simple tensors, he starts the tensor product with the vectors first and the dual vectors last. Is this not incorrect, because the tensor product does not, in general, commute? Shouldn't he have put the dual vectors first and the vectors last in that tensor product?

• Maybe you should try to read up on the subject with another book. Using $\{v^{\nu^*}\}$ as notation for the dual basis is an abomination. Especially if you're trying to introduce the reader to a new concept. Commented May 4, 2016 at 18:51
• I agree. I hate the notation in Wald. But I'm studying this book as part of a summer project with a professor, so I have to stick with this book. Plus from the reviews of the book, he is one of the most mathematically rigorous for GR, which is something that I value. Commented May 4, 2016 at 22:19

An element of the domain of $T$ is of the form $(f_1,\dots, f_k, v_1, \dots, v_l),\, f_1,\dots,f_k\in V^*, v_1,\dots,v_l \in V$, so the first $k$ elements of $T$ should be able to take elements of $V^*$ and the next $l$ terms should be able to take elements of $V$, hence $v_{\mu_1} \otimes \cdots \otimes v_{\mu_k}(f_1,\dots, f_k)$ first, and $v^{{\nu_1}^*} \otimes \cdots \otimes v^{{\nu_l}^*} (v_1,\dots,v_l)$ later.
• Thank you. I keep forgetting that $V$ and $V^{**}$ are isomorphic. Commented May 4, 2016 at 22:19