Sean Carrol in his book of general relativity, he defines a tensor to be a multilinear map from a collection of dual vectors and vectors to $\mathbb{R}$: $T:T^*_p \times...\times T^*_p \times T_p \times...\times T_p \to \mathbb{R}$.
He then goes on to say that a scalar is a type $(0,0)$ tensor, a vector is a type $(1,0)$ tensor and a dual vector is a type $(0,1)$. However, I thought that dual vectors are elements of a dual space, $T^*_p$, so shouldn't it be that dual vectors are rank $(1,0)$ tensors and vectors rank $(0,1)$ tensors?