Are $(1, 0)$ tensors always vectors? (resolved)

Question

An $(r, s)$ tensor $T$ is defined to be an element of the tensor product of a vector space and its dual: $$T \in T^r_sV := V^{\otimes r}\otimes V^{* \otimes s}.$$ However, when $V$ is finite dimensional, we have an isomorphism $$V \cong T^1_0V$$in which case every vector is considered a tensor, i.e., and element of a tensor space. However, according to Simon Rea's notes from Frederic Schuller's Geometric Anatomy of Theoretical Physics, this is not true in infinite dimensions. Does this mean that the physicists conception of a vector as a $(1, 0)$ tensor is not always true?

Answer 1

The lecture notes actually state that $$T^0_1V \cong V$$ only in finite dimensions. However, $T^1_0V := V$, and so every vector is a rank $(1, 0)$ tensor, since the vector space $V$ is equal to $T^1_0V$. So, no the physicists are not wrong, every vector is indeed a tensor.