Two different definitions of tensor product space?

Question

I'm currently taking a course on the mathematical foundations of QM and we're formalizing tensor products. The definition I'm used to which was taught in my differential geometry course seems to be different than the one I'm seeing now.

Here's the definition I'm used to: Let $V$ be a vector field over a field $F$. Then a $(p,q)$ tensor is a multi linear map $T: V \times \cdots \times V \times V^* \times \cdots \times V^* \rightarrow F$ where we have $p$ copies of $V$ and $q$ copies of $V^*$. We denote the space of all such $T$ as $V^* \otimes \cdots \otimes V^* \otimes V \cdots \otimes V$.

The definition I encounter now is $V_1 \otimes \cdots \otimes V_k = (\text{k-Lin}(V_1, \ldots, V_k; F))^*$ where $\text{k-Lin}(V_1, \ldots, V_k; F)$ is the set of k-linear maps into $F$.

What's going on here? If $V$ is infinite dimensional then it is not necessarily the case that $V \cong (V^*)^*$ so these definitions don't actually match up?

EDIT:

The point I was trying to make was that the first definition implies that elements of $V \otimes \cdots \otimes V$ are multi linear maps on $V^* \times \cdots \times V^*$ whereas the second definition says that the elements are maps on $(V_1 \times \cdots \times V_k)^*$.

An answer suggested that both these definitions apply only to finite dimensional space in which case this could be resolved if $$ V^* \times \cdots \times V^* \cong (V_1 \times \cdots \times V_k)^* $$ is this correct?

Answer 1

Both of these definitions assume finite-dimensionality of $V$, if you want them to agree with what most mathematicians might say. For example, one might say that $V_1\otimes\dots\otimes V_k$ is an $F$-vector space along with an $F$-multilinear map $\mu: V_1\times\dots\times V_k \to V_1\otimes\dots\otimes V_k$ so that for any $F$-vector space $W$ and $F$-multilinear map $A: V_1\times\dots\times V_k \to W$, there is a unique $F$-linear map $\alpha:V_1\otimes\dots\otimes V_k \to W$ such that $\alpha \circ \mu = A$. (Such $F$-vector-space-map pairs exist, as one can construct such a thing, and they are unique up to unique isomorphism, in a precise way).

One can prove that if $B_1,\dots,B_k$ are bases of $V_1,\dots,V_k$, then the dimension of $V_1\otimes\dots\otimes V_k$ is $|B_1\times\dots\times B_k|$. This will rule your definitions out.

However: I don't know whether these definitions are just extended to infinite-dimensional vector spaces in physics, ignoring that it will no longer match the above definition. That could well be.

In order to see whether the two definitions at least agree for infinite-dimensional spaces, I would want a more precise statement of the first definition. Are you saying that the first definition would say that $V\otimes\dots\otimes V$ (the space of $(k,0)$-tensors) is the space of $F$-multilinear maps $V^*\times\dots\times V^* \to F$, where $V$ is repeated $k$ times in the first and $V^*$ is repeated $k$ times in the second?