Tensor Products and Von Neumann's Direct Multiplication As part of a recent conversation over at the History of Mathematics and Science Stack Exchange (https://hsm.stackexchange.com/questions/13894/history-of-direct-sums-and-direct-products) I was informed that the earliest known use of the symbol $\otimes$ for the tensor product in mathematics comes from a paper written by Murray and Von Neumann in 1936 where they define what they term the direct multiplication of vector spaces, and which, if I had understood it correctly, did give the definition of the tensor product (albeit under another name) for vector spaces.
Having found the paper now myself, I find myself unable to either prove or disprove if this indeed is equivalent to the definition of the tensor product for vector spaces as we now have it in modern textbooks.
If someone could help me clear this out, it would be much appreciated.
Letting $\mathfrak{H}$ be either a Hilbert space or a finite dimensional Euclidean space. Murray's and Von Neumann's definition reads as follows:

Let $n=1,2, \dots$ spaces $\mathfrak{H}_1, \dots , \mathfrak{H}_n$ be given. Consider all functionals $\Phi (f_1 , \dots , f_n)$, which are defined for all systems
$$f_i \in \mathfrak{H}_i, \quad i = 1, \dots , n,$$
and have complex values, and which are conjugate linear in each $f_i$:
(i) $\Phi (\dots , \alpha f_i , \dots) = \overline{\alpha} \Phi (\dots , f_i , \dots)$.
(ii) $\Phi (\dots , f_i + g_i , \dots) = \Phi (\dots , f_i , \dots) + \Phi (\dots , g_i , \dots)$.
Call their set $\prod_{i=1}^n \otimes \mathfrak{H}_i$.

Is this equivalent to the modern definition of a tensor product of vector spaces?
 A: Given a complex vector space $V$, let me write $\overline{V}$ for $V$ with its scalar multiplication replaced by its complex conjugate (that is, $\alpha\cdot_\overline{V}v=\overline{\alpha}\cdot_V v$).  Then the "direct multiplication" of $V_1,\dots,V_n$ is just the space of multilinear maps $\overline{V_1}\times\dots\times \overline{V_n}\to\mathbb{C}$.  By the universal property of the tensor product, these can be identified with linear maps $\overline{V_1}\otimes\dots\otimes \overline{V_n}\to\mathbb{C}$.  That is, the "direct multiplication" can be identified with the dual space $(\overline{V_1}\otimes\dots\otimes \overline{V_n})^*$.  If the $V_i$ are finite-dimensional, this is further naturally isomorphic to $\overline{V_1}^*\otimes\dots\otimes \overline{V_n}^*$.
Now if $V$ is not just a complex vector space but a finite-dimensional complex inner product space, there is a canonical isomorphism $V\cong\overline{V}^*$, since the inner product is a perfect pairing $V\times \overline{V}\to\mathbb{C}$.  So when the $V_i$ all have the structure of a finite dimensional-complex inner product space, their "direct multiplication" can be canonically identified with the tensor product $V_1\otimes\dots\otimes V_n$.  Explicitly, this identification sends a simple tensor $v_1\otimes\dots\otimes v_n\in V_1\otimes\dots\otimes V_n$ to the functional $\Phi$ given by $\Phi(f_1,\dots,f_n)=\langle v_1,f_1\rangle\cdots\langle v_n,f_n\rangle$.
From a modern perspective, this may seem a rather roundabout way to define the tensor product: we're essentially taking the dual of the tensor product of the duals.  This only works for finite-dimensional vector spaces, so that the two duals end up cancelling.  (Or it works in an analytic setting with something like Hilbert spaces, if you take continuous duals and are constructing some kind of completed tensor product.)  My understanding, though, is that in 1936 the distinction between a vector space and its dual was only really just starting to be understood (driven by developments in differential geometry and algebraic topology), and definitions of this sort that made such circuitous use of duality were common.
