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We are given vector spaces V_{1}, V_{2}, ..., V_{n}of dimensions d_{1}, d_{2}, ..., d_{n}respectively.

Let V = V_{1}[tex]\otimes[/tex] V_{2}[tex]\otimes[/tex] ... [tex]\otimes[/tex] V_{n}

Claim: Any element v [tex]\in[/tex] V can be represented in the following form:

[tex]\sum[/tex]_{i=1...R}(v_{1,i}[tex]\otimes[/tex] ... [tex]\otimes[/tex] v_{n,i})

Where R = - MAX {d_{j}} + [tex]\sum[/tex]_{j=1..n}d_{j}

And where v_{j,i}[tex]\in[/tex] V_{j}.

In other words, there is an upper bound of R on the number of "elementary" tensors w_{i}needed needed to represent a particular tensor v in V (where an "elementary" tensor w_{i}is one which can be written in the form w_{i}= v_{1}[tex]\otimes[/tex] v_{2}[tex]\otimes[/tex] ... v_{n}, where v_{j}[tex]\in[/tex] V_{j}). The upper bound R is simply the sum of all the d_{j}except for the d_{j0}which is maximal.

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# Tensor Product of Spaces

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