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What does one call the first (or second) factor of an element of a tensor product? For example, if $V,W$ are vector spaces, and $v \in V$, $w \in W$, with $v \otimes w \in V \otimes W$, how would one refer to $v$? First tensor factor?

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I think any denomination other than $v$ is confusing. The proof is that you are asking here, so it is hardly standard. –  Andrea Ferretti Mar 9 '10 at 23:07
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Tensorand? I've heard something like that before. –  tkr Mar 9 '10 at 23:08
    
Call it the first projection of a representative of the tensor $v\otimes w=\otimes((v,w))$, so $(v,w)\in \otimes^{-1}(v\otimes w)\subset V\times W$. –  Harry Gindi Mar 9 '10 at 23:14
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@ Andrea: But I've got a collection of about 116 $v \otimes w$'s, and I really don't want to refer to them individually. I need a way to say "Take all the first tensor factors and consider the ideal they generate". –  Aston Smythe Mar 9 '10 at 23:18
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The single element $v\otimes w$ is often called a "pure tensor", so you could talk about the "first component(s)/factor(s) of the pure tensors". "First tensor factor" seems okay to me, since you'll have to define whatever locution you choose anyway. –  Arturo Magidin Mar 9 '10 at 23:26

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This issue is similar to what someone faces when dealing with a polynomial expression $$ c_n\alpha^n + c_{n-1}\alpha^{n-1} + \cdots + c_1\alpha + c_0 $$ where $\alpha$ actually satisfies an equation of degree smaller than $n$. Logically speaking such expressions can be written in multiple ways (consider a quartic polynomial expression in $\sqrt{3}$), but nobody has a problem speaking about the $i$th term in the expression.

Just do the same thing when you write down an elementary tensor $v_1 \otimes v_2 \otimes \cdots \otimes v_k$: call $v_i$ the $i$th component (or $i$th term, or perhaps even the $i$th factor). Now comes an issue of who your audience is (which you didn't indicate). If your audience is experts, then it would be clear to your audience that whatever you're doing with $v_i$ is eventually leading to some well-defined result in terms of the tensor itself, so there's nothing more to say.

If your audience is students, to whom the tensor product is still somewhat new, then be sure to remind them that mathematically an elementary tensor does not have well-defined components, since an elementary tensor could be written as an elementary tensor in multiple ways. You might then mention the example of polynomial expressions as above which could be written in multiple ways, as an analogy.

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Aston, now that you indicated in your comments what it is you actually wanted to do with the tensors, I can response more directly: say you want to form the ideal generated by the first components of all the elementary tensors in your particular expression. –  KConrad Mar 9 '10 at 23:29

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