I was browsing wikipedia the other day when I came across the following (paraphrased) claim: $$ \exists f_{ij}:\mathbb{C}^2\to \mathbb{C} \mbox{ s.t. } f(x_1,\dots,x_n)=\sum_{i,j} f_{ij}(x_i,x_j) $$ Now for the life of me, I can't find that page and I'm not sure how I would search the literature for such a claim. I'm having trouble believing it unconditionally, but I would like to know under which conditions it is true.
Reduction of $f:\mathbb{C}^n\to \mathbb{C}$ into sum of $f_{ij}:\mathbb{C}^2\to\mathbb{C}$
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1Is there an assumption about $f$? – 2012-05-24
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0I would assume $f$ to be (sufficiently) smooth and square integrable. – 2012-05-24
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0It seems to be not true, for example with $f(x_1,x_2,x_3)=x_1x_2x_3$. – 2012-05-24
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0Can you give a link for this claim? – 2012-05-24
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0@DavideGiraudo: The lack of a link is part of my problem ;-) – 2012-05-25
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0I think the claim is equivalent to the tensor algebra $T(V)$ on $V=L^2(\mathbb{C})$ is just $\oplus V\otimes V$, which I doubt is true. So the crux must be in the properties of $f$. – 2012-05-25