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Suppose the map $u:(\mathbb{R}^{|A|})^{n} \rightarrow \mathbb{R}^{|A|}$ can be written in an additive form, i.e. there exist real-valued functions $g_{i}$ s.t.

$u(x_{1},\dots,x_{n})=\sum g_{i}(x_{1},\dots,x_{n})x_{i}$

where each $x_{i}\in \mathbb{R}^{|A|}$. We can think of each $g_{i}(x_{1},\dots,x_{n})$ as being the "weight" corresponding to $x_{i}$.

When is it the case that each weight depends only on the vector to which the weight corresponds? That is, when is it the case that we can also write, for some real-valued $f_{i}$ (for $i=1,\dots ,n$)

$u(x_{1},\dots,x_{n})=\sum f_{i}(x_{i})x_{i}$ ?

Thanks in advance

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    (Thanks for the clarifying question.) $|A|$ is finite and yes if you'd like, take |A|>n.2012-10-31

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