# When is a vector-valued function additively separable?

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 $i=1,\dots ,n$)

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

Is $|A|$ finite? And is it $>n$? –  ronno Oct 27 '12 at 19:04
(Thanks for the clarifying question.) $|A|$ is finite and yes if you'd like, take $|A|>n$. –  Kenny LJ Oct 31 '12 at 16:48