Let $V$ be a vector space over the complex numbers. Let $f_{1\leq i \leq n}$ and $g_{1\leq i \leq n}$ be two sets of nonzero linear functionals on $V$.
Suppose we have
$$f_1(v) f_2 (v)\ldots f_n (v) = g_1(v) g_2 (v)\ldots g_n (v) $$
for arbitrary $v\in V $. It is conjectured that $f_i \propto g_{\sigma(i)}$ for some permutation $\sigma \in S_n$. How to prove it?
I can show that $f_i $ appears among the $g$'s. For example, consider $f_1$. Take $v\in Ker(f_1)$. Then $v\in \bigcup_i Ker(g_i) . $ As this holds for all $v\in Ker(f_1)$, we have
$$Ker(f_1) \subseteq \bigcup_i Ker(g_i) $$
Or
$$ Ker(f_1) = \bigcup_i \left(Ker(f_1) \bigcap Ker(g_i ) \right) . $$
By the well known fact that a vector space cannot be the union of a finite number of proper subspaces, we know $Ker(f_1) = Ker(g_i )$ for some $i$, which means $f_1 \propto g_i $.
The difficulty is to handle the case of repeated factors. Can anyone provide a way out? I thought of using the unique factorization property of multi-variable polynomials. But $V$ could be an infinite dimensional space.