Let $A = \{a_{i}\}_{i=1}^{n}$ and Let $B = \{b_{i}\}_{i=1}^{n}$ be two finite sequences of real numbers. Is there any functions $f$, $g$, and $h$ to fulfill the following equalities?
$$\sum_{i}a_{i}b_{i} = f([\sum_{i}a_{i}], [\sum_{i}b_{i}])$$
$$\sum_{i}a_{i}b_{i} = g([\sum_{i}a_{i}])h([\sum_{i}b_{i}])$$
No, because there are two different choices for the $a$s and the $b$s such that $\sum a_i$ and $\sum b_i$ are the same for both choices, yet $\sum a_ib_i$ depends on the choice. The simplest example is $(a_1,a_2)=(0,1)$ (for both choices) and $(b_1,b_2)=(1,0)$ or $(b_1,b_2)=(0,1)$.