Consider the following so-called bi-linear functional equation $$\sum_{i=1}^{n}f_i(x)g_i(y)=f_1(x)g_1(y)+f_2(x)g_2(y)+\cdot\cdot\cdot+f_n(x)g_n(y)=0 \tag{1}$$ where $f_i(x)$ and $g_i(y)$ are arbitrary functions. What conditions are required for such a functional equation to hold?
For example, for $n=2$ we have
$$\begin{align} f_1(x)g_1(y)+f_2(x)g_2(y) &= 0 \\ \frac{f_1(x)}{f_2(x)}+\frac{g_2(y)}{g_1(y)} &= 0 \end{align} \tag{2}$$
where I assumed that $f_2(x) \ne 0$ and $g_1(y) \ne 0$. Consequently, we can conclude that Eq.$(2)$ will hold if and only if
$$\begin{align} f_1(x) &= +\lambda f_2(x) \\ g_2(y) &= -\lambda g_1(y) \end{align} \tag{3}$$
where $\lambda$ is some constant. Considering other cases, I ended up with
$$\boxed{ \begin{array}{ll} 1.\, \text{if} \, f_2(x)=0 \, \text{and} \, g_1(y)=0 \, \text{then} & 0=0 \\ 2.\, \text{if} \, f_2(x)\ne0 \, \text{and} \, g_1(y)=0 \, \text{then} & g_2(y)=0 \\ 3.\, \text{if} \, f_2(x)=0 \, \text{and} \, g_1(y)\ne0 \, \text{then} & f_1(x)=0 \\ 4.\, \text{if} \, f_2(x)\ne0 \, \text{and} \, g_1(y)\ne0 \, \text{then} & f_1(x) = +\lambda f_2(x) \, \text{and} \, g_2(y) = -\lambda g_1(y) \end{array} }\tag{4}$$
How can I generalize the result in $(4)$ for functional equation $(1)$?
Any help is appreciated.