Let $C={\cal C}([0,1],(0,\infty))$ denote the set of all continuous maps $[0,1]\to (0,\infty)$. Let $g_1,g_2 \in C$ ; one can then define
$$ \begin{array}{rcl} \Phi &: C& \to (0,\infty)^2 \\ f &\mapsto& \bigg(\int_{[0,1]} fg_1,\int_{[0,1]} fg_2\bigg) \\ \end{array} $$
Obviously, when $g_1$ and $g_2$ are not linearly independent, say $g_1=ag_2$ for some $a>0$, the image of $f$ is the diagonal $\lbrace (i,ai) | i > 0\rbrace$. I believe that when $g_1$ and $g_2$ are linearly independent, $\Phi$ is always surjective. Can anyone help me to show this ?