Answer
The set of all positive functions: $f(x)>0$ is not a vector subspace of $C(-\infty, \infty)$.
Work Step by Step
The set of all positive functions: $f(x)>0$ is not a vector subspace of $C(-\infty, \infty)$ because it is not closed under scalar multiplication. For example, $f(x)=x^2+1$ and $c=-2z\in R$. One can see that $cf(x)=-2x^2-2$ which is not a positive function.