In Linear Algebra via Exterior Products by S. Winitzki the following example of a dual vector space is given:
With emphasis on the word linear, which of course defines the dual vector space.
The question is whether the reason it is linear is because the coefficients in the polynomials are unchanged.
I presume this is true of all integrable $p$, i.e. $f^*(p+q)=f^*(p)+f^*(q)$. The mechanics of linearity are that multiplication by $e^{-x}$ distributes over addition and that integrals are linear. You could replace $e^{-x}$ by any function and the integral by any linear function and the result would be linear without any care to what the functions are describing.
The concept not to be lost here is that all integrals are in a strong sense elements of a dual space, the Reisz representation theorem says that more accurately.