Why cannot a linear function of a dual space have a "constant term"?
Your definition of $V^*$ is simply incorrect. $V^*$ only contains linear functionals, i.e. $f(x+\lambda y) = f(x) + \lambda f(y)$ for all $x, y \in V$, $\lambda \in \mathbb{K}$. Assuming this property for any $f:V \rightarrow \mathbb{K}$, observe: $$ f(0) = f(0+0) = f(0)+f(0) $$ Subtract $f(0)$ from both sides to get $0=f(0)$. So check whether your functional $\phi$ can be linear.
I wish you a fun math career.