Let X be n-dimensional vector space. Z a proper substance of X.
Let x_0 be element of X - Z (complement of Z).
Show there is a linear functional f on X such that:
f(x_0)=1 and f(x)=0 for all x element of Z.
Construct a basis for Z, say $\displaystyle \{u_1, u_2, ..., u_n\}$. Extend that to a basis for X: $\displaystyle \{u_1, u_2, ..., u_n, v_1, v_2, ..., v_m\}$.
Define $\displaystyle f(u_i)= 0$, $\displaystyle f(v_i)= 1$, and extend to all of X by "linearity".