[nooonuii]

Let $\{z_1,...,z_m\}$ be a basis of $Z$. It is not hard to see that $\{z_1,...,z_m,x_0\}$ is a linearly independent set. Thus we can extend this set to a basis of $X$, say $B$. Define a function $$f:X\to\mathbb{R}$$ by $f(b)=0$ for all $b\in B-\{x_0\}$ and $f(x_0)=1$ and then extend $f$ to be a linear functional by defining $$f(\sum_{b\in B} c_b\cdot b)=\sum_{b\in B}c_b\cdot f(b).$$