How can I prove that $d = dist(z,L) = \sup_{f \in C[0,1]*} f(z)$. So far I constructed extension $ f$ in $ L_1 = span\{z, L\}$ where $ L = \{ x \in C[0,1]: x(0) = 0 \} $ and $ z(t) = e^t$. I got $f(y+\alpha*z) = \alpha d$ but I do not know how to prove the distance equation.
Any help would be appreaciated!
P.S. I'm not 100% sure that my solution is correct.