$F$ is defined by $F: C[0,1] \rightarrow \mathbb{R}$ as $F(f)=f(0)$. I solved the same function with sup norm, but I found difficult to solve with this norm,
The hint in the textbook is $f_n(x)=(1-x)^n, f=0, ||f_n - f|| \rightarrow 0$ I don't how this hint helps to solve this! Can someone help with this
If $F$ were continuous, then $\lim_n F(f_n) = F(\lim_n f_n)$ where the limit on the left is understood in the usual real line sense and the limit on the right is understood in the $L^1$ sense. $f_n$ converges to $f$ in $L^1$ (do this yourself by showing that $\lim_n \int f_n(x) \,dx = 0$).
What is $F(f_n)$? What is $F(f)$?