Example of a non-continuous functional on C[0,1] with L2-Norm do You know some simple example of a non-continuous functional on $C[0,1]$ with the $L^2$-Norm? Surely we can build such example using the Hamel basis but I ask myself if there is some simpler example. On $C^1[0,1]$ we could simply take differentiation. But on $C[0,1]$?
Thanks in advance.

If you take $\phi(f)=f(1)$, this is unbounded. Define
$$
f_n(t)=\begin{cases}
0,&\ 0\leq t\leq 1-\tfrac1{n^2}\\[0.3cm]
nt+\tfrac1n-n,&\ t>1-\tfrac1{n^2}
\end{cases}
$$
Then $\phi(f_n)=n$, while
$$
\|f_n\|_2^2=\int_{1-\tfrac1n}^1(nt+\tfrac1n-n)\,dt=\tfrac1{3n^4}.
$$
