For $L: C[0,1] \to \mathbb{C}$ denote the linear functional by $L(f) = f(0)$. Show $L \notin (C[0,1] , || . ||_2)* $
If I were to show $L \in (C[0,1] , || . ||_2)^* $ I would need to show that $L$ is continuous for all $f \in C[0,1]$ in $|| . ||_2$. To do this I could show it was bounded and thus try and show $^{sup}_{||f||_2 \leq 1} ||Lf||<\infty$. However as we are trying to prove by counterexample I need to find some unbounded sequence for $||f||_2 \leq 1$ but I can't think of any ideas, could I have some help please?
Take $f_n = (n-n^3x)1_{x\in [0, \frac{1}{n^2}]}$, then $\|f_n\|_2 \leq 1$, but $Lf_n = n \to +\infty$