So I have the following statement to prove
Let $L:C\:[0,1]\to \mathbb{C}$ be a linear functional defined by $ Lf=f(0)$ Show that $L\notin(C[0,1],||\cdot||_2)^*$, where $||\cdot||_2$ is the usual 2-norm.
This functional is definitely linear, so I guess I need to show that it is not continuous, I understand that if I show that it is not continuous at one point, then it's not continuous anywhere.
But suppose I take $f=0\in C\:[0,1]$, then if I get $\varepsilon>0$ by taking $\delta=\epsilon$ I obtain $ ||0||_2<\delta \: \Rightarrow\: |0|<\epsilon $ So it seems that $L$ is in this dual space, where is my reasoning wrong and what would be the correct way of proving this statement?