Show that $ T \notin X' $ if $ X = C([0,1]) $ is equipped with the norm $ \| f \|_{L^{2}} \stackrel{\text{df}}{=} \sqrt{\int_{0}^{1} |f|^{2}} $. Let $ T $ be an operator on $ X = C([0,1]) $ defined by $ T(f) \stackrel{\text{df}}{=} f(0) $. I want to show that $ T \notin X' $ (the dual space of $ X $) if $ X $ is equipped with the norm
$$
\| f \|_{L^{2}} \stackrel{\text{df}}{=} \sqrt{\int_{0}^{1} |f|^{2}}.
$$
I know that if $ T $ isn’t bounded, then it can’t be an element of $ X' $, but I couldn’t find an $ f $ so as to make $ T $ unbounded. Can you help me? Thank you for your help.

You need to construct a sequence of functions $(f_n)$ such that $\sup_n\lVert f_n\rVert<\infty$ but $f_n(0)\geq n$ for each $n\in\mathbb N$. Try the functions $f_n$ which take the value $n$ at $x=0$ and decreases to $0$ at $1/n^2$. Then show that $(f_n)$ has the desired properties.
