Someone can I help about the question:
Let $E=C[0,1]$ be the space of continous functions defined on unital interval. Consider the sequence $\{f_n\}$ in $E'$ by $\langle f_n,\varphi \rangle = n \int_{0}^{\frac{1}{n}}\varphi(t) dt$ with $\varphi \in E$. Let $f \in E'$ defined by $\langle f,\varphi \rangle=\varphi(0)$. How to show that $f_n$ doesn't converges strong to $f$?
Recall that
$$\lVert f_n - f\rVert = \sup \{ \lvert \langle f_n - f,\varphi\rangle\rvert : \lVert\varphi\rVert_\infty \leqslant 1\}.$$
So for any $n$, find a $\varphi_n \in C([0,1])$ with $\lVert\varphi_n\rVert_\infty \leqslant 1$ and
$$\left\lvert n\int_0^{1/n}\varphi_n(t)\,dt - \varphi_n(0)\right\rvert \geqslant \frac{1}{2}$$
(for example, the constant $\frac12$ could be replaced with other constants $c$, as long as $0 < c < 2$).
To achieve that, choose $\varphi_n$ such that its value at $0$ differs much from the value(s) on the largest part of the interval $[0,1/n]$, a piecewise linear $\varphi_n$ is a simple possibility.
