How to construct a function in $C_{wk}([0,T];L^2(\Omega))$, but may not in $C([0,T];L^2(\Omega))$?

Question

The question is in the third chapter of this book http://www.ams.org/publications/authors/books/postpub/gsm-192

The notation $C_{wk}([0,T];L^2(\Omega))$ mean: A function f(t):$[0,T]\to L^2(\Omega)$ is said to be in $C_{wk}([0,T];L^2(\Omega))$ if for any $t_0\in [0,T]$ and $w\in (L^2(\Omega))^*,$ $\int_{\Omega}f(x,t)\cdot w(x)\,dx\to\int_{\Omega}f(x,t_o)\cdot w(x)\,dx$ as $t\to t_0.$

I have thought about using a concept similar to the Riemann-Lebesgue Lemma to try, but there is something wrong.

This may be an overly simple question, but I really can’t figure it out,Can someone provide any good textbooks or reference ? Thank you for your help!