Let $\Omega \subset \mathbb{R}^n$ be a bounded set and $\{f_n\}_{n \in \mathbb{N}}$ be a sequence which converges to $f$ in $C([0,T];L^2(\Omega))$. I need to prove that $\int_\Omega f_n^2 \phi \,dx \to \int_\Omega f^2\phi\,dx$ in $L^\infty([0,T])$ as $n \to \infty$ for any $\phi \in C^\infty_c(\Omega)$. And does boundedness really matter, can we choose $\Omega =\mathbb{R}^n$ as well?
We have $\lim_{n \to \infty} \sup_{[0,T]} \int_\Omega |f_n(t)-f(t)|^2\,dx = 0$. We therefore consider $\left| \int \left(f^2_n\phi-f^2\phi\right) \,dx \right|\le \int \left|f^2_n-f^2\right| \phi\,dx \le \|\phi\|_\infty \int |f^2_n-f^2|\,dx $. But I do not understand how this dominating part tends to zero.
Let's continue with your argument. So far you have shown \begin{align} &\left|\int_{\Omega}(f_n^2 -f^2)\phi\, \mathrm{d}x\right| \\ \leq &\lVert\phi\rVert_{L^\infty(\Omega)}\int_{\Omega}|f_n^2-f^2|\, \mathrm{d}x \\ = & \lVert\phi\rVert_{L^\infty(\Omega)} \int_{\Omega}(f-f_n)(f+f_n)\, \mathrm{d}x \\ \leq & \lVert\phi\rVert_{L^\infty(\Omega)} \left(\lVert f_n\rVert_{L^2(\Omega)} + \lVert f\rVert_{L^2(\Omega)}\right) \lVert f_n- f\rVert_{L^2(\Omega)} \, . \end{align} Taking the limit, the result follows