Convergence of the square of a positive sequence to the square of a positive function

Question

Let $\;f_n,f\in L^2(\mathbb R;\mathbb R^m)\;$ and assume $\;{\vert \vert f_n \vert \vert}^2_{L^2} \to {\vert \vert f \vert \vert}^2_{L^2}\;$.

Is it true that $\;{\vert \vert f_n \vert \vert}_{L^2} \to {\vert \vert f \vert \vert}_{L^2}\;$ and why?

I thought that since $\;{\vert \vert \cdot \vert \vert}_{L^2}\;$ is a positive function then it must be true. However, this "explanation" doesn't convince me...

I believe it's something absolutely obvious I 'm missing here, but I've been stuck. Any help would be valuable.

Thanks in advance!

Answer 1

The map $u\rightarrow u^{1/2}$ is continuous on $[0,\infty)$, so $(\|u_{n}\|_{L^{2}}^{2})^{1/2}\rightarrow(\|u\|_{L^{2}}^{2})^{1/2}$.