if I have a sequence $(u_n)_n$ that converges to $u$ in $L^2(]0,T[,v)$ , then why $ ||u_n(.)||^2_v \rightarrow ||u(.)||^2_v$ in $L^1(]0,T[)$? Remark that $L^2(]0,T[,v)$ is the space of measurable functions $t \rightarrow ||f(t)||_v$ in $L^2(]0,T[)$ with the norm $||f||_{L^2v}=$$(\int_0^T ||f(t)||^2_v\,dt)^\frac{1}{2}$ , $v$ is a Hilbert space.
I tried to use the assumption that in general I can use : $| ||x||^2-||y||^2 | \le ||x-y||^2$ , but as answered below, it is not true.
No. Let $||\cdot|| $ be the usual absolute value in $ \mathbb R$ and take $x=2$ and $y=3.$