Implication of $L^p$ convergence

Question

Take $U$ as open subset of $\mathbb{R}^{n}$. If $u_{m} \rightarrow u$ in $L^{p}(U)$ then does it follow that $||u_{m}||_{L^{p}(U)} \rightarrow ||u||_{L^{p}(U)}$?

Answer 1

Yes, by the triangle inequality (dropping the subscript on the norms): $\|u\|=\|u-u_m+u_m\|\le\|u-u_m\|+\|u_m\|$, so $\|u\|-\|u_m\|\le\|u-u_m\|$. Interchanging $u$ and $u_m$ in this and combining the two inequalities leads to $\bigl|\|u\|-\|u_m\|\big|\le\|u-u_m\|$, and the result follows.