Question about error analysis, and why the choice of the norm is important

Question

I'm currently studying approximation methods for variational formulation of some elliptic problem for example $-\Delta u = f$.
When describing error estimation for a spectral method, we have the following estimation if $u\in H^m$ and $f\in H^r$ $$\| u - u_n \|_{H^1} \leq c (n^{1-m} \|u\|_{H^m} + n^{-r} \|f\|_{H^r}). $$ But my issues is when $m=1$, then we know that $u_n$ cannot converge to $u$ in $H^1$ (the estimation is known to be optimal).
Now we have this second estimation error, in $L^2$-norm $$\|u-u_n\|_{L^2} \leq c(n^{-m}\|u\|_{H^m} + n^{-r} \|f\|_{H^r}). $$ So we always have that $u_n$ converge to $u$ in $L^2$. Now my question is : Does this really matter in practice when we perform the actual computation on a computer ? I know that we are looking for a solution in $H^1$, so it should be natural to measure the error with the $H^1$ norm. But if we compute an approximate solution $u_n$ that converge to $u$ in $L^2$, isn't it enough to do our practical work when we want to compute the solution of a pde ?