Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\lambda_n e_n$ with zero boundary conditions. We know that $\lambda_n\to+\infty$.
For $s\geq 0$ let me denote the "spectral" $H^s$ norm $$ \|f\|^2_{H^s}=\sum\limits_{n\geq 0} \lambda_n^{s}|f_n|^2 \hspace{1cm}\mbox{for }f=\sum\limits_{n\geq 0} f_ne_n, $$ and finally for $N\geq 1$ let me denote the projector onto $E_N=span(e_0,\dots,e_{N-1})$ as $$ P_N f=\sum\limits_{n\leq N-1} f_n e_n. $$
Fact 1: for any fixed $f\in H^s$ there holds $(1-P_N)f\to 0$ in $H^s$, which can be stated as "$1-P_N\to 0$ pointwise on $H^s$". This is easy to check, since $$ \|(1-P_N)f\|_{H^s}^2=\sum\limits_{n\geq N}|\lambda_n|^{s}|f_n|^2\to 0 $$ as the remainder of a convergent series.
Fact 2: for any $r<s$ there holds $$ \|(1-P_N)f\|_{H^r}\leq \lambda_{N}^{(r-s)/2}\|f\|_{H^s}, $$ which can be stated as "$(1-P_N)\to 0$ in the $H^r$ norm uniformly on any $H^s$ ball." Indeed, one can write immediately $$ \|(1-P_N)f\|_{H^r}^2 =\sum\limits_{n\geq N}|\lambda_n|^{r-s}|\lambda_n|^s|f_n|^2 \leq |\lambda_N|^{r-s}\sum\limits_{n\geq N}|\lambda_n|^s|f_n|^2 \leq |\lambda_N|^{r-s} \|f\|_{H^s}^2 $$
Question: can we extend the uniform convergence on arbitrary $H^r$-compact sets? More explicitly, is it true that $$ \sup\limits_{f\in K}\|(1-P_N)f\|_{H^r}\to 0 \qquad\mbox{as }N\to\infty $$ for any $H^r$-compact set $K$? Fact 2 ecactly guarantees that this holds at least if $K$ is an $H^s$ ball, which is indeed $H^r$-compact classical Sobolev embedding since $r<s$. What about more generic compact sets?
I suspect this is well-known but for some reason I could not find anything on the subject (me not being a specialis in spectral analysis certainly does not help).
