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
$$
P\_N f=\sum\limits\_{n\leq N-1} f\_n e\_n.
$$




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**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.




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**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
$$
\|(1-P\_N)f\|\_{H^r}^2
=\sum\limits\_{n\geq N}|\lambda\_n|^{r-s}|\lambda\_n|^s|f\_n|^2
\leq |\lambda\_N|^{r-s}\sum\limits\_{n\geq N}|\lambda\_n|^s|f\_n|^2
\leq |\lambda\_N|^{r-s} \|f\|\_{H^s}^2
$$




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> 
> **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).



uniform convergence of $H^r$ projectors on compact sets?
If $(T\_n)$ is a sequence of uniformly bounded, linear operators in a Banach space $X$ nd $T\_nx→0$ for every $x∈X$, then the convergence is uniform on a compact set $K$. Just fix $ϵ>0$ and cover $K$ with a finite number of balls $B(x\_i,ϵ)$ and use $∥T\_nx∥≤∥T\_n(x−x\_i)∥+∥T\_n x\_i∥$.


