A queston in Rellich's embedding theorem

Question

Let $(e_j)$ be an orthonormal basis of $H^1_0(\Omega)$ and define $X_k=\oplus_{j\geq k}X(j)$ where $X(j)=span\{e_j\}$. Then $$\sup_{u\in X_k-0}\dfrac{\|u\|_{L^q(\Omega)}}{\|u\|_{H^1_0(\Omega)}}\to 0, \text{ as } k\to \infty.$$ This is a result used in this article. It is written that it follows from Rellich's embedding theorem. By Rellich's embedding theorem one can obtain $$\dfrac{\|u\|_{L^q(\Omega)}}{\|u\|_{H^1_0(\Omega)}}\leq C, $$ for some constant $C$. But how to use $k\to \infty$ here.

Answer 1

Note that $$\sup_{u\in X_k\setminus\{0\}}\frac{\|u\|_{L^q(\Omega)}}{\|u\|_{H_0^1(\Omega)}}=\|i\rvert_{X_k}\|$$ where $i:H_0^1(\Omega)\hookrightarrow L^q(\Omega)$ is the inclusion (this is actually well-defined because $\Omega$ has finite measure and $q<2$ in this case). Assume that $\|i\rvert_{X_k}\|\not\to0$. Then by passing to a subsequence there are $u_k\in X_k$ such that $\|u_k\|=1$ but $\|i(u_k)\|>\varepsilon$ for some $\varepsilon>0$ for all $k$. Note that for any $j$ we have $\langle e_j,u_k\rangle\to0$ as $k\to\infty$. Since the $u_k$ are bounded, this implies that $u_k\rightharpoonup0$. Now $i$ is compact by Rellich's theorem, hence $i(u_k)\to0$ which is a contradiction. We get $\|i\rvert_{X_k}\|\to0$.