I found a remark in Adams' book Sobolev spaces (Acad. Press) which I cannot understand completely. It is on page 34 and says:
We remark that Theorem 2.22 is just a setting, suitable for our purposes, of a well-known theorem stating that the operator-norm limit of a sequence of compact operators is compact.
Theorem 2.22 is stated as follows:
Theorem Let $1\le p <+\infty$ and let $K\subset L^p(\Omega)$. Suppose there exists a sequence $(\Omega_j)$ of subdomains of $\Omega$ having the following properties:
a. For each $j$, $\Omega_j \subset \Omega_{j+1}$;
b. For each $j$ the set of restrictions to $\Omega_j$ of the functions in $K$ is precompact in $L^p(\Omega_j)$;
c. For every $\varepsilon > 0$ there exists $j$ such that
$\int_{\Omega \setminus \Omega_j} \lvert u(x)\rvert^p\, dx <\varepsilon \qquad \text{for every}\ u\in K.$
Then $K$ is precompact in $L^p(\Omega)$.
What "sequence of operators" is he talking about?
Thank you.