Today, let us summarize steps during the standard proof of the Rellich embedding theorem on a bounded domain . This theorem says that

Theorem(Rellich). Suppose that is an open, bounded domain with boundary, and that . Then is compactly embedded in for all .In particular, for any sequence , there exists a subsequence such that strongly in for some .

In order to prove the Rellich theorem, we need the so-called Arzela-Ascoli theorem.

Lemma(The Arzela-Ascoli theorem). Suppose that , , and is equicontinuous. Then there exists a subsequence uniformly on .

The Arzela-Ascoli theorem is well-known. To prove the Rellich theorem, we shall use the standard mollifiers. To do that, we have to extend a little bit.

Step 1. Assume that is also an open, bounded domain with boundary with . By the Sobolev extension theorem, we can define by with

By the Gagliardo-Nirenberg inequality,

For , let denote the standard mollifiers and set . By choosing sufficiently small, .