In the study of spaces, there is a well-known theorem that used frequently in PDE. Its statement is the following
Theorem. Let be a sequence of functions in with . If there is a positive constant and a function such that
then and weakly in .
For a proof, we refer the reader to Brezis’s book. The idea is as follows
Step 1. If weakly in and almost everywhere in then almost everywhere.
The idea of the proof of Step 1 is well-known. If weakly in , then we can select a new sequence such that strongly in . Moreover, almost everywhere.
Step 2. The boundedness and the reflexivity of imply there is a subsequence of converging to .
This is a standard property.
Step 3. For any subsequence, if we can find a subsubsequence converging to the same limit, the whole sequence also converges to the limit.
This can be proved by contradiction.
The advantage of the theorem is very clear. If we want to prove something like
as , , , and for certain higher enough that we don’t have any compactness property, the uniformly bounded of in some space, , will help us.