I suddenly think that I should post this entry ‘cos sometimes I don’t remember these stuffs. These results appear frequently in solving PDEs especially when using the direct method. For example, the simplest case is the following eigenvalue problem

over a bounded domain with Dirichlet boundary condition. We assume . Our aim is to show the existence of the first eigenvalue . Obviously, our problem is to solve the following optimization

.

The direct method says that we firstly select a minimizing sequence, say , then we need to prove is convergent. There are two steps in the above argument which lead to this entry. Our first claim is the following.

**Boundedness implies weakly convergence**. The first result says that

If is a reflexive Banach space and is a bounded sequence. Then up to a subsequence converges weakly to some in .

The proof of this claim can be found in a book due to Brezis (Theorem III.27). Interestingly, its converse also holds by the Eberlein-Šmulian theorem.

**Theorem **(Eberlein-Šmulian). Suppose is a Banach space such that every bounded sequence contains a weakly convergent subsequence. Then is reflexive.

There was an elementary proof of this theorem. We refer the reader to a paper due to Whitley [here]. Let us get back to our optimization problem. Once we have a minimizing sequence it is clear to see that is bounded in since

and

.

By using the first claim, converges weakly to some . It is worth noticing that by saying in we mean in and in . Now we need further argument

**Weakly convergence becomes strongly convergence via compact operator**. This second result says that

A compact operator between Banach spaces maps every weakly convergent sequence in into one that converges strongly in .

The proof of this relies on the contradiction argument and the fact that once a sequence converges strongly to some limit, this limit is unique. However, the converse is no long true. For example, by the Schur theorem, a sequence in converges weakly, it also converges strongly. We take to be the identity in . Since has infinity dimensional, by using the Riesz theorem, cannot be compact.

Using this claim we deduce that converges strongly to in for any . By using the Minkowski and Holder inequalities we can show that satisfies the constraint. It now follows from the weakly lower semi-continuous of norm that indeed satisfies the equation. The proof follows.

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