One of my favorite PhD exam questions is: ``What is your favorite compactness result?'' In \(\RR^n\) we have the theorem of Bolzano-Weierstrass, i.e. if \(U\subset \RR^n\) is bounded and closed, then it is compact. This, unfortunately, is no longer true in infinite dimensions, since, as we will show that even unit balls in general Banach spaces do not need to be compact. Another well known compactness result is the theorem of Arzela-Ascoli:
Theorem1.4.1.Arzela-Ascoli.
Consider a sequence of real-valued continuous functions \(\{f_n(x)\}_{n\in\NN} \) defined on a closed and bounded interval \([a, b]\) of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a uniformly convergent subsequence. Compactness results are rather essential in functional analysis as they allow us to find limits in function spaces. In the chapters on dual spaces (Chapter Chapter 5) and on Sobolev spaces (Chapter Chapter 6) we will add a few new compactness results to our menu, such as the Rellich-Kondrachov compactness, weak*-compactness, reflexive weak compactness, and compact Sobolev embeddings.
