Compactness in standard spaces

U uvodnom poglavlju dan je pregled nekih općenitih rezultata iz područja topologije i funkcionalne analize potrebnih u ostatku rada. Poseban naglasak je na kompaktnosti u različitim prostorima od koje su navedene četiri vrste za koje se ispostavlja da su ekvivalentne u metričkim prostorima. U širokoj klasi beskonačnodimenzionalnih normiranih prostora zatvorena jedinična kugla nije kompaktna, što nas motivira na promatranje nekih drugih topologija na tom prostoru. U drugom poglavlju dokazani su Arzela-Ascolijev i Frechet-Kolmogorovljev teorem kojima su dane karakterizacije relativne kompaktnosti u prostorima C(K), odnosno \(L^p(\mathbb{R}^n)\). Zatim je promatran suodnos konveksnosti i kompaktnosti, posebno u vidu ekstremnih točaka takvih skupova. Tu smo dokazali poznati Krein-Milmanov teorem o egzistenciji ekstremnih točaka. U trećem poglavlju su promatrane slabe topologije inducirane familijom preslikavanja i dokazana su brojna svojstva tako dobivenih topologija. Zatim smo proučili posebne slučajeve kada je ta familija jednaka dualu normiranog prostora, odnosno kada je jednaka slici kanonskog ulaganja prostora u njegov bidual, čime redom dobivamo slabu, odnosno slabu* topologiju. Posljednje poglavlje posvećeno je najvažnijim rezultatima o kompaktnosti u slabim i slabim* topologijama. Prvo smo dokazali Banach-Alaogluov teorem o slaboj* kompaktnosti zatvorene jedinične kugle duala bilo kojeg normiranog prostora. Pokazane su brojne posljedice tog teorema poput činjenice da slaba* topologija duala Banachovog prostora ima Heine-Borelovo svojstvo, kao i Kakutanijev teorem koji daje karakterizaciju refleksivnosti preko slabe kompaktnosti zatvorene jedinične kugle. Nakon toga smo se dotaknuli Eberlein-Šmulianovog teorema kojim smo pokazali ekvivalenciju četiri tipa kompaktnosti u slabim topologijama (koje općenito nisu metrizabilne) te Krein-Šmulianovog teorema kojim smo pokazali da se u Banachovim prostorima svojstvo slabe kompaktnosti ne gubi uzimanjem zatvorene konveksne ljuske. Rad smo završili Jamesovim teoremom kojim je dana zanimljiva karakterizacija refleksivnosti i slabe kompaktnosti pomoću svojstva dostizanja supremuma funkcionala dualnog prostora.In the opening chapter we gave an overview of some general results in topology and functional analysis required in the rest of the thesis. The emphasis was on compactness in different settings: we considered four types which turn out to be equivalent in metric spaces. We showed that in the wide class of infinite-dimensional normed spaces closed unit ball is not compact, which motivated us to consider some other topologies on these spaces. The second chapter is devoted to compactness criteria in some commonly used normed spaces. We began by proving the Arzela-Ascoli and Frechet-Kolmogorov theorems that give characterizations of relative compactness in C(K) and \(L^p(\mathbb{R}^n)\) spaces, respectively. Then we took a look at the relationship between convexity and compactness, especially with respect to extreme points of such sets. Here we proved the famous Krein-Milman theorem on the existence of extreme points. In the third chapter we started by defining weak topologies induced by a family of mappings and proved many properties of these topologies. Then we considered special cases when this family is either the dual of a normed space or the image of the space under the canonical embedding into its bidual which gave us the weak and weak* topology. In the final chapter we covered the main results regarding compactness in weak and weak* topologies. First we proved the Banach-Alaoglu theorem on weak* compactness of the closed unit ball in the dual of any normed space. We showed many consequences of this theorem, such as the fact that the weak* topology has the Heine-Borel property, as well as Kakutani’s theorem which gives us a characterization of reflexivity in terms of weak compactness of the closed unit ball. We continued by proving the Eberlein- Šmulian theorem which states that the four types of compactness are equivalent in the weak topology (which isn’t generally metrizable) and the Krein-Šmulian theorem which shows that taking closed convex hulls preserves weak compactness in Banach spaces. We concluded the thesis with James’ theorem which gives us an interesting criterion for reflexivity and weak compactness by considering the supremum attaining functionals of the dual space