Strogo konveksni i glatki normirani prostori

master's thesis

• Title (english): Strictly convex and smooth standard spaces
• Author: Matea Jakovljević
• Mentor: Tomislav Berić (mentor)
• Committee member: Tomislav Berić (predsjednik povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu smo uveli strogo konveksne i glatke normirane prostore, te izveli brojne karakterizacije i svojstva. Prvo smo promatrali strogo konveksne prostore i vidjeli njihovu vezu s Hahn-Banachovim teoremom, kao i s potpornim funkcionalima te ekstremalnim skupovima. Vidjeli smo da na refleksivnim strogo konveksnim prostorima vrijedi analogon Rieszovog teorema o projekciji. Međutim, pokazali smo da refleksivnost nije nužno svojstvo strogo konveksnih prostora, te smo uveli uniformno konveksne prostore koji su nužno refleksivni. Potom smo promatrali glatke prostore i uveli pojam Gˆateauxove derivacije norme. Pokazali smo da je normiran prostor strogo konveksan ako mu je dualni prostor gladak, te je gladak ako mu je dualni prostor strogo konveksan. Nadalje, vidjeli smo da refleksivnost nije nužno svojstvo ni glatkih prostora, ali je uniformno glatkih. Također, pokazali smo da je normiran prostor uniformno gladak ako i samo ako mu je dualni prostor uniformno konveksan, te je uniformno konveksan ako i samo ako mu je dualni prostor uniformno gladak.

Abstract (english)

In this thesis, we have introduced strictly convex and smooth normed spaces, and performed numerous characterizations and properties. We first observed strictly convex spaces and saw their connection with the Hahn-Banach theorem, as well as with support functionals and extreme sets. We have seen that an analogue of Riesz’s projection theorem holds on reflexive strictly convex spaces. However, we have shown that reflexivity is not necessarily a property of strictly convex spaces, and we have introduced uniformly convex spaces that are necessarily reflexive. We then observed smooth spaces and introduced the notion of Gˆateaux derivation of the norm. We have shown that a normed space is strictly convex if its dual space is smooth, and it is smooth if its dual space is strictly convex. Furthermore, we have seen that reflexivity is not necessarily a property of smooth spaces either, but it holds for uniformly smooth spaces. Also, we have shown that a normed space is uniformly smooth if and only if its dual space is uniformly convex, and it is uniformly convex if and only if its dual space is uniformly smooth.

• Keywords: Hahn-Banachov teorem; Rieszov teorem; Gˆateauxova derivacija norme
• Keywords (english): Hahn-Banach theorem; Riesz’s projection theorem; Gˆateaux derivation of the norm
• Language: croatian
• URN:NBN: urn:nbn:hr:217:431238
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-09-15 09:59:20