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master's thesis
Strogo konveksni i glatki normirani prostori
Zagreb: University of Zagreb, Faculty of Science, 2021. urn:nbn:hr:217:431238
Jakovljević, Matea
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Jakovljević, M. (2021). Strogo konveksni i glatki normirani prostori (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:431238

Jakovljević, Matea. "Strogo konveksni i glatki normirani prostori." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:431238

Jakovljević, Matea. "Strogo konveksni i glatki normirani prostori." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:431238

Jakovljević, M. (2021). 'Strogo konveksni i glatki normirani prostori', Master's thesis, University of Zagreb, Faculty of Science, accessed 30 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:431238

Jakovljević M. Strogo konveksni i glatki normirani prostori [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2021 [cited 2024 December 30] Available at: https://urn.nsk.hr/urn:nbn:hr:217:431238

M. Jakovljević, "Strogo konveksni i glatki normirani prostori", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2021. Available at: https://urn.nsk.hr/urn:nbn:hr:217:431238

Metadata
TitleStrogo konveksni i glatki normirani prostori
Title (english)Strictly convex and smooth standard spaces
AuthorMatea Jakovljević
MentorTomislav Berić (mentor)
Committee memberTomislav Berić (predsjednik povjerenstva)
Committee memberEduard Marušić-Paloka (član povjerenstva)
Committee memberMarko Erceg (član povjerenstva)
Committee memberMarcela Hanzer (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2021-07-22, 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
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:431238
Study programmeTitle: Applied Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2021-09-15 09:59:20