Topološke baze normiranih prostora

master's thesis

• Author: Ivana Bobinac
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu su najprije navedene osnovne definicije i rezultati iz teorije normiranih prostora te su potom izložene osnovne činjenice o bezuvjetnoj konvergenciji redova. U centralnom dijelu rada su prikazani glavni rezultati o topološkim bazama normiranih, odnosno Banachovih prostora. Pokazana je neprekidnost koeficijentnih funkcionala i pokazano je da je svaka baza Banachovog prostora Schauderova baza tog prostora; štoviše, da je ona i slaba baza i slaba Schauderova baza tog prostora. Također su opisani različiti oblici nezavisnosti nizova te su pokazani odnosi među različitim oblicima nezavisnosti. Opisana je biortogonalnost i minimalnost nizova. Posebno, opisane su bezuvjetne baze kao posebno važna klasa topoloških baza. Na kraju su pokazane osnovne činjenice o Rieszovim bazama Hilbertovih prostora.

Abstract (english)

In this thesis, some of the fundamental definitions and results from the theory of normed spaces are first outlined, and then the basic facts about unconditional convergence of series are presented. In the central part of the thesis, the main results about topological bases in normed, that is Banach spaces, are shown. Continuity of coefficient functionals is shown and it is also shown that every basis in a Banach space is Schauder basis in that space, and that it is, in fact, weak basis and weak Schauder basis in that space. Various forms of sequence independence are described and some basic relations among them are shown. Biorthogonality and minimality of sequences are also described. Especially, unconditional bases, which are important class of topological bases, are described. Finally, some basic facts about Riesz bases in Hilbert spaces are shown as well.

• Keywords: normirani prostori; bezuvjetna konvergencija redova; Banachovi prostori; Schauderova baza; biortogonalnost nizova; minimalnost nizova; Rieszove baze Hilbertovih prostora
• Keywords (english): normed spaces; unconditional convergence of series; Banach spaces; Schauder basis; biorthogonality of sequences; minimality of sequences; Riesz bases in Hilbert spaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:143840
• 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: 2017-12-18 13:57:14