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master's thesis
Karakterizacije unitarnih prostora
Zagreb: University of Zagreb, Faculty of Science, 2018. urn:nbn:hr:217:252085
Cinek, Sanela
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Cinek, S. (2018). Karakterizacije unitarnih prostora (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:252085

Cinek, Sanela. "Karakterizacije unitarnih prostora." Master's thesis, University of Zagreb, Faculty of Science, 2018. https://urn.nsk.hr/urn:nbn:hr:217:252085

Cinek, Sanela. "Karakterizacije unitarnih prostora." Master's thesis, University of Zagreb, Faculty of Science, 2018. https://urn.nsk.hr/urn:nbn:hr:217:252085

Cinek, S. (2018). 'Karakterizacije unitarnih prostora', Master's thesis, University of Zagreb, Faculty of Science, accessed 21 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:252085

Cinek S. Karakterizacije unitarnih prostora [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2018 [cited 2024 December 21] Available at: https://urn.nsk.hr/urn:nbn:hr:217:252085

S. Cinek, "Karakterizacije unitarnih prostora", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2018. Available at: https://urn.nsk.hr/urn:nbn:hr:217:252085

Metadata
TitleKarakterizacije unitarnih prostora
AuthorSanela Cinek
MentorDijana Ilišević (mentor)
Committee memberDijana Ilišević (predsjednik povjerenstva)
Committee memberPavle Pandžić (član povjerenstva)
Committee memberVjekoslav Kovač (član povjerenstva)
Committee memberMarcela Hanzer (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2018-09-26, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Cilj ovog rada je izložiti različite karakterizacije unitarnih prostora među normiranim prostorima. Osnovna karakterizacija je dana Jordan - von Neumannovim teoremom prema kojem je norma izvedena iz skalarnog produkta ako i samo ako ona zadovoljava jednakost paralelograma \[ ||x+y||^2+||x-y||^2=2||x||^2+2||y||^2. \] Poopćenjem te jednakosti jezikom funkcijskih jednadžbi dobiju se brojne funkcijske jednadžbe koje karakteriziraju unitarne prostore, a najvažnija je kvadratna funkcijska jednadžba \[ f(x+y)+f(x-y)=2f(x)+2f(y). \] Jedno poopćenje jednakosti paralelograma je i jednakost \[ ||x+y+z||^2=||x+y||^2-||z||^2+||y+z||^2-||x||^2+||z+x||^2-||y||^2, \] a iz nje je izvedena Frechétova funkcijska jednadžba: \[ f(x+y+z)=f(x+y)-f(z)+f(y+z)-f(x)+f(z+x)-f(y). \]
Abstract (english)
The aim of this thesis is to expose various characterizations of inner product spaces among normed spaces. A basic characterization is given by the Jordan - von Neumann theorem according to which a norm is derived from an inner product if and only if it satisfies the parallelogram identity \[ ||x+y||^2+||x-y||^2=2||x||^2+2||y||^2. \] Numerous functional equations characterizing inner product spaces are obtained by generalizing this identity to the language of functional equations. The most important is the quadratic functional equation \[ f(x+y)+f(x-y)=2f(x)+2f(y). \] Another generalization of the parallelogram identity is the identity \[ ||x+y+z||^2=||x+y||^2-||z||^2+||y+z||^2-||x||^2+||z+x||^2-||y||^2, \] from which the Frechét’s functional equation is derived: \[ f(x+y+z)=f(x+y)-f(z)+f(y+z)-f(x)+f(z+x)-f(y). \]
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:252085
Study programmeTitle: Mathematics Education; specializations in: Mathematics Education
Course: Mathematics Education
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-03-05 13:46:48