Karakterizacije unitarnih prostora

master's thesis

• Author: Sanela Cinek
• Mentor: Dijana Ilišević (mentor)
• Committee member: Dijana Ilišević (predsjednik povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, 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: unitarni prostori; normirani prostori; Jordan - von Neumannov teorem; Frechétova funkcijska jednadžba
• Keywords (english): inner product spaces; normed spaces; Jordan - von Neumann theorem; Frechét functional equation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:252085
• Study programme: Title: 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 resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-05 13:46:48