Stoneov reprezentacijski izrek in vektorske mreže s krepko enoto

delo diplomskega seminarja

David Čadež (Author), Marko Kandić (Mentor)

Abstract

Cilj tega diplomskega dela je predstaviti reprezentacijo Banachovih mrež s krepko enoto s prostori funkcij C(K) na kompaktnih topoloških prostorih K. V ta namen je vpeljan pojem Boolove algebre in dokazan Stoneov reprezentacijski izrek, ki služi kot močno orodje pri reprezentaciji Banachovih mrež. Nato je definiran Rieszov prostor, ki je vektorski prostor in hkrati mreža. Dokazanih je nekaj osnovnih lastnosti Rieszovih prostorov. Vpeljani so pojmi ideala, pasu, glavne projekcijske lastnosti, komponent pozitivnega vektorja in polnosti. Brez dokaza je naveden Freudenthalov spektralni izrek. Na koncu je dokazan glavni izrek, ki pravi, da je vsaka Banachova mreža s krepko enoto Rieszovo izomorfna nekemu prostoru funkcij C(K).

Keywords

matematika;Booleova algebra;delno urejeni vektorski prostori;Rieszovi prostori;Stoneov reprezentacijski izrek;ideali;Banachova mreža;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[D. Čadež]
UDC:	512
COBISS:	115567875
Views:	523
Downloads:	65
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Stone representation theorem and vector lattices with a strong unit
Secondary abstract:	The aim of this thesis is to introduce the representation of Banach lattices with a strong unit by function spaces C(K) on compact topological spaces. To this end, the notion of Boolean algebra is introduced and Stone’s representation theorem is proven, which serves as a powerful tool in representation of Banach lattices. The Riesz space is defined, which is both a vector space and a lattice. After that, some basic properties of Riesz spaces are shown and the notions of the ideal, band, principal projection property, components of a positive vector and completeness are introduced. Freudenthal’s spectral theorem is stated without proof. In the end, the main theorem is proven, which states that every Banach lattice with a strong unit is Riesz isomorphic to some function space C(K).
Secondary keywords:	mathematics;Boolean algebra;partially ordered vector spaces;Riesz spaces;Stone representation theorem;ideals;Banach lattice;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	36 str.
ID:	15899403