20.500.12556/RUL-138238 Stoneov reprezentacijski izrek in vektorske mreže s krepko enoto delo diplomskega seminarja Stone representation theorem and vector lattices with a strong unit 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). 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). Boolova algebra delno urejen vektorski prostor Rieszov prostor Stoneov reprezentacijski izrek ideal Banachova mreža Boolean algebra partially ordered vector space Riesz space Stone representation theorem ideal Banach lattice true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2022-07-13 08:15:02 2022-07-13 08:15:09 2024-05-29 12:20:02 0000-00-00 00:00:00 2022 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 123141 115567875 4446.pdf 4446.pdf 1 AFBB237773DAFD316EB970D6AF9A486B 91911c230e56e171fd7762d29e03b30e38b4c5ab895a640ab97ef848fea5ac2a ea584ac9-0272-11ed-8aca-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=158468 Fakulteta za matematiko in fiziko 0 0 0