Associate space with respect to a semi-finite measure

master's thesis

Filip Talimdjioski (Author), Marko Kandić (Mentor)

Abstract

We introduce the basic theory of vector lattices and order bounded operators. Order continuous operators, the order dual space, normed lattices and several variants of the Fatou property of normed lattices are studied. We then introduce normed function spaces, saturated function seminorms and the associate space to a normed function space, as well as semi-finite and localizable measures. We prove that the associate space $E'$ of an arbitrary saturated Banach function space $E$ with respect to a semi-finite measure $\mu$ equals the order continuous dual space $E_n^{^\sim}$ if and only if $E'$ has the strong Fatou property. If $E$ is furthermore $\sigma$-order continuous we prove that $E_n^{^\sim}$, the sigma-order continuous dual space $E_c^{^\sim}$ and the norm dual $E^*$ are equal which implies that in the previously mentioned result the equality $E' = E_n^{^\sim}$ can be replaced with $E' = E^*$. Also, if $\mu$ is localizable, we prove that $E'$ has the strong Fatou property which implies that $E' = E_n^{^\sim}$. Finally, we give an example where the aforementioned equality fails.

Keywords

semi-finite and localizable measures;Banach function spaces;associate space;order continuinity;Fatou property;

Data

Language:	English
Year of publishing:	2019
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[F. Talimdjioski]
UDC:	517.9
COBISS:	18718297
Views:	1208
Downloads:	200
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Pridruženi prostor glede na semi-končno mero
Secondary abstract:	Uvedemo osnovno teorijo vektorskih mrež in urejenostno omejenih operatorjev. Preučujemo urejenostno zvezne operatorje, urejenostni dualni prostor normiranih mrež in več različic Fatoujeve lastnosti normiranih mrež. Nato uvedemo normirane funkcijske prostore, nasičene funkcijske polnorme in pridruženi prostor normiranemu funkcijskemu prostoru, na koncu pa tudi semi-končne in lokalizabilne mere. Dokažemo, da je pridruženi prostor $E'$ poljubnega nasičenega Banachovega funkcijskega prostora $E$ glede na semi-končno mero $\mu$ enak urejenostno zveznemu dualnemu prostoru $E_n^{^\sim}$ natanko tedaj, ko ima $E'$ krepko Fatoujevo lastnost. Če je $E$ nadalje sigma-urejenostno zvezen, dokažemo, da so $E_n^{^\sim}$, $\sigma$-urejenostno zvezni dualni prostor $E_c^{^\sim}$ in normirani dualni prostor $E^$ enaki, kar pomeni, da lahko v prej omenjenem rezultatu enakost $E' = E_n^{^\sim}$ nadomestimo z $E' = E^$. Če je $\mu$ lokalizabilna, dokažemo, da ima $E'$ krepko Fatoujevo lastnost, kar pomeni, da je $E' = E_n^{^\sim}$. Na koncu navedemo primer, ko pravkar omenjena enakost ne drži.
Secondary keywords:	semi-končne in lokalizabilne mere;Banachovi funkcijski prostori;pridruženi prostor;urejenostna zveznost;Fatoujeva lastnost;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages:	XI, 63 str.
ID:	11221925