Citation
Chow, Theresa Kee Yu (1969) The Egoroff property and its relation to the order topology in the theory of Riesz spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/T2KVBF37. https://resolver.caltech.edu/CaltechETD:etd10072002143502
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is order convergent to an element [...] whenever there exists a sequence [...] in L such that [...] holds for all n. Sequential order convergence defines the order topology on L. The closure of a subset S in this topology is denoted by cl(S). The pseudo order closure S' of a subset S is the set of all [...] such that there exists a sequence in S which is order convergent to f. If S' = cl(S) for every convex subset S, then S' = cl(S) for every subset S. L has the Egoroff property if and only if S' = cl(S) for every order bounded subset S of L. A necessary and sufficient condition for L to have the property that S' = cl(S) for every subset S of L is that L has the strong Egoroff property. A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is ruconvergent to an element [...] whenever there exists a real sequence [...] and an element [...] such that [...] holds for all n. Sequential ruconvergence defines the rutopology on L. The closure of a subset S in this topology is denoted by [...]. The pseudo ruclosure S'[subscript ru] of a subset S is the set of all [...] such that there exists a sequence in S which is ruconvergent to f. If L is Archimedean, then [...] for every convex subset S implies that [...] for every subset S. A characterization of those Archimedean Riesz spaces L with the property that [...] for every subset S of L is obtained. If [...] is a monotone seminorm on a Riesz space L, then a necessary and sufficient condition for [...] in L implies [...] is that the set [...] is order closed. For every monotone seminorm [...] on L, the largest [...]Fatou monotone serninorm bounded by [...] is the Minkowski functional of the order closure of [...]. A monotone seminorm p on a Riesz space L is called strong Fatou whenever [...]. A characterization of those Riesz spaces L which have the following property is given: "For every monotone seminorm [...], the largest strong Fatou monotone seminorm bounded by [...] : [...]." A similar characterization for Boolean algebras is also obtained.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  7 April 1969 
Record Number:  CaltechETD:etd10072002143502 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd10072002143502 
DOI:  10.7907/T2KVBF37 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3955 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  07 Oct 2002 
Last Modified:  21 Dec 2019 04:20 
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