Citation
Moore, Lawrence Carlton, Jr. (1966) Locally Convex Riesz Spaces and Archimedean Quotient Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/15GPR248. https://resolver.caltech.edu/CaltechTHESIS:10052015142308267
Abstract
A Riesz space with a Hausdorff, locally convex topology determined by Riesz seminorms is called a locally convex Riesz space. A sequence {x_{n}} in a locally convex Riesz space L is said to converge locally to x ϵ L if for some topologically bounded set B and every real r ˃ 0 there exists N (r) and n ≥ N (r) implies x – x_{n} ϵ r^{b}. Local Cauchy sequences are defined analogously, and L is said to be locally complete if every local Cauchy sequence converges locally. Then L is locally complete if and only if every monotone local Cauchy sequence has a least upper bound. This is a somewhat more general form of the completeness criterion for Riesz – normed Riesz spaces given by Luxemburg and Zaanen. Locally complete, bound, locally convex Riesz spaces are barrelled. If the space is metrizable, local completeness and topological completeness are equivalent.
Two measures of the nonarchimedean character of a nonarchimedean Riesz space L are the smallest ideal A_{o} (L) such that quotient space is Archimedean and the ideal I (L) = { x ϵ L: for some 0 ≤ v ϵ L, n x ≤ v for n = 1, 2, …}. In general A_{o} (L) ᴝ I (L). If L is itself a quotient space, a necessary and sufficient condition that A_{o} (L) = I (L) is given. There is an example where A_{o} (L) ≠ I (L).
A necessary and sufficient condition that a Riesz space L have every quotient space Archimedean is that for every 0 ≤ u, v ϵ L there exist u_{1} = sup (inf (n v, u): n = 1, 2, …), and real numbers m_{1} and m_{2} such that m_{1} u_{1} ≥ v_{1} and m_{2} v_{1} ≥ u_{1}. If, in addition, L is Dedekind σ – complete, then L may be represented as the space of all functions which vanish off finite subsets of some nonempty set.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  (Mathematics)  
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:  28 March 1966  
Funders: 
 
Record Number:  CaltechTHESIS:10052015142308267  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:10052015142308267  
DOI:  10.7907/15GPR248  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9200  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  05 Oct 2015 22:45  
Last Modified:  05 Mar 2024 22:39 
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