Citation
Holbrook, John Arthur Rankin (1965) The Egoroff Property and Related Properties in the Theory of Riesz Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BN4B-9460. https://resolver.caltech.edu/CaltechETD:etd-01142003-101748
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A Riesz space L is said to be Egoroff, if, whenever [...] and [...], there is a sequence [...] in L such that [...] and, for each n,m, there exists an index k(n,m) such that [...]. This notion was introduced, in rather a different form, by Nakano. Banach function spaces are Egoroff, and Lorentz showed that, for any function seminorm [...], the maximal seminorm [...] among those which are dominated by [...] and which are [...] (a monotone seminorm [...] is [...] if [...]) is precisely the "Lorentz seminorm" [...], where [...]. In this thesis the extent to which [...] holds in general Riesz spaces is determined. In fact, [...] for every monotone seminorm [...] on a Riesz space L if, and only if, L is "almost-Egoroff". The almost-Egoroff property is closely related to the Egoroff property and, indeed, coincides with it in the case of Archimedean spaces. Analogous theorems for Boolean algebras are discussed. The almost-Egoroff property is shown to yield a number of results which ensure that, under certain conditions, a monotone seminorm is [...] when restricted to an appropriate super order dense ideal. Riesz spaces L possessing an integral, Riesz norm [...](i.e., a Riesz norm such that [...] are considered also, since in many cases these are known to be Egoroff. In particular if [...] is normal on L (i.e., [...] a directed system, [...] ), then L is Egoroff. In this connection, a pathological space, possessing an integral Riesz norm which is nowhere normal, is constructed.
Item Type: | Thesis (Dissertation (Ph.D.)) |
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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): |
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Thesis Committee: |
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Defense Date: | 5 April 1965 |
Record Number: | CaltechETD:etd-01142003-101748 |
Persistent URL: | https://resolver.caltech.edu/CaltechETD:etd-01142003-101748 |
DOI: | 10.7907/BN4B-9460 |
Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. |
ID Code: | 157 |
Collection: | CaltechTHESIS |
Deposited By: | Imported from ETD-db |
Deposited On: | 15 Jan 2003 |
Last Modified: | 08 Feb 2024 22:06 |
Thesis Files
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PDF (Holbrook_jar_1965.pdf)
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