Nabla spaces, the theory of the locally convex topologies (2-norms, etc.) which arise from the mensuration of triangles.

Persistent Link:
http://hdl.handle.net/10150/184510
Title:
Nabla spaces, the theory of the locally convex topologies (2-norms, etc.) which arise from the mensuration of triangles.
Author:
Griesan, Raymond William.
Issue Date:
1988
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Metric topologies can be viewed as one-dimensional measures. This dissertation is a topological study of two-dimensional measures. Attention is focused on locally convex vector topologies on infinite dimensional real spaces. A nabla (referred to in the literature as a 2-norm) is the analogue of a norm which assigns areas to the parallelograms. Nablas are defined for the classical normed spaces and techniques are developed for defining nablas on arbitrary spaces. The work here brings out a strong connection with tensor and wedge products. Aside from the normable theory, it is shown that nabla topologies need not be metrizable or Mackey. A class of concretely given non-Mackey nablas on the ℓp and Lp spaces is introduced and extensively analyzed. Among other results it is found that the topological dual of ℓ₁ with respect to these nabla topologies is C₀, one of the spaces infamous for having no normed predual. Also, a connection is made with the theory of two-norm convergence (not to be confused with 2-norms). In addition to the hard analysis on the classical spaces, a duality framework from which to study the softer aspects is introduced. This theory is developed in analogy with polar duality. The ideas corresponding to barrelledness, quasi-barrelledness, equicontinuity and so on are developed. This dissertation concludes with a discussion of angles in arbitrary normed spaces and a list of open questions.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Locally convex spaces.; Topological spaces.; Vector spaces.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Lomont, John

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleNabla spaces, the theory of the locally convex topologies (2-norms, etc.) which arise from the mensuration of triangles.en_US
dc.creatorGriesan, Raymond William.en_US
dc.contributor.authorGriesan, Raymond William.en_US
dc.date.issued1988en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractMetric topologies can be viewed as one-dimensional measures. This dissertation is a topological study of two-dimensional measures. Attention is focused on locally convex vector topologies on infinite dimensional real spaces. A nabla (referred to in the literature as a 2-norm) is the analogue of a norm which assigns areas to the parallelograms. Nablas are defined for the classical normed spaces and techniques are developed for defining nablas on arbitrary spaces. The work here brings out a strong connection with tensor and wedge products. Aside from the normable theory, it is shown that nabla topologies need not be metrizable or Mackey. A class of concretely given non-Mackey nablas on the ℓp and Lp spaces is introduced and extensively analyzed. Among other results it is found that the topological dual of ℓ₁ with respect to these nabla topologies is C₀, one of the spaces infamous for having no normed predual. Also, a connection is made with the theory of two-norm convergence (not to be confused with 2-norms). In addition to the hard analysis on the classical spaces, a duality framework from which to study the softer aspects is introduced. This theory is developed in analogy with polar duality. The ideas corresponding to barrelledness, quasi-barrelledness, equicontinuity and so on are developed. This dissertation concludes with a discussion of angles in arbitrary normed spaces and a list of open questions.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectLocally convex spaces.en_US
dc.subjectTopological spaces.en_US
dc.subjectVector spaces.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorLomont, Johnen_US
dc.contributor.committeememberSuchanek, Amyen_US
dc.contributor.committeememberWright, A. Larryen_US
dc.contributor.committeememberBenson, Clarken_US
dc.contributor.committeememberLaetsch, Theodoreen_US
dc.identifier.proquest8902343en_US
dc.identifier.oclc701542647en_US
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